Classical liquids: exact results, integral equations

(in particular the virial-free energy one), and uniqueness of the solution, in the integral equation theory of liquids. We propose a new approach for ...

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` DEGLI STUDI DI TRIESTE UNIVERSITA Facolt`a di Scienze Matematiche, Fisiche e Naturali Dipartimento di Fisica Teorica

XVI CICLO DEL DOTTORATO DI RICERCA IN FISICA

Classical liquids: exact results, integral equations theory, and Monte Carlo simulations

DOTTORANDO

COORDINATORE DEL COLLEGIO DEI DOCENTI

Riccardo Fantoni CHIAR.MO PROF. Gaetano Senatore Dipartimento di Fisica Teorica dell’ Universit`a degli Studi di Trieste FIRMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

RELATORE CHIAR.MO PROF. Giorgio Pastore Dipartimento di Fisica Teorica dell’ Universit`a degli Studi di Trieste FIRMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents 1 Introduction

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2 Background i How to make a model . . . . . . . . . . . . . i.1 The interaction potential . . . . . . . ii Experimental methods . . . . . . . . . . . . . ii.1 Measurements on a macroscopic scale ii.2 Measurements on a microscopic scale . iii Numerical simulations . . . . . . . . . . . . . iii.1 Molecular dynamics . . . . . . . . . . iii.2 Monte Carlo . . . . . . . . . . . . . .

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3 The theory of classical fluids i Grand canonical formalism . . . . . . . . . . . . . . i.1 Free energy as the Legendre transform of ln Θ i.2 Correlation functions generated by ln Θ[u] . . ¯ i.3 Correlation functions generated by A[ρ] . . . ii Percus method . . . . . . . . . . . . . . . . . . . . . ii.1 The Percus-Yevick (PY) approximation . . . ii.2 The hypernetted chain approximation . . . . iii The mean spherical approximation . . . . . . . . . .

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4 MSA and PY analytic solutions i Restricted primitive model for charged hard spheres i.1 Method of solution . . . . . . . . . . . . . . . i.1.1 Relationship between c0 and Q . . . i.1.2 Relationship between U and Q . . . ii PY solution for non additive hard spheres . . . . . . ii.1 The Wiener-Hopf factorization is ill defined . ii.2 Symmetric binary mixture . . . . . . . . . . . ii.3 The Widom-Rowlinson model . . . . . . . . . iii Hard spheres with surface adhesion . . . . . . . . . . iii.1 Compressibility Pressure . . . . . . . . . . . . iii.2 Virial pressure . . . . . . . . . . . . . . . . . 3

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4

CONTENTS

5 Generating functionals, consistency, and uniqueness in the integral equation theory of liquids i Thermodynamic consistency and uniqueness of the solution of integral equations ii Extensions of HNC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii.1 The HNC/H2 approximation . . . . . . . . . . . . . . . . . . . . . . . . . ii.2 The HNC/H3 approximation . . . . . . . . . . . . . . . . . . . . . . . . . iii Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii.1 Inverse power potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii.1.1 The inverse 12th power potential . . . . . . . . . . . . . . . . . . iii.1.2 The inverse 6th power potential . . . . . . . . . . . . . . . . . . iii.1.3 The inverse 4th power potential . . . . . . . . . . . . . . . . . . iii.2 The spinodal line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Improving the closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv.1 The reference HNC/H2 approximation . . . . . . . . . . . . . . . . . . . . iv.1.1 Results from the RHNC/H2 approximation . . . . . . . . . . . . iv.2 Optimized HNC/H3 approximation . . . . . . . . . . . . . . . . . . . . . . iv.3 Functionals of γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 54 57 57 58 58 59 59 60 61 62 65 65 67 67 69 71

6 Stability of the iterative solutions of integral criterion i Introduction . . . . . . . . . . . . . . . . . . . ii Technical details . . . . . . . . . . . . . . . . iii Numerical results . . . . . . . . . . . . . . . . iii.1 Three dimensional systems . . . . . . iii.2 The one dimensional hard spheres . . iii.3 The Floquet matrix . . . . . . . . . . iv Conclusions . . . . . . . . . . . . . . . . . . .

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73 73 75 76 76 77 77 78

equations as one phase freezing . . . . . . .

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7 Direct correlation functions of the Widom-Rowlinson model i Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Monte Carlo simulation and PY solution . . . . . . . . . . . . . iii Fit of the data . . . . . . . . . . . . . . . . . . . . . . . . . . . iv From WR to non additive hard spheres . . . . . . . . . . . . . v Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Pressures for a One-Component Plasma on a pseudosphere i Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i.1 The pseudosphere . . . . . . . . . . . . . . . . . . . . . i.2 The one component Coulomb plasma . . . . . . . . . . . ii Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii.1 Calculation of the curvature of M . . . . . . . . . . . . ii.2 Ergodicity of the semi-ideal Coulomb plasma . . . . . . ii.3 The thermodynamic limit . . . . . . . . . . . . . . . . . iii Pressures of the one component Coulomb plasma . . . . . . . . iii.1 The virial theorem . . . . . . . . . . . . . . . . . . . . . iii.2 Equivalence of virial and kinetic pressures . . . . . . . . iii.3 The thermal pressure in the Canonical ensemble . . . .

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5

CONTENTS

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108 109 111 112 115 116 117 117 118 118 121 122 122 123 125 128

9 Functional integration in one dimensional classical statistical mechanics i The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Averaging over a general Gaussian Random Process . . . . . . . . . . . . . . iii Kac’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv.1 The Ornstein-Uhlenbeck process . . . . . . . . . . . . . . . . . . . . . iv.2 The Wiener process . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Characteristic value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . vii General potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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130 131 131 132 134 134 135 136 137 139

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iii.4 Difference between thermal and kinetic pressure . . . . . . . . . . . iii.5 Non neutral system and the mechanical pressure . . . . . . . . . . . iii.6 Thermal and mechanical pressures in the Grand Canonical ensemble The Yukawa fluid and the Maxwell tensor pressure . . . . . . . . . . . . . . iv.1 Calculation of the self part of the excess pressure . . . . . . . . . . . iv.2 Calculation of the non-self part of the excess pressure . . . . . . . . iv.3 The Coulomb limit on the pseudosphere . . . . . . . . . . . . . . . . iv.4 Range of validity of the equation of state . . . . . . . . . . . . . . . Exact results at βq 2 = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v.1 The grand potential . . . . . . . . . . . . . . . . . . . . . . . . . . . v.2 The density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v.3 Large domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v.3.1 The case 4πnb a2 = 1 . . . . . . . . . . . . . . . . . . . . . . v.3.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . v.4 Relations between the different pressures . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Conclusions 140 i Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 A The Born-Green approximation

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B The Wiener-Hopf factorization

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C The i ii iii

PY solution for Hard Compressibility pressure Virial pressure . . . . . Hard rods . . . . . . . .

spheres 152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

D Mixtures i The static structure factor . . . . ii The OZ equation . . . . . . . . . iii The grand canonical formalism . iv The Kirkwood and Buff equation

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6

CONTENTS

E Very tedious algebra for the MSA solution i Calculation of D b and D c . . . . . . . . . . . . . . . . . . . ii Relationship between aj and Jij . . . . . . . . . . . . . . . . iii Calculation of Qij (λji ) . . . . . . . . . . . . . . . . . . . . . iv The equimolar binary mixture . . . . . . . . . . . . . . . . . v Calculation of the charge density direct correlation function

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F Thermodynamic consistency

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G Strict convexity of FOZ [h]

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H Green function of Helmholtz equation

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I

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Density near the wall

J Electrostatic potential of the background

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K The flat limit

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Bibliography

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Chapter 1

Introduction The relation between interaction and structure and thermodynamic properties in condensed matter is a central issue for the theory. Liquid phases have often been considered very difficult to describe by theory since they lack the possibilities of controlled approximations like in dilute (gas) or symmetric (crystalline) phases. However, the source of the difficulties, the partial disorder of the relevant configurations, makes them ideal for studying the interactions in condensed matter phases. Notwithstanding many theories have been developed in the last half century, this remains an active field of research witnessed by recent progress in colloidal science or in confined liquid theory. In the present thesis work we have studied issues related to some unsolved problems in this area. The final thesis has been confined to those problems for which we could give either a definite solution or an original discussion. The outline of the thesis is as follows. In the second chapter we briefly present some background notions necessary to have a clear picture of where the arguments treated in the thesis are collocated in the realm of theoretical physics: a classical liquid is defined, the creation of a mathematical model of a real fluid is discussed, and a brief outline of the experimental methods used to analyze a real fluid and of the simulation methods used to analyze its mathematical models is given. In chapters 3, 4, 5, and 6 we study various aspects of the integral equations theory. These are approximate theories which allow to gain some insight into the structure and thermodynamics of a given model. In chapter 4 we give particular emphasis to the analytic solutions of such theories. While in chapter 5 and 6 we concentrate on their numerical solutions. In chapter 7 we carry out a Monte Carlo simulation aimed to study the structure of some simple models. Up until 1961 the statistical mechanics of the one dimensional Coulomb gas was an unsolved problem. At more or less the same time the problem was solved by Lenard [1] and by Prager [2] independently. A powerful alternative method of solution using functional integration was subsequently found by Edwards and Lenard [3]. In chapter 9 we give a review of their method of solution and show how it is suitable to study other one dimensional fluid models. The two dimensional Coulomb gas may also be solved exactly at a temperature βq 2 = 2 [4]. In chapter 8 we find some exact results for a particular two dimensional Coulomb gas: one in a disk on the surface of a pseudosphere. We will now give a brief abstract for each of the chapters three to nine, stressing whether they contain original results or not. In chapter 3 there are no original results. We try to give a unified introduction to the theory of classical fluids, introducing the density functional theory as a development of the grand canonical formalism of statistical mechanics, defining the free energy and the most commonly used correlation functions, as well as the Ornstein-Zernike (OZ) equation. We use Percus method to introduce the most commonly known integral equations [like the Random Phase Approximation (RPA), the Percus-Yevick (PY) approximation, and the hypernetted chain (HNC) approxima7

CHAPTER 1. INTRODUCTION

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tion]. At the end of the chapter we present the mean spherical approximation (MSA) which (together with PY) admits an analytic solutions for a number of fluid models of physical interest. In chapter 4 there are no published results. We present the analytic solutions of the MSA for the three dimensional mixture of charged hard spheres, obtained using the Wiener-Hopf factorization (see appendix B). The original subject of this chapter, presented in the last section, is the discussion of the impossibility to use the Wiener-Hopf technique to find an analytic solution of the PY approximation for the three dimensional non additive hard spheres. Such solution has not yet been found. Chapter 5 contains original results published on [5]. We discuss and illustrate through numerical examples the relations between generating functionals, thermodynamic consistency (in particular the virial-free energy one), and uniqueness of the solution, in the integral equation theory of liquids. We propose a new approach for deriving closures automatically satisfying such characteristics. Results from a first exploration of this program are presented and discussed. In chapter 6 we present some original results published on [6]. A recently proposed connection between the threshold for the stability of the iterative solution of integral equations for the pair correlation functions of a classical fluid and the structural instability of the corresponding real fluid is carefully analyzed. Direct calculation of the Lyapunov exponent of the standard iterative solution of HNC and PY integral equations for the 1D hard rods fluid shows the same behavior observed in 3D systems. Since no phase transition is allowed in such 1D system, our analysis shows that the proposed one phase criterion, at least in this case, fails. We argue that the observed proximity between the numerical and the structural instability in 3D originates from the enhanced structure present in the fluid but, in view of the arbitrary dependence on the iteration scheme, it seems uneasy to relate the numerical stability analysis to a robust one-phase criterion for predicting a thermodynamic phase transition. Chapter 7 contains original results which are in course of publication on Physica A [7]. We calculate, through Monte Carlo numerical simulations, the partial total and direct correlation functions of the three dimensional symmetric Widom-Rowlinson mixture. We find that the differences between the partial direct correlation functions from simulation and from the Percus-Yevick approximation (calculated analytically by Ahn and Lebowitz) are well fitted by Gaussians. We provide an analytical expression for the fit parameters as function of the density. We also present Monte Carlo simulation data for the direct correlation functions of a couple of non additive hard sphere systems to discuss the modification induced by finite like diameters. In chapter 8 we present original results published on [8]. The classical (i.e. non-quantum) equilibrium statistical mechanics of a two-dimensional one-component plasma (a system of charged point-particles embedded in a neutralizing background) living on a pseudosphere (an infinite surface of constant negative curvature) is considered. In the case of a flat space, it is known that, for a one-component plasma, there are several reasonable definitions of the pressure, and that some of them are not equivalent to each other. In this chapter, this problem is revisited in the case of a pseudosphere. General relations between the different pressures are given. At one special temperature, the model is exactly solvable in the grand canonical ensemble. The grand potential and the one-body density are calculated in a disk, and the thermodynamic limit is investigated. The general relations between the different pressures are checked on the solvable model. In chapter 9 there are no published results. Following Edwards and Lenard paper [3] we describe a way of simplifying the calculation of the grand canonical partition function of an ensemble of classical particles living in a one dimensional world and interacting with a given pair potential. Using the notion of a general Gaussian random process and of Kac’s theorem, we show how it is possible to express the grand partition function as a one dimensional integral of

CHAPTER 1. INTRODUCTION

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the fundamental solution of a given partial differential equation. The kind of partial differential equation will be fixed by the kind of diffusion equation satisfied by the Gaussian random process. Following [3] we see how the Wiener process allows to treat the “Edwards model”. We then show how other stochastic processes can be used to treat other fluid models: we use the OrnsteinUhlenbeck process to simplify the calculation of the grand partition function of the “Kac-Baker model” and the generalized Ornstein-Uhlenbeck process to treat a fluid with a “general” pair potential.

Chapter 2

Background This thesis presents results on a few issues of the theory of classical liquids. A liquid is a particular phase of matter which occurs at intermediate values of pressure, temperature, and volume. In figure 2.1 we draw the typical phase diagram of a monatomic

(a)

F+S

Pressure

(b)

F

S

Critical point

Critical point

F 1111 0000 0000 1111 0000 1111 0000 1111 L 0000 1111 0000 1111 0000 1111 0000 G+L 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111

L+S

G

1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 L 0000000 1111111 0000000 1111111 0000000 1111111 0000000 Triple point 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111

S

G

Triple point

G+S Volume

Temperature

Figure 2.1: Phase diagram of a typical monatomic substance. The solid lines indicate the boundaries between solid (S), liquid (L), vapor (G) or fluid (F) phases. (a) is the projection in the pressure-temperature plane and (b) is that in the pressure-volume plane. The shaded regions indicate the part of these diagrams considered as a dense liquid. The limits of the liquid state are marked triple and critical points.

substance (for example argon). From the figure we can see that the liquid phase (bounded above by the critical point and below by the triple point) occupies a relatively small region of the phase diagram. The temperatures at which most substances are liquid are high enough that the system may be considered classical. From the point of view of the dynamical and structural properties a liquid distinguishes itself from a solid by the presence of an important diffusion and by the lack of long range order, and 10

CHAPTER 2. BACKGROUND

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from a gas by the importance of collisional processes and short range correlations. The work done by physicists when studying liquids can be described by the block diagram shown in figure 2.2. To understand theoretically the behavior of a liquid we need a model for the

make models

real liquids

model liquids

statistical mechanics

D perform experiment

carry out analytic solution computer of the model simulations A

construct approximate theories

numerical solution of the approximation

analytic solution of the approximation B

C experimental results

exact results for the model

theoretical predictions

compare

compare

tests of model

tests of theories

Figure 2.2: Block diagram showing the processes needed to test a particular model of a liquid and the ones needed to test a particular theory for a given model. In the thesis we will present some examples to illustrate those parts of the diagram which have a circle beneath them.

interactions. Then we need to compare the results obtained from the experiments with the ones obtained from the numerical simulations or the rarely available analytic solution of the model. While to test theories constructed from a particular model we need to compare the latter with the results from numerical or analytical (when available) solutions of the theories. In this thesis we will give some examples of “analytic solution of a model” (see chapters 8 and 9), some examples of “analytic solution of the approximation” (see chapter 4), some examples of “numerical solution of the approximation” (see chapters 5 and 6), and some examples of “computer simulations” (see chapter 7). We will also see how to “construct approximate theories” (see chapters 3, 5, and 6). Since we will not talk about how to “make models”, how to “perform experiment”, and how to “carry out computer simulations” we will spend some words on this arguments in this introductory chapter.

CHAPTER 2. BACKGROUND I. HOW TO MAKE A MODEL

i

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How to make a model

When creating a physical model of a real liquid one usually makes two basic hypothesis: the system, in absence of an external potential, has to be homogeneous and isotropic, in the bulk. The main ingredient to be chosen when creating a physical model of a real liquid is the type of interaction amongst the particles.

i.1

The interaction potential

The most general potential energy for a system of N pointwise particles is N

VN (r ) =

N X i=1

v1 (r1 ) +

X

v2 (ri , rj ) +

1≤i (Rα + Rβ )/2) non additivity is crucial to model several experimental results on binary systems (compound forming alloys, aqueous electrolyte solutions as well as molten salts are good examples for negative non additivity. Positive non-additivity could be used to model the tendency to phase separation in liquid alloys, some alkali metals alloys, or supercritical aqueous solutions of NaCl. See [52] for the references). Ordering phenomena (compound alternation or segregation) can often be interpreted in terms of the excluded volume effects due to non additivity in the repulsive cores. In this section we will show which are the main difficulties in trying to generalize to NAHS the work of Baxter [44] on the analytic solution of the PY approximation for a mixture of additive hard spheres. Let us consider an homogeneous and isotropic fluid of density ρ made of n different types of hard spheres of diameter R1 , R2 , . . . , Rn . Let ρα be the density of the hard spheres of type α = 1, 2, . . . , n. Moreover let the distance of closest approach between two spheres be  Rα α=β , (4.ii:1) Rαβ = 1 (R + R )(1 + ∆ ) α 6= β α β αβ 2 where ∆αβ are the parameters which rules the non additivity: for ∆αβ = 0 the spheres are said to be additive. The Ornstein-Zernike (OZ) equation (D.ii:1) give a relationship between the partial total correlation functions hαβ and the partial direct correlation functions cαβ X Z (4.ii:2) hαβ (r) = cαβ (r) + ργ cαγ (s)hγβ (|r − s|) , γ

where r = |r| and s = |s|. Since the pair interaction potential φαβ (r) is infinite for r < Rαβ we must have hαβ (r) = −1

for r < Rαβ

.

(4.ii:3)

Since φαβ (r) is zero for r > Rαβ , the PY approximation states that we must have cαβ (r) = 0

for r > Rαβ

.

(4.ii:4)

The problem consists in the solution of the system of equations (4.ii:2), (4.ii:3), and (4.ii:4).

ii.1

The Wiener-Hopf factorization is ill defined

Multiplying the OZ equation (4.ii:2) times dr we find in matricial form



ρα ρβ eik·r and integrating over the whole space in

˜ ˜ ˜ H(k) ˜ H(k) = C(k) + C(k)

,

(4.ii:5)

CHAPTER 4. MSA AND PY ANALYTIC SOLUTIONS II. PY SOLUTION FOR NON ADDITIVE HARD SPHERES

40

where k = |k| and  Z ∞   ˜ eikr Uαβ (r) dr  Hαβ (k) = 2π Z ∞ −∞ √   ρα ρβ shαβ (s) ds  Uαβ (r) =

,

(4.ii:6)

.

(4.ii:7)

r

 Z ∞  ˜  eikr Vαβ (r) dr  Cαβ (k) = 2π Z ∞ −∞ √   ρα ρβ scαβ (s) ds  Vαβ (r) = r

We follow the convention of choosing hαβ (−r) = hαβ (r) and cαβ (−r) = cαβ (r) so that Uαβ and Vαβ are even functions. ˜ Now we note that I − C(k), where I is the identity matrix, is a symmetric matrix and an even function of k. So following Baxter we perform the following Wiener-Hopf factorization −1 ˜ ˜ ˜ T (−k)Q(k) ˜ I − C(k) = [I + H(k)] =Q

,

where we use the superscript T to denote the transposed matrix and Z Rαβ ˆ αβ (k) . ˜ αβ (k) = δαβ − eikr Qαβ (r) dr ≡ δαβ − Q Q

(4.ii:8)

(4.ii:9)

Sαβ

The Qαβ (r) are real functions with support in [Sαβ , Rαβ ] and zero everywhere else. The parameters Sαβ are for the moment unknowns. We will now prove that they are ill defined for the NAHS. We first rewrite equation (4.ii:8) as follows X ˆ αβ (k) + Q ˆ βα (−k) − ˆ γα (−k)Q ˆ γβ (k) . C˜αβ (k) = Q Q (4.ii:10) γ

Then we multiply both sides by e−ikr and integrate in dk/(2π) over the whole real axis. We find for r > Rαβ XZ 2πVαβ (r) = − Qγα (t)Qγβ (t + r) dt , (4.ii:11) γ

where the integration in dt is over the interval [Sγα , Rγα ] ∩ [Sγβ − r, Rγβ − r] .

(4.ii:12)

Now from (4.ii:4) and (4.ii:7) follows that Vαβ (r) = 0 for r > Rαβ . So also the sum of the n integrals in (4.ii:11) must vanish. One can readily verify that each one of the n integrals vanishes if we choose Sγα = Rγβ − Rαβ

.

(4.ii:13)

For additive hard sphere this choice reduces to Sγα = (Rγ − Rα )/2 independent from β, which is Baxter’ s choice. We readily realize looking at (4.ii:1) that using (4.ii:13) for NAHS, would lead to a dependence of Sγα from an intermediate index β, occurrence which is not admissible.

CHAPTER 4. MSA AND PY ANALYTIC SOLUTIONS II. PY SOLUTION FOR NON ADDITIVE HARD SPHERES

ii.2

41

Symmetric binary mixture

For a symmetric binary mixture n = 2, ∆11 = ∆22 = ∆12 = ∆21 = ∆, R1 = R2 = R, and ρ1 = ρ2 = ρ/2. The problem reduces to determine the direct correlation function for like species c11 (r) and for unlike species c12 (r). A good approximation to the true direct correlation functions has been given by Gazzillo [53]. In this case the determination of the 4 Sαβ is much simplified. Following the argument given in the previous subsection we have studied the interval (4.ii:12) for r = Rαβ in the 8 cases of interest, determining in each case the Sαβ for which the intersection vanishes for r > Rαβ . We found: • α=β=1 γ = 1 [S11 , R] ∩ [S11 − R, 0]. The intersection vanishes for S11 = 0. γ = 2 [S21 , (1 + ∆)R] ∩ [S21 − R, ∆R]. The intersection vanishes for S21 = ∆R. • α=β=2 γ = 1 [S12 , (1 + ∆)R] ∩ [S12 − R, ∆R]. The intersection vanishes for S12 = ∆R. γ = 2 [S22 , R] ∩ [S22 − R, 0]. The intersection vanishes for S22 = 0. • α = 1, β = 2 γ = 1 [S11 , R] ∩ [S12 − (1 + ∆)R, 0]. γ = 2 [S21 , (1 + ∆)R] ∩ [S22 − (1 + ∆)R, −∆R]. • α = 2, β = 1 γ = 1 [S12 , (1 + ∆)R] ∩ [S11 − (1 + ∆)R, −∆R]. γ = 2 [S22 , R] ∩ [S21 − (1 + ∆)R, 0]. We see that with the choice S11 = S22 = 0 and S12 = S21 = ∆R each integral in (4.ii:11) vanishes. Once the Sαβ have been chosen the determination of the partial direct correlation function through Baxter’ s method is straightforward. With our choice of the Sαβ we had to restrict 0 < ∆ < 1/2 and we found for the like direct correlation function c11 (r) = a1 + a2 r + a4 r 3

for 0 < r < R

,

(4.ii:14)

where a1 , a2 , and a4 are functions of ∆ and R. The solution has the correct limit (4.i:15) as ∆ → 0. Unfortunately despite having the correct functional form and the correct ∆ → 0 limit this solution does not compare well with Gazzillo’s approximation or with the numerically generated function. For the unlike direct correlation function we found c12 (r) = c12 (r) =

c1

for 0 < r < ∆R

a0 /r + a1 + a2 r + a4

c12 (r) = b0 /r + b1 +

1 2 a2 r

+

r3

1 3 2 a4 r

,

for ∆R < r < (1 − ∆)R

(4.ii:15) ,

for (1 − ∆)R < r < (1 + ∆)R

(4.ii:16) ,

(4.ii:17)

where c1 , a0 , b0 , and b1 are functions of ∆ and R. c12 (r) is discontinuous at r = ∆R, lim c′12 (r) = 0 ,

r→∆R+

(4.ii:18)

and c′12 (r) is discontinuous at r = (1 − ∆)R. The discontinuities are unphysical. So it looks as if the Wiener-Hopf factorization does not give any good result for the NAHS. An element in favor of this conclusion come from the analysis of the Widom-Rowlinson model in the next subsection.

CHAPTER 4. MSA AND PY ANALYTIC SOLUTIONS II. PY SOLUTION FOR NON ADDITIVE HARD SPHERES

ii.3

42

The Widom-Rowlinson model

The Widom-Rowlinson model [54] is obtained from the binary mixture of hard spheres by setting ∆ = d/R

,

(4.ii:19)

and letting R → 0. Ahn and Lebowitz find the following solution in one dimension C˜11 (k) = 0 , C˜12 (k) = 2µ

(4.ii:20)

sin[d(k2

4µ2 )1/2 ]

+ (k2 + 4µ2 )1/2

,

(4.ii:21)

where µ is a parameter which can be determined from the following equation µ=



ρ1 ρ2 cos(2µd)

(4.ii:22)

It is possible to find a factorization like in (4.ii:10) and (4.ii:9). For example choosing ˆ ˆ 22 , Q ˆ 12 = Q ˆ 21 , and Q ˆ 11 , Q ˆ 12 real functions one finds Q11 = Q r q q 2 2 )/2]3/2 ˜ ˜ C12 − 2 (1 − 1 − C12 )/2 + 2[(1 − 1 − C˜12 ˜12 →0 2 C 4 ˆ Q11 = −→ C˜12 /8 + O[C˜12 ] , ˜ C12 r q ˜12 →0 C ˆ 12 = C˜12 /2 + O[C˜ 3 ] , (1 − 1 − C˜ 2 )/2 −→ Q 12

12

ˆ 11 = where amongst the 4 solutions to (4.ii:10) we have chosen the one for which limC˜12 →0 Q ˆ 12 = 0. Notice that the solution found indeed give for Q ˆ 11 (k) and Q ˆ 12 (k) even, limC˜12 →0 Q real functions. Infact for a symmetric binary mixture, the partial structure factors Sαβ (k) = √ −1 } ˜ ( ρα ρβ /ρ){[I − C(k)] αβ are given by 1 1 S(k) = 2 (k) ˜ 2 1 − C12



1 C˜12 (k) C˜12 (k) 1



.

But since limk→∞ C˜12 (k) = 0 and the Sαβ (k) has to remain finite for all k then we must have 2 (k) < 1 for all k. C˜12 However even if a factorization is possible, in the Widom-Rowlinson model the h11 (r) is unknown for all r. For the Wiener-Hopf factorization technique to be useful it is necessary that in the relationship between the Q(r) and the h(r), the h(r) are involved only over the interval where they are known. But this is not possible in the Widom-Rowlinson model for what we just said. So it seems as if the Wiener-Hopf factorization is not a useful technique to solve the Widom-Rowlinson model or the more general non additive hard spheres model in the PY approximation. In chapter 7 the structure of the three dimensional Widom-Rowlinson model will be studied in detail starting from Monte Carlo simulation results. We will also point out several misprints in the portion of the paper of Ahn and Lebowitz [54] dealing with the three dimensional WidomRowlinson model.

CHAPTER 4. MSA AND PY ANALYTIC SOLUTIONS III. HARD SPHERES WITH SURFACE ADHESION

iii

43

Hard spheres with surface adhesion

In studies on colloidal suspensions of neutral particles with adhesive interaction, Baxter’s “sticky hard spheres” model [30, 31] has often been used. In Baxter’s original formulation and its extension to the multicomponent case [32, 33] (refereed in the literature as SHS1 model) the pair potential, in addition to a hard sphere repulsion, contains an infinitely deep and narrow attractive square well, obtained according to a particular limiting procedure (“sticky limit”) that keeps the second virial coefficient finite. This model has the following positive features: • when studied within the PY approximation an analytic solution can be found, • such analytic solution exhibits a gas-liquid phase transition, • it is appropriate for the description of some properties of colloidal suspensions, micelles, microemulsions, and protein solutions with short range interactions. It can also describe some aspects of adsorption, flocculation, and percolation phenomena, solvent-mediated forces, ionic mixtures, solutions with a small degree of size polydispersity, and fluid of chain-like molecules (see [34] for a list of references). Its main drawbacks are: • Stell found [35] that sticky hard spheres of the same diameter, in the Baxter limit, when treated exactly, are not thermodynamically stable, • more importantly, colloidal suspensions are rather commonly polydisperse. Polydispersity means that mesoscopic suspended particles of a same chemical specie are not necessarily identical, but some of their properties (size, charge, etc.)may exhibit a discrete or continuous distribution of values. Even when all macroparticles belong to a unique chemical specie, a polydisperse fluid must therefore be treated as a multicomponent mixture. The number n of components may be of order 101 ÷ 103 or more (discrete polydispersity) or infinite (continuous polydispersity). Now the SHS1 in the PY approximation, when applied to mixtures, requires the solution of a set of n(n + 1)/2 coupled quadratic equations [32] a task which cannot be accomplished analytically. More recently there have been attempts to find an alternative SHS model which could be analytically tractable even in the general multicomponent case (see [34] for the references). In particular Brey and co-workers [36] proposed to start from a hard sphere Yukawa potential  +∞ r 0 and θ(x) = 0 otherwise, is the Heaviside step function, and δ(x) is the Dirac delta function. We then find lim g1∆ (R) Z Z R ds s = −2π

g1 (R) =

∆→0

R+s

dt tf (t) + 2π

R−s

0

R



R2 12τ

Z

2R

dt tf (t) 0

   2   1 2 R2 R R2 R2 ds s s − 2sR + = −2π + + 2π − 2 12τ 12τ 2 12τ 0 # "   2 R2 R4 5R4 − + 0> η+ η+ > η− >0

0> η+ > η−

η+ > η− >0 a>0 d>0

τ4

1/3

τ3

D0

for all η

τ2

1

τ1

Figure 4.3: Sign of the discriminant D.

τc = τ4 ≃ 0.09763

.

We have the following cases τ < τc τc < τ < τ2 τ2 < τ < τ1 τ1 < τ

(4.iii:29) has no solutions for η− < η < η+ (4.iii:29) has two real solutions for all η

(4.iii:29) has no solutions for η > η− > 1

(4.iii:29) has no solutions for η+ > η > η− > 1

(4.iii:34)

CHAPTER 4. MSA AND PY ANALYTIC SOLUTIONS III. HARD SPHERES WITH SURFACE ADHESION

49

√ Notice that the value physically accessible for the density are η ∈ [0, η0 ] where η0 = π/(3 2) is the maximum packing density. Corresponding to the critical temperature τc we have a critical density √ 2 ≃ 0.12132 . (4.iii:35) ηc = η± (τc ) = 6 + 25/2 We are left with the problem of deciding which one of the two solutions λ± is the physically acceptable one. From the virial expansion for g (4.iii:10), follows that 1 τ

lim λ(τ, η) =

η→0

.

(4.iii:36)

It follows then from equation (4.iii:29) that the physically acceptable solution is λ+ the other one diverging as η goes to 0. There is one more physical constraint that must be fulfilled, namely the radial distribution function has to be a bounded function. Multiplying equation (4.iii:16) times e−tr and integrating over r from 0 to ∞ (we change variables from (r, s) to (y = r − s, s) and integrate first over y) we find ¯′ (t) + 2πρQ(t) ¯ ¯ H(t) ¯ H(t) = −Q + 2πρI(t) Z R Z R I(t) = dy yety ds Q(s)e−ts , 0 y Z ∞ f (r)e−tr dr . f¯(t) ≡

,

(4.iii:37)

0

From which follows ¯′ (t) + 2πρI(t) −Q ¯ H(t) = ¯ 1 − 2πρQ(t)

.

(4.iii:38)

Moreover we have ¯ Q(t) = E(t) + F (t)e−Rt 2

E(t) = (a + bt + ct )/t

3

,

(4.iii:39)

,

(4.iii:40)

2

2

F (t) = (aR /2 + bR + c)/t + (Ra + b)/t + a/t

3

.

(4.iii:41)

¯ The function 1 − 2πρQ(t) tends to 1 for t → ∞ and behaves as −2πρ(−Ra)/t2 near t = 0. Then if a < 0 it must have at least a zero on the positive real axis. We thus conclude that for µ > 1 + 2η, H(r) (or g(r)) is an unbounded function. In figure 4.4 we plot the function λ+ (τ, η) (bold lines) and λ− (τ, η) for different values of τ . For τ < τc , λ± end on the curve s 6(2 + η) , (4.iii:42) A(η) = λ+ (τ− (η), η) = η(1 − η)2 where τ± (η) are the solutions to D = 0. For solutions above the curve B(η) =

2η + 1 η(1 − η)

,

(4.iii:43)

the radial distribution function is unbounded. The physical region is the one below the two curves A and B. This two curves meet at η = ηc .

CHAPTER 4. MSA AND PY ANALYTIC SOLUTIONS III. HARD SPHERES WITH SURFACE ADHESION

20

50

A(η) B(η) τ=0.13 τ=0.105 τ=0.0975 τ=0.092

18 16

λ

14 12 10 8 6 0

0.05

0.1

0.15 η

0.2

0.25

0.3

Figure 4.4: Plot of A(η) (4.iii:42), B(η) (4.iii:43), and λ± (τ, η) for τ = 0.13, 0.105, 0.0975, and 0.092. The physical region for the solutions to equation (4.iii:29) is the one below the two curves A and B.

iii.1

Compressibility Pressure

The pressure from the compressibility equation is given by 1 1 ∂P = = = 1 − ρˆ c(0) β ˆ ∂ρ T S(0) 1 + ρh(0) Z R sC(s) ds = 1 − 4πρ 0

= −C1

= [1 + 2η − λ− (τ, η)η(1 − η)]2 /(1 − η)4

,

(4.iii:44)

where S(k) is the static structure factor and a hat denotes a Fourier transform. Note that the isothermal compressibility is positive for all η and τ , and diverges on λ = B(η). We find the pressure integrating over η. If we call v0 = πR3 /6 the volume of a sphere we find Z η (1 + 2η ′ − µ(τ, η ′ ))2 /(1 − η ′ )4 dη ′ (4.iii:45) βv0 P (τ, η) = 0

=

1 [−3 η (1 + η (−11 + 19 η)) + 18 (−1 + η) η 3 (−1 + η)3

× (−2 + 11 η) τ − 216 (−1 + η)2 η τ 2 + 72 (−1 + η)3 τ 3 − 3 √  2 2 6 η (−2 + 5 η) − 12 (−1 + η) η τ + 6 (−1 + η)2 τ 2 ]

We see from figure 4.5 that, with the Percus-Yevick closure, there is a non accessible region on the plane (pressure,volume). That region is delimited by the curve C(η) = βv0 P (τ− (η), η) p √ √ η 2 = [−3 − 3η − 3η + η(2 + η)( 6η + 24)] 3(η − 1)3

.

(4.iii:46)

CHAPTER 4. MSA AND PY ANALYTIC SOLUTIONS III. HARD SPHERES WITH SURFACE ADHESION

0.1

51

C(1/η) τ=0.2 τ=0.13 τ=0.105 τ=0.0975 τ=0.092

0.08

βv0P

0.06

0.04

0.02

0 0

5

10 1/η

15

20

Figure 4.5: Plot of several isotherms at τ = 0.2, 0.13, 0.105, 0.0975, and 0.092. The equation of state is obtained from the compressibility equation (4.iii:44). We plot βv0 P against 1/η, the dimensionless volume per particle. The region below the curve C(1/η) (4.iii:46) is unaccessible with the Percus-Yevick closure.

At the critical point we find

√ 2(4 + 3 2) √ βv0 Pc = ≃ 0.04602 9(2 + 2)3

,

(4.iii:47)

where Pc = P (τc , ηc ). The critical isotherm Pc (η) = P (τc , η) has the following behavior near the critical point  ω+ (η − ηc )3 + O[(η − ηc )4 ] η > ηc (4.iii:48) Pc (η) − Pc = ω− (η − ηc )3 + O[(η − ηc )4 ] η < ηc where ω± = limη→ηc ± ∂ 3 Pc /∂η 3 with ω− ≈ 34ω+ .

iii.2

Virial pressure

From the virial theorem we have   Z ∞ dφ∆ (r) βP 2 3 ∆ r g (r) = lim 1 − πβρ dr ∆→0 ρ 3 dr 0

.

(4.iii:49)

Because this equation involves the derivative of the pair potential, it cannot be used directly in the calculation of the equation of state. This problem can be overcome rewriting the equation in terms of the function N (r)∆ defined as N ∆ (r) = reβφ

∆ (r)

g∆ (r) .

(4.iii:50)

Indeed it can be shown that this function must be a continuous function even when g∆ and φ∆ are discontinuous. Using equation (4.iii:50) in (4.iii:49) we find   Z ∞ ∆ dφ∆ (r) 2 βP r 2 N ∆ (r)e−βφ (r) = lim 1 − πβρ dr ∆→0 ρ 3 dr 0 ( ) Z ∞ ∆ 4η de−βφ (r) = lim 1 + 3 r 2 N ∆ (r) dr . (4.iii:51) ∆→0 R 0 dr

CHAPTER 4. MSA AND PY ANALYTIC SOLUTIONS III. HARD SPHERES WITH SURFACE ADHESION

52

The discontinuities in the pair potential are such (see figure 4.1) that the derivative of the Boltzmann factor is ∆ (r)

de−βφ dr

−ǫ(∆)

=e

  −ǫ(∆) δ(r − R ) − e − 1 δ(r − R) , ′

where R′ = R − ∆ and e−ǫ(∆) = R/(12τ ∆) (see equation (4.iii:4)). We find then     βP R 4η R 4η 2 = lim 1 + 3 R′ N ∆ (R′ ) − 3 − 1 R2 N ∆ (R) ∆→0 ρ R 12τ ∆ R 12τ ∆ " #) ( η R2 N ∆ (R) − R′ 2 N ∆ (R′ ) 4η ∆ . = lim 1 + N (R) − ∆→0 R 3τ R2 ∆

(4.iii:52)

(4.iii:53)

We now observe that from the Percus-Yevick closure follows N ∆ (r) = r + H ∆ (r) − C ∆ (r). ∆→0 ˜ From equations (4.iii:13) and (4.iii:24) then follows N ∆ (r) −→ N (r) = −C(r) for r ≤ R.Using equation (4.iii:28) we further find that the second term on the right hand side of equation (4.iii:53) equals 4ηλτ . To calculate the third term one has to carefully evaluate the ∆ → 0 limit. Assuming we can evaluate it as follows # " ˜ ˜ ′) η R2 C(R) − R′ 2 C(R η = lim [(2/R)C0 + 3C1 + 4RC2 + 6R3 C4 ] , (4.iii:54) 2 3τ ∆→0 R ∆ 3τ we find for the virial pressure the following expression   o n βP η βv0 P = η = η 1 + 4ηλ+ τ + [(2/R)C0 + 3C1 + 4RC2 + 6R3 C4 ] ρ 3τ

. (4.iii:55)

This expression coincides with equation (38) in [30] apart from the lack of the additional term η 2 λ3 /24 within the square brackets, present in Baxter’s equation. Indeed the function N ∆ (r) is not differentiable in r = R. So our assumption was wrong: we cannot substitute N ∆ with N and then take the limit ∆ → 0. The task of calculating N ∆ is a little bit more laborious, since one cannot work with delta functions, and is left as an exercise to the reader.

Chapter 5

Generating functionals, consistency, and uniqueness in the integral equation theory of liquids In chapter 3 we introduced the most common integral equation theories (IET). IET of the liquid state statistical mechanics are valuable tools for studying structural and thermodynamic properties of pairwise interacting fluid systems [12, 55]. Many of these approximations to the exact relation between pair potential and pair correlation functions have been proposed in the last half century, starting from the pioneering works [56–58] to the most refined and modern approximations [59–63] which may approach the accuracy of computer simulation with a negligible computational cost. The functional method in statistical mechanics [12] provides the most general and sound starting point to introduce IET as approximations of the exact functional relations and it is the classical statistical mechanics counterpart of the quantum density functional theory. Notwithstanding the success of present IET to describe the structure of simple one component systems, considerable work is still devoted to derive improved approximations which could accurately describe the thermodynamics as well. Also applications to non simple or multicomponent systems are still subject of current studies. Actually, the description of thermodynamics is one weak point of IET approaches: reasonable and apparently harmless approximations to the potential-correlation relations usually result in a dramatically inconsistent thermodynamics where many, if not all, among the exact sum rules derived from statistical mechanics, are violated. The problem of thermodynamic inconsistency, i.e. the inequivalence between different routes to thermodynamics, actually plagues the IET approach to the point that the degree of inconsistency between different formulae for the same quantity is used as an intrinsic measurement of the quality of a closure. In the past, some discussion of the thermodynamic consistency appeared in the literature. Hypernetted chain approximation (HNC) was recognized as a closure directly derivable from an approximation for the free energy functional [64] , thus exhibiting consistency between the virial formula and the thermodynamic expression for the pressure. However, this limited consistency is not enough to guarantee a unique and faithful description of the phase diagram. Apart the problem of the remaining inconsistencies, the descriptions of the critical points and spinodal lines are seriously inadequate. Extensive work on HNC [65–67] showed that in place of a true spinodal line, it is more 53

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS I. THERMODYNAMIC CONSISTENCY AND UNIQUENESS OF THE SOLUTION OF INTEGRAL EQUATIONS 54

appropriate to describe the numerical results as due to a region in the thermodynamic plane where no real solution of the integral equation exists. In particular, Belloni[65] showed that the disappearance of the solution originates from a branching point where two solutions merge, instead than from a line of diverging compressibility. Thus, we have direct evidence that HNC may have multiple solutions, at least in part of the phase diagram. Empirical improvements on HNC have been proposed [59, 62, 63] providing in many cases excellent results for one-component simple fluids. However, although reduced, the thermodynamic inconsistency problem remains and the multiple solution problem is completely untouched. In this work we start an investigation of a new approach to IET directly addressing the two points of uniqueness of the solution and thermodynamic consistency. The basic idea is to constrain the search for new closures within the class of generating functionals which are strictly convex free-energy functionals, thus enforcing the virial-energy consistency as well as the uniqueness of the solution. In particular, in the present chapter we try to answer the following questions: i) does at least one strictly convex free-energy functional of the pair correlation function exist? ii) what is the nature of the resulting spinodal line (if any), iii) what is the quality of the resulting thermodynamic and structural results? iv) does the simultaneous requirement of consistency and uniqueness automatically provide improved results? As we will show, we have a positive answer for i), a thorough and interesting characterization for ii), some interesting indications for iii), and a partly negative answer for iv). However, we can show that it is possible to exploit the control provided by the generating functional approach to easily generate new closures and we feel our procedure could be the basis of a more systematic approach to IET. In section i we recall the connections between closures, generating functionals, thermodynamic consistency and uniqueness of solutions and we illustrate them in the well known case of HNC approximation. In section ii we introduce two straightforward extensions of HNC intended to cure its problems. In Section iii numerical results are presented and discussed. In section iv we show two possible improvements of the closures studied.

i

Thermodynamic consistency and uniqueness of the solution of integral equations

Since the work by Olivares and McQuarrie [68] it is known the general method to obtain the generating functional whose extremum with respect to variations of the direct (c(r)) or total (h(r)) correlation functions results in the closure relation, provided the Ornstein-Zernike equation is satisfied. In a way, here we address the inverse problem of the derivation of a closure: given a closure (whatever was the way of deriving it) what is the functional of the correlation functions which has the closure as extremum value equation? For example, if we have a closure of the form ρ2 c(r) = Ψ{h(r), βφ(r)}

,

(5.i:1)

where φ(r) is the pair interaction potential and Ψ is an arbitrary function, the functional Z dk 1 ˆ ˆ {ρh(k) − ln[1 − ρh(k)]}− Q[h(r), βφ(r)] = 2βρ (2π)3  Z Z 1 dt Ψ{th(r), βφ(r)} + constant , (5.i:2) dr h(r) 0

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS I. THERMODYNAMIC CONSISTENCY AND UNIQUENESS OF THE SOLUTION OF INTEGRAL EQUATIONS 55

is such that the extremum condition

δQ =0 , δh(r)

(5.i:3)

is equivalent to 2

ρ h(r) = Ψ{h(r), βφ(r)} + ρ

Z

h(|r − r′ |)Ψ{h(r ′ ), βφ(r ′ )} dr′

.

(5.i:4)

Olivares and McQuarrie also showed how to find the generating functional if the closure is expressed in the form ρ2 h(r) = Ψ{c(r), βφ(r)} . (5.i:5) In section iv.3 we discuss the extension of their method to the case of a closure written as ρ2 c(r) = Ψ{γ(r), βφ(r)}

,

(5.i:6)

where γ(r) = h(r) − c(r) is the indirect correlation function. Notice that most of the modern closures correspond to this last case. The possibility of translating the original integral equation into an extremum problem allows to get an easy control on two important characteristics of the approximation: thermodynamic consistency between energy and virial routes to the thermodynamics and uniqueness of the solution. Indeed, once we get the generating functional Q, due to the approximations induced by the closure, there is no guarantee that its value at the extremum is an excess free energy. In order to be a free energy, the functional should satisfy the condition ρ δQ = g(r) δφ(r) 2

,

(5.i:7)

where g(r) = h(r) + 1 is the pair distribution function. Even if this condition is not new, and mention to it is present in the literature [69], we discuss it in appendix F as well as its consequences on the thermodynamic consistency between the virial pressure and the density derivative of the free energy. Another issue where the generating functional approach is useful is the problem of multiple solutions of the integral equations [65]. In particular, the analysis of the convexity properties of the generating functional is a very powerful tool [70, 71]. Let us illustrate this techniques in the case of HNC closure. It is well known [64, 68] that the HNC equation with closure i h , (5.i:8) c(r) = h(r) − ln g(r)eβφ(r) can be derived from the variational principle δF[h] =0 , δh(r)

(5.i:9)

F[h] = FOZ [h] + FHN C [h] ,

(5.i:10)

where

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS I. THERMODYNAMIC CONSISTENCY AND UNIQUENESS OF THE SOLUTION OF INTEGRAL EQUATIONS 56

with Z  dk  ˆ ˆ F [h] = {ρh(k) − ln[1 + ρh(k)]} ,    OZ (2π)3 Z   i o n h     FHN C [h] = ρ2 dr 1 + g(r) ln g(r)eβφ(r) − 1 − h2 (r)/2

(5.i:11) .

¯ Let us call h(r) the extremum of F, solution of the variational principle (5.i:9). It can be ¯ shown (see appendix F) that, within an additive constant, F[h]/(2βρ) is the excess Helmholtz free energy per particle of the liquid. This ensure thermodynamic consistency between the route to the pressure going through the partial derivative of the free energy and the one going through the virial theorem (see appendix F). In addition, it allows to get a closed expression for the excess chemical potential without further approximations [72, 73]. This feature is highly desirable for applications of IET to the determination of the phase diagrams. Moreover if we can prove that F, defined on some convex set of trial correlation functions Dc , is a strictly convex functional, then we know that if a solution to (5.i:9) exists, it corresponds to a minimum and is unique. A functional F is strictly convex if for all y(r) ∈ Dc and y(r) 6= 0, we have Z δ2 F[h] y(r ′ ) dr dr′ > 0 . (5.i:12) A = y(r) δh(r)δh(r ′ ) We calculate the second functional derivatives as follows Z  2 δ FOZ [h] 1 dk −ik·(r+r′ )  2  =ρ e  ′ 3  2 ˆ δh(r)δh(r ) (2π)  [1 + ρh(k)]      1 δ2 FHN C [h] 2 ′   = ρ δ(r − r ) −1 δh(r)δh(r ′ ) g(r)

, (5.i:13)

.

ˆ Recalling that the static structure factor S(k) = 1 + ρh(k), we find for A   Z Z dk yˆ2 (k) 1 2 2 A/ρ = + dr y (r) −1 . (2π)3 S 2 (k) g(r)

(5.i:14)

Now, the most interesting results would be to show the strict convexity of the HNC functional over the convex set of all the admissible pair correlation functions (all the h(r) ≥ −1 and properly decaying to zero at large distance. However, this is not the case for HNC. It has not been possible to show the positive definiteness of equation (5.i:14) and it has been shown [65] that in some region of the thermodynamic plane HNC does exhibit multiple solutions. The best we can do is to obtain a more limited result. Calling g1 = sup g(r) (g1 > 1 is the height of the first peak of the pair distribution function) and using Parseval theorem, we find   Z 1 1 dk 2 2 yˆ (k) −1+ , (5.i:15) A/ρ > (2π)3 S 2 (k) g1 from which we deduce that A > 0 on the following set of functions o n p . D = h(r) | 0 < S(k) < g1 /(g1 − 1) ∀k

(5.i:16)

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS II. EXTENSIONS OF HNC 57

We conclude that F defined on any convex set of functions Dc ⊂ D is strictly convex. Near the triple point we are sure we are out from such set since the first p peak of the pair distribution function for the Lennard-Jones fluid is g1 ≃ 3 [74], so that g1 /(g1 − 1) ≃ 1.2. The first peak of the static structure factor is also close to 3. Then we are not inside D and the HNC approximation may have multiple solutions [65]. Instead, if we are in the weak coupling regime, the previous conditions tells us that there is a range where the branch of solutions going to the perfect gas limit is unique and quite isolated from other solutions.

ii

Extensions of HNC

The generating functional approach can be used in a systematic way to look for better closures. We think that this way, we can obtain a less empirical search method for improving closures. In the following we report some preliminary analysis we have done. As a first test of our program, we have restricted our investigations to simple modifications of HNC functional. As we will discuss later, such a choice is certainly not optimal. However, we can learn enough to consider the approach worthwhile of further investigations and we feel the results are interesting in order to reveal more details about the characteristics of the solutions of the highly non linear IET.

ii.1

The HNC/H2 approximation

We want to modify the HNC closure in order to have an integral equation with a generating functional which is strictly convex without having to restrict its definition domain. We choose as our modified HNC (HNC/H2) closure 1 c(r) = h(r) − ln[g(r)] − βφ(r) − αh2 (r) ,

(5.ii:1)

with α a parameter to be determined. The new closure generating functional is Z  i o n h  2 FHN C/H2 [h] = ρ dr 1 + g(r) ln g(r)eβφ(r) − 1 − h2 (r)/2 + αh3 (r)/3 .

Its second functional derivative with respect to h is   δ2 FHN C/H2 [h] 1 2 ′ = ρ δ(r − r ) − 1 + 2αh(r) δh(r)δh(r ′ ) g(r)

.

(5.ii:2)

(5.ii:3)

Recalling that h = g − 1 and g(r) > 0 for all r, we see that for α = 1/2 (1 − g)2 1 − 1 + 2αh = ≥ 0 ∀g g g

.

(5.ii:4)

Then FHN C/H2 is a convex functional and since FOZ is unchanged and strictly convex (see appendix G), their sum, the generating functional of the integral equation, is strictly convex. ¯ + FHN C/H2 [h]}/(2βρ) ¯ Moreover {FOZ [h] continues to be the excess Helmholtz free energy per particle of the liquid since equation (5.i:7) holds (see appendix F). 1

Our first trial should really be c = − ln g − βφ. Which should be called HNC/H1. We have tested numerically this closure and we found that it performed worst than HNC/H2 both for the structure and for the thermodynamics of the system under exam.

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS III. NUMERICAL RESULTS 58

We have then an integral equation which is both thermodynamically consistent (the pressure calculated from the virial theorem coincides with that one calculated from the Helmholtz free energy) and with a solution which, when it exists, is unique.

ii.2

The HNC/H3 approximation

In the same spirit as in subsection ii.1 we can try to add a term h3 in the HNC/H2 closure c(r) = h(r) − ln[g(r)] − βφ(r) − αh2 (r) − γh3 (r) ,

(5.ii:5)

with α and γ parameters to be determined. We call this approximation HNC/H3. The closure generating functional is Z  i n h  2 FHN C/H3 [h] = ρ dr 1 + g(r) ln g(r)eβφ(r) − 1 − h2 (r)/2+ αh3 (r)/3 + γh4 (r)/4 . (5.ii:6) Its second functional derivative with respect to h is   δ2 FHN C/H3 [h] 1 2 ′ 2 = ρ δ(r − r ) − 1 + 2αh(r) + 3γh (r) δh(r)δh(r ′ ) g(r) 1 − g(r) {1 − 2αg(r) + 3γg(r)[1 − g(r)]} = ρ2 δ(r − r′ ) g(r)

.

(5.ii:7)

In order to have the right hand side of this expression positive for g > 0 the only choice we have is to set α = 1/2. In this way (1 − g)[1 − 2αg + 3γg(1 − g)] = (1 − g)2 (1 + 3γg)

,

(5.ii:8)

and we see that FHN C/H3 is a convex functional if we additionally choose γ > −1/[3 sup g(r)]. ¯ + FHN C/H3 [h]}/(2βρ) ¯ Once again {FOZ [h] is the excess Helmholtz free energy per particle of the liquid and the thermodynamic consistency virial-free energy is ensured.

iii

Numerical results

To solve numerically the OZ plus closure system of nonlinear equations we used Zerah’ s algorithm [75]. We performed Fourier transforms using a fast Fourier transform routine taken from CERN library. In the code we always work with adimensional thermodynamic variables T ∗ = 1/(βǫ),ρ∗ = ρσ 3 , and P ∗ = P σ 3 /ǫ, where σ and ǫ are the characteristic length and characteristic energy of the system respectively.We always used 1024 grid points and a step size ∆r = 0.025σ. The thermodynamic quantities were calculated according to the statistical mechanics formulae for: the excess internal energy per particle Z ∞ exc φ(r)g(r)r 2 dr , (5.iii:1) U /N = 2πρ 0

the excess virial pressure 2 βP /ρ − 1 = − πβρ 3 v

Z

∞ 0

dφ(r) g(r)r 3 dr dr

,

(5.iii:2)

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS III. NUMERICAL RESULTS 59

the bulk modulus calculated from the compressibility equation Bc =

1 β = ρχT S(k = 0)

,

(5.iii:3)

where χT is the isothermal compressibility, and the bulk modulus calculated from the virial equation ∂P v ∂ρ Z ∞ Z ∞ dφ(r) dφ(r) ∂g(r) 3 2 4 g(r)r 3 dr − πβρ2 r dr = 1 − πβρ 3 dr 3 dr ∂ρ 0 0

Bp = β

.

(5.iii:4)

For the calculation of Bp once g(r) and c(r) had been calculated, Lado’ s scheme for Fourier transforms [76] was used to determine ∂ˆ g (k)/∂ρ. Even if slow, this allows us to explicitly calculate and later invert the coefficients matrix of the linear system of equations which enters the calculation of ∂ˆ g (k)/∂ρ.

iii.1

Inverse power potentials

The general form of the inverse power potential is  σ n φ(r) = ǫ r

,

(5.iii:5)

where 3 < n < ∞. For this class of fluids the thermodynamics depends only from the dimensionless coupling parameter √ z = (ρσ 3 / 2)(βǫ)3/n

.

(5.iii:6)

In this subsection we choose to fix ρ∗ = 1 so that equation (5.iii:6) gives the relation between z and T ∗ . We performed our calculations on the n =12, 6, and 4 fluids at the freezing point. We compared three kind of closures: the thermodynamically consistent one of Rogers and Young [77] (RY) with thermodynamic consistency virial-compressibility and known to be very close to the simulation results, the hypernetted-chain (HNC) closure, and the HNC/H2 described in subsection ii.1. In each case we compared our data with the Monte Carlo (MC) results of Hansen and Schiff [78]. iii.1.1

The inverse 12th power potential

The freezing point for this fluid is at z = 0.813. In figure 5.1 we compare the MC and RY results for the pair distribution function. The RY α parameter to achieve thermodynamic consistency at this value of z is 0.603. Notice that we express α in units of σ and not of a = (3/4πρ)1/3 as in the original Rogers and Young’ s paper [77]. In figure 5.2 we compare the MC, the HNC, and the HNC/H2 results for the pair distribution function. In table 5.1 we compare various thermodynamic quantities (the excess internal energy per particle, the excess virial pressure, the bulk moduli) obtained from the RY, the HNC, and the HNC/H2 closures. In the MC calculation of Hansen and Schiff the excess internal energy per particle is 2.675, the excess virial pressure is 18.7, and the bulk modulus 72.7.

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS III. NUMERICAL RESULTS 60

3

MC RY (α=0.603)

g(x)

2

1

0

0

1

2

3

x=r/σ

Figure 5.1: Comparison of the Monte Carlo (MC) and Rogers Young (RY) results for the pair distribution function of the inverse 12th-power fluid at z = 0.813. 3

MC HNC HNC/H2

g(x)

2

1

0

0

1

2

3

x=r/σ

Figure 5.2: Comparison of the Monte Carlo (MC), the HNC, and HNC/H2 results for the pair distribution function of the inverse 12th-power fluid at z = 0.813.

iii.1.2

The inverse 6th power potential

The freezing point for this fluid is at z = 1.54. In figure 5.3 we compare the MC and RY results for the pair distribution function. The RY α parameter to achieve thermodynamic consistency at this value of z is 1.209. In figure 5.4 we compare the MC, the HNC, and the HNC/H2 results for the pair distribution function.

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS III. NUMERICAL RESULTS 61

closure RY (α = 0.603) HNC HNC/H2

U exc /(N ǫ) 2.626 3.009 3.200

βP (v) /ρ − 1 18.359 21.036 22.372

Bc 69.782 45.278 52.661

Bp 70.125 80.430 87.255

Table 5.1: We compare various thermodynamic quantities as obtained from the RY, the HNC, and the HNC/H2 closure, for the inverse 12th-power fluid at the freezing point (z = 0.813). U exc /(N ǫ) is the excess internal energy per particle, βP (v) /ρ − 1 the excess virial pressure, Bc and Bp are the bulk moduli from the compressibility and the virial equation respectively. 3

MC RY (α=1.209)

g(x)

2

1

0

0

1

2

3

x=r/σ

Figure 5.3: Comparison of the Monte Carlo (MC) and Rogers Young (RY) results for the pair distribution function of the inverse 6th-power fluid at z = 1.54.

In table 5.2 we compare various thermodynamic quantities (the excess internal energy per particle, the excess virial pressure, the bulk moduli) obtained from the RY, the HNC, and the HNC/H2 closures. In the MC calculation of Hansen and Schiff the excess internal energy per particle is 4.090, the excess virial pressure is 38.8 and the bulk modulus 110.1.

iii.1.3

The inverse 4th power potential

The freezing point for this fluid is at z = 3.92. In figure 5.5 we compare the MC and RY results for the pair distribution function. The RY α parameter to achieve thermodynamic consistency at this value of z is 1.794. In figure 5.6 we compare the MC, the HNC, and the HNC/H2 results for the pair distribution function. In table 5.3 we compare various thermodynamic quantities (the excess internal energy per particle, the excess virial pressure, the bulk moduli) obtained from the RY, the HNC, and the HNC/H2 closures. In the MC calculation of Hansen and Schiff the excess internal energy per particle is 8.233, the excess virial pressure is 107.7 and the bulk modulus 156.

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS III. NUMERICAL RESULTS 62

3

MC HNC HNC/H2

g(x)

2

1

0

0

1

2

3

x=r/σ

Figure 5.4: Comparison of the Monte Carlo (MC), the HNC, and HNC/H2 results for the pair distribution function of the inverse 6th-power fluid at z = 1.54.

closure RY (α = 1.209) HNC HNC/H2

U exc /(N ǫ) 4.114 4.235 4.283

βP (v) /ρ − 1 39.027 40.178 40.635

Bc 110.952 84.016 88.289

Bp 111.420 113.733 115.757

Table 5.2: We compare various thermodynamic quantities as obtained from the RY, the HNC, and the HNC/H2 closure, for the inverse 6th-power fluid at the freezing point (z = 1.54). U exc /(N ǫ) is the excess internal energy per particle, βP (v) /ρ − 1 the excess virial pressure, Bc and Bp are the bulk moduli from the compressibility and the virial equation respectively.

iii.2

The spinodal line

In this subsection we study a pair potential with a minimum In particular we chose the LennardJones potential    σ 12  σ 6 φ(r) = 4ǫ − , (5.iii:7) r r where ǫ and σ are positive parameters. The critical point for this fluid is at [79] Tc∗ = 1.3120 ± 0.0007 ρ∗c = 0.316 ± 0.001

Pc∗ = 0.1279 ± 0.0006 Integral equations fail to have a solution at low temperature and intermediate density, i.e. in the two-phase unstable region of the phase diagram. In particular it is well known that the HNC approximation is unable to reproduce the spinodal line, the locus of points of infinite compressibility in the phase diagram [65]. This is due to the loss of solution as one approaches

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS III. NUMERICAL RESULTS 63

3

MC RY (α=1.794)

g(x)

2

1

0

0

1

2

3

x=r/σ

Figure 5.5: Comparison of the Monte Carlo (MC) and Rogers Young (RY) results for the pair distribution function of the inverse 4th-power fluid at z = 3.92. 3

MC HNC HNC/H2

g(x)

2

1

0

0

1

2

3

x=r/σ

Figure 5.6: Comparison of the Monte Carlo (MC), HNC, and HNC/H2 results for the pair distribution function of the inverse 4th-power fluid at z = 3.92.

the spinodal line on an isotherm from high or from low densities. The line of loss of solution, in the phase diagram, is called termination line. The loss of solution for the HNC approximation is due to the loss of strict convexity of the generating functional [80]. Indeed, using HNC approximation, we computed the bulk modulus from the compressibility equation Bc , on several isotherms as a function of the density. At low temperatures we found that both at high density and at low density we were unable to continue the isotherm at low values of Bc . Zerah’ s

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS III. NUMERICAL RESULTS 64

closure RY (α = 1.794) HNC HNC/H2

U exc /(N ǫ) 8.001 8.047 8.068

βP (v) /ρ − 1 104.664 105.277 105.542

Bc 250.106 223.328 226.966

Bp 242.948 244.212 257.678

Table 5.3: We compare various thermodynamic quantities as obtained from the RY (notice that the bulk moduli were not given in the Rogers and Young’ s paper and the value of the virial pressure as reported in our table was not corrected to take into account the long range nature of the potential), the HNC and the HNC/H2 closure, for the inverse 4thpower fluid at the freezing point (z = 3.92). U exc /(N ǫ) is the excess internal energy per particle, βP (v) /ρ − 1 the excess virial pressure, Bc and Bp are the bulk moduli from the compressibility and the virial equation respectively.

algorithm either could not get to convergence or it would converge at a non physical solution (with a structure factor negative at some wavevector k). Since HNC/H2 has, by construction, an always strictly convex generating functional, we expect it to be able to reproduce a spinodal line (there should be no termination line). In Figure 5.7 we show the behavior of Bc on several isotherms as a function of density,

0.5

T**=1.4 T =1.3 T**=1.2 T =1.1 T**=1.0 T =0.9 T**=0.8 T*=0.7 T =0.6

0.45 0.4 0.35

Bc

0.3 0.25 0.2 0.15 0.1 0.05 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ρ∗

Figure 5.7: Behavior of Bc of the Lennard-Jones fluid, on several isotherms as a function of the density for the HNC/H2 approximation.

calculated with the HNC/H2 approximation. We see that now there are no termination points. Bc never becomes exactly zero and the low temperature isotherms develop a bump in the intermediate density region. The same plot for the bulk modulus calculated from the virial pressure Bp , shows that at low temperatures this bulk modulus indeed becomes zero along the isotherms both at high and low densities. In figure 5.8 the pressure is plotted as a function of the density on several isotherms for the HNC/H2 approximation. Apart from the fact that we find negative pressures, the isotherms has a Van der Waals like behavior.

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS IV. IMPROVING THE CLOSURES 65

0.4 0.2

P*

0 T*=1.4 T*=1.3 T**=1.2 T*=1.1 T =1.0 T*=0.9 T**=0.8 T*=0.7 T =0.6

-0.2 -0.4 -0.6 -0.8 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ρ∗

Figure 5.8: Behavior of the pressure of the Lennard-Jones fluid, on several isotherms as a function of the density for the HNC/H2 approximation.

iv

Improving the closures

The numerical results for HNC/H2 exhibit interesting features as far as the coexistence region is concerned but show unambiguously a worst agreement with the MC structural data in correspondence with a marginal improvement in the thermodynamics. We feel that the main problem is the difficulty of an accurate description of the bridge functions in terms of powers of the pair correlation function. Recent investigations on improved closures seem to point to the indirect correlation function γ(r) or some renormalized version of it, as the best starting point for progress. However, before moving to more complex relations or functional dependences, we have explored two possible directions for improving the HNC/H2 closure. In the first approach we have tried to follow the MHNC approach by Lado et al. [81]. In the second we have explored the possibilities of optimization offered by the numerical coefficient of the cubic term in the generating functional.

iv.1

The reference HNC/H2 approximation

From the graphical analysis of the pair distribution function it is known [12] that g(r) may be written as g(r) = e−βφ(r)+γ(r)+B(r)

,

(5.iv:1)

where γ(r) = h(r) − c(r) is the sum of all the series type diagrams and B(r) the sum of bridge type diagrams. If we take 1 B(r) = − h2 (r) + G(r) 2

,

(5.iv:2)

we have that our HNC/H2 approximation amounts to setting G(r) = 0. Rosenfeld and Ashcroft [59] proposed that B(r) should be essentially the same for all potentials φ(r). We now make the same proposal for the G function. In the same spirit of the RHNC approximation of Lado [81] we will approximate G(r) with the G function of a short range (reference) potential φ0 (r).

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS IV. IMPROVING THE CLOSURES 66

Assuming known the properties of the reference system, we can calculate the G function as follows i h 1 (5.iv:3) G0 (r) = ln g0 (r)eβφ0 (r) − γ0 (r) + h20 (r) . 2 The reference HNC/H2 (RHNC/H2) approximation is then 1

g(r) = e−βφ(r)+γ(r)− 2 h

2 (r)+G

0 (r)

.

(5.iv:4)

An expression for the free energy functional can be obtained turning on the potential φ(r) in two stages: first, from the noninteracting state to the reference potential φ0 (r) and then from there to the full potential φ(r). To this end we write φ(r; λ0 , λ1 ) = λ0 φ0 (r) + λ1 ∆φ(r) ,

(5.iv:5)

with ∆φ(r) = φ(r) − φ0 (r). Following the same steps as in [60] we obtain for the excess free energy per particle (0)

f exc = f1 + f2 + f3 + ∆f3

(5.iv:6)

where the first two terms were already encountered in section i Z  i o n h  1 ρ dr 1 + g(r) ln g(r)eβφ(r) − 1 − h2 (r)/2 + h3 (r)/6 βf1 = 2 Z 1 dk ˆ ˆ βf2 = {ρh(k) − ln[1 + ρh(k)]} . 2ρ (2π)3

,

(5.iv:8)

The third term is assumed known Z Z 1 ∂g(r; λ0 , 0) 1 (0) (0) (0) dλ0 G(r; λ0 , 0) = β(f (0) − f1 − f2 ) , βf3 = − ρ dr 2 ∂λ 0 0 (0)

(5.iv:7)

(5.iv:9)

(0)

here f (0) is the excess free energy per particle of the reference system and f1 , f2 are defined as in equations (5.iv:7), (5.iv:8) for the reference potential and its corresponding correlation functions. The last term is Z Z 1 1 ∂g(r; 1, λ1 ) β∆f3 = − ρ dr dλ1 G(r; 1, λ1 ) . (5.iv:10) 2 ∂λ1 0 According to our proposal, G is insensitive to a change in potential from φ0 to φ. We may then approximate this last term as follows Z 1 β∆f3 ≈ − ρ drG0 (r)[g(r) − g0 (r)] . (5.iv:11) 2 Now that we have the free energy we may consider it as a functional of both h(r) and G0 (r) and take its variation with respect to these functions. We find, Z i o n h 1 exc ρ dr c(r) − h(r) + h2 (r)/2 + ln g(r)eβφ(r) − G0 (r) δh(r) − β δf = 2 Z 1 ρ dr[g(r) − g0 (r)]δG0 (r) . (5.iv:12) 2

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS IV. IMPROVING THE CLOSURES 67

It follows that the free energy is minimized when both the RHNC/H2 closure (equation (5.iv:4)) is satisfied and when the following constraint Z dr[g(r) − g0 (r)]δG0 (r) = 0 , (5.iv:13) is fulfilled. Taking the second functional derivative of f exc with respect to h(r) we find that also this free energy is a strictly convex functional of the total correlation function. This property was lacking in the RHNC theory and constitutes the main feature of the RHNC/H2 closure. As already stressed in section ii.1 it ensures that if a solution to the integral equation exists it has to be unique. The constraint, as for RHNC, gives a certain thermodynamic consistency to the theory (see [60]). If we choose a reference potential φ0 (r) = φ0 (r; σ, ǫ) which depends on some length and energy parameters σ and ǫ, the optimum values of the parameters that minimize the free energy can be determined by the constraint (5.iv:13) which becomes Z ∂G0 (r) =0 , (5.iv:14) dr[g(r) − g0 (r)] ∂σ and Z

dr[g(r) − g0 (r)]

∂G0 (r) =0 , ∂ǫ

(5.iv:15)

However, neither the hard-sphere pseudo bridge functions nor some empirical attempt to model the unknown function via a Yukawa function provided useful results. iv.1.1

Results from the RHNC/H2 approximation

For the Lennard-Jones fluid near its triple point (ρ∗ = 0.85 and T ∗ = 0.719) we tried to mimic the G function with a Yukawian. We chose G0 (r) = −A

e−r/λ r

,

(5.iv:16)

where A and λ are two positive constants. Setting λ equal to the first minimum of the pair distribution function obtained from a molecular dynamics simulation [74] (λ ≃ 1.5), we varied A to fit the excess internal energy obtained in the simulation [82] (U exc /(N ǫ) = −6.12). The resulting value for A was around 124. The values of the pressure and of the bulk modulus did not match with the ones of the simulation and the pair distribution function had a lower first peak and the successive peaks shifted forward respect to the g(r) of the simulation as is shown in figure 5.9.

iv.2

Optimized HNC/H3 approximation

For γ = 0 HNC/H3 reduces to HNC/H2. For γ > 0 the first peak of the pair distribution function is dumped respect to the one of the pair distribution function calculated with HNC/H2. For γ < 0 the first peak increases giving in general a better fit to the simulation data. In figure 5.10 we compare the pair distribution function of the Lennard-Jones fluid near its triple point, calculated with a molecular dynamic simulation [74], the HNC/H2 approximation, the approximation HNC/H3 with γ = −0.203 (at lower values of γ Zerah’ s algorithm would fail

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS IV. IMPROVING THE CLOSURES 68

3

MD RHNC/H2

g(x)

2

1

0

0

1

2

3

x=r/σ

Figure 5.9: Comparison of the pair distribution function computed from the RHNC/H2 with a yukawian G function and from a molecular dynamics (MD) simulation, for a LennardJones fluid at ρ∗ = 0.85 and T ∗ = 0.719. 3 MD HNC/H2 HNC/H3 (γ=−0.1) HNC/H3 (γ=−0.203)

g(x)

2

1

0 0.7

1.7

x=r/σ Figure 5.10: Comparison of the pair distribution function of a Lennard-Jones fluid at ρ∗ = 0.85 and T ∗ = 0.719 computed from the molecular dynamic (MD) simulation of Verlet, the HNC/H2 approximation, and the HNC/H3 approximation. For HNC/H3 we present results obtained setting γ = −0.1 (when the generating functional of the approximation is still strictly convex) and γ = −0.203 (which gives the best fit possible to the simulation data but does not ensure the strict convexity of the generating functional).

to converge), and the approximation HNC/H3 with γ = −0.1 (when the generating functional of HNC/H3 is still strictly convex). As we can see HNC/H3 fits the simulation data better than

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS IV. IMPROVING THE CLOSURES 69

HNC/H2 even if the first peak is still slightly displaced to the left of the simulation data, a well known problem of the HNC approximation [59]. The best results are given by HNC/H3 with γ = −0.203. Note that the HNC/H3 generating functional at this value of γ is not strictly convex (strict convexity is lost for γ . −1/9). The first peak of the static structure factor is at kσ ≃ 6.75 and has a magnitude of 2.41, a quite low value for a liquid near the triple point. We have calculated the pressure and the internal energy. We found βP/ρ ≃ 3.87 and U exc /(N ǫ) ≃ −5.72 (very close to the HNC results βP/ρ ≃ 3.12 and U exc /(N ǫ) ≃ −5.87) to be compared with the simulation results [82] 0.36 and −6.12 respectively. The bulk moduli are Bc ≃ 11.74 and Bp ≃ 36.61 which shows that at the chosen value of γ we do not have the thermodynamic consistency virial-compressibility and we do not improve on HNC inconsistency (using HNC we find Bc ≃ 7.09 and Bp ≃ 32.72).

iv.3

Functionals of γ

Often in the numerical solution of the OZ + closure integral equation use is made of the auxiliary function γ(r) = h(r) − c(r). Suppose that the closure relation can be written as ρ2 c(r) = −Ψ{γ(r)}

,

(5.iv:17)

where Ψ is a function of a function. We want to translate the integral equation into a variational principle involving functionals of γ(r). Then we introduce a closure functional Fcl [γ] such that δFcl [γ] = Ψ{γ(r)} δγ(r)

,

(5.iv:18)

and an OZ functional FOZ,c [γ] such that, when c(r) and γ(r) satisfy the OZ equation, we have δFOZ,c [γ] = ρ2 c(r) . δγ(r)

(5.iv:19)

Then when both the closure and the OZ relations are satisfied, the functional F = Fcl + FOZ,c is stationary with respect to variations of γ(r), i.e. δF[γ] =0 . δγ(r)

(5.iv:20)

This is the variational principle sought. We want now find FOZ,c . The OZ equation in k space is ρˆ c2 (k) + ρˆ γ (k)ˆ c(k) − γˆ(k) = 0 . When we solve it for cˆ we find two solutions p ˆ± Γ ˆ 2 + 4Γ ˆ −Γ cˆ = 2ρ

(5.iv:21)

,

(5.iv:22)

ˆ where Γ(k) = ρˆ γ (k) is always positive since ˆ = ρ2 ˆhˆ Γ c = ρ2

ˆ2 h ˆ 1 + ρh

= ρ2

ˆ2 h S(k)

,

(5.iv:23)

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS IV. IMPROVING THE CLOSURES 70

S(k) being the liquid static structure factor which is positive definite for all k. Since cˆ(k) is a function which oscillates around 0, where cˆ is negative we have to choose the solution with the minus sign, where it is positive the one with the plus sign. In particular if the isothermal compressibility of the liquid χT , is smaller than the one of the ideal gas χ0T , we have that   χ0T 1 0 and -1 when h(k) ≤ 0. Note that since hˆ we must have sh = sc . The second functional derivative of FOZ,h is p Z ˆ 2 + 4Γ ˆ + sh (k)(2 + Γ) ˆ δ2 FOZ,h [γ] dk −ik·(r+r′ ) Γ p , (5.iv:33) = e δΓ(r)δΓ(r ′ ) (2π)3 ˆ 2 + 4Γ ˆ 2 Γ

which shows that due to the presence of the sign sh the functional FOZ,h is neither convex nor concave. Thus, any check of the convexity properties of generating functionals of the γ(r) function should be done on the full functional.

v

Conclusions

In this chapter we have analyzed the relations between generating functionals, thermodynamic consistency and uniqueness of the solution of the integral equations of liquid state theory. We think that the requirement of deriving from a free energy and the uniqueness of the solution are two important ingredients to enforce in the quest for better closures. The former requirement is of course crucial to get virial-energy consistency. But it is also important to get integral equations able to provide a closed formula for the chemical potential without additional approximations. This last issue looks highly desirable for applications of IET to the determination of phase diagrams. The latter is certainly a useful constraint from the numerical point of view but it is also a very strong condition, probably able to avoid some non physical behavior in the coexistence region, although this point would deserve further investigation. Most of the existing closures fail to satisfy the condition of uniqueness of the solution. Among them, only the Optimized Random Phase Approximation by Andersen and Chandler [71, 83] satisfies both constraints although they were not used in the original derivation of the approximation. One obvious question is whether the enforcement of these constraints automatically results in improved closures. In this work, we have started an exploration of the capabilities of the combined requirement of consistency and uniqueness, starting with simple modifications to the HNC closure, corresponding to the addition of a square and a cubic power of h(r) in the HNC functional. We found a couple of approximations (HNC/H2 and HNC/H3), which have built in the virial-free energy thermodynamic consistency and have a unique solution. We numerically tested these closures on inverse power and the Lennard-Jones fluid. From the tests on the inverse power potential fluids one can see that the HNC/H2 approximation is comparable to HNC for the thermodynamic quantities and performs worst than RY and even HNC for structural properties. The tests on the Lennard-Jones fluid revealed as this approximation does not suffer from the presence of a termination line (present in HNC and almost all the existing closures). This allowed us to follow isotherms from the low density to the high density region and this behavior would be very useful in the study of the phase coexistence. However, the thermodynamic results show only a marginal improvement on HNC and the structure is definitely worse. Our trials to improve HNC/H2 in the same spirit of the modified HNC approaches did not succeed. We feel that the main reason is in the difficulty of modeling the real bridge functions through a polynomial in the function h(r). In this respect, approaches based on generating functionals depending on the indirect correlation function γ(r) look more promising but we have not tried them yet. Much better results for the structure are found with HNC/H3 as is shown in figure 5.10. However, probably for the same reasons just discussed, one has to renounce to have an approximation with a strictly convex generating functional depending on h(r). The thermodynamics

CHAPTER 5. GENERATING FUNCTIONALS, CONSISTENCY, AND UNIQUENESS IN THE INTEGRAL EQUATION THEORY OF LIQUIDS V. CONCLUSIONS 72

reproduced by HNC/H3 is not yet satisfactory: due to the slight left shift of the main peak of the g(r) the calculated pressure misses the simulation result. Nonetheless the presence of the free parameter γ in HNC/H3 leaves open the possibility of imposing the thermodynamic consistency virial-compressibility. If the value of the parameter needed to have the consistency is bigger than −1/[3 sup g(r)] then we would have an approximation which is completely thermodynamically consistent and have a unique solution. This strategy may eventually lead to discover that the price we have to pay to have a completely thermodynamically consistent approximation is the loss of strict convexity of the generating functional.

Chapter 6

Stability of the iterative solutions of integral equations as one phase freezing criterion A recently proposed connection between the threshold for the stability of the iterative solution of integral equations for the pair correlation functions of a classical fluid and the structural instability of the corresponding real fluid is carefully analyzed. Direct calculation of the Lyapunov exponent of the standard iterative solution of HNC and PY integral equations for the 1D hard rods fluid shows the same behavior observed in 3D systems. Since no phase transition is allowed in such 1D system, our analysis shows that the proposed one phase criterion, at least in this case, fails. We argue that the observed proximity between the numerical and the structural instability in 3D originates from the enhanced structure present in the fluid but, in view of the arbitrary dependence on the iteration scheme, it seems uneasy to relate the numerical stability analysis to a robust one-phase criterion for predicting a thermodynamic phase transition.

i

Introduction

When studying the structure and thermodynamics of classical fluids one is often faced with the task of solving the nonlinear integral equation which stems out of the combination of the Ornstein-Zernike equation and an approximate relation between pair potential and correlation functions (the closure) [12]. Integral equations can be generally written in the form γ(r) = Aγ(r)

,

(6.i:1)

where γ(r) ∈ S may be the total correlation function h(r), the direct correlation function c(r), or a combination of the two, S is a set of a metric space of functions, and A : S → S is a non linear operator mapping S into itself. Numerical analysis of integral equations suggests the use of the following combination γ(r) = h(r) − c(r)

,

(6.i:2)

since γ is a much smoother function than h or c, especially in the core region. It has been pointed out by Malescio et. al. [84–86] that, amongst the different numerical schemes that one may choose to solve (6.i:1), the simple iterative scheme of Picard plays a special 73

CHAPTER 6. STABILITY OF THE ITERATIVE SOLUTIONS OF INTEGRAL EQUATIONS AS ONE PHASE FREEZING CRITERION I. INTRODUCTION 74

role. Picard scheme consists in generating successive approximations to the solution through the relationship γn+1 = Aγn

,

(6.i:3)

starting from some initial value γ0 . If the sequence of successive approximations {γn } converges toward a value γ ⋆ , then γ ⋆ is a fixed point for the operator A, i.e. it is a solution of Eq. (6.i:1), γ ⋆ = Aγ ⋆ . Banach’ s fixed point theorem (see chapter 1 in [87] especially theorem 1.A) states that, given an operator A : S → S, where S is a closed nonempty set in a complete metric space, the simple iteration (6.i:3) may converge toward the only fixed point in S (A is k contractive) or it may not converge (A is non expansive). So the simple iterative method can be used to signal a fundamental change in the properties of the underlying operator. The operator A will in general depend on the thermodynamic state of the fluid. In order to determine the properties of the operator at a given state we can proceed as follows. First, we find the fixed point γ ⋆ using a numerical scheme (more refined then Picard’ s) capable of converging in the high density region. Next, we perturb the fixed point with an arbitrary initial perturbation δ0 (r) so that ∂A ⋆ ⋆ δ0 = γ ⋆ + M δ0 , (6.i:4) A(γ + δ0 ) ≃ Aγ + ∂γ γ ⋆

where we have introduced the Floquet matrix M . Now δ1 = M δ0 may be considered as the new perturbation. We then generate the succession {δn } where δn = M δn−1

.

(6.i:5)

If the succession converges to zero then the operator A is k contractive, if it diverges the operator is non expansive. Malescio et. al. call {δn } fictitious dynamics and associate to the resulting fate of the initial perturbation the nature of the structural equilibrium of the fluid. If the succession converges to zero they say that the fluid is structurally stable and structurally unstable otherwise. We will call ρinst the density where the transition between a structurally stable and unstable fluid occurs. Following Malescio et. al. it is possible to define a measure for the structural stability of the system as follows. We define Si =

||M δi (r)|| ||δi (r)||

,

(6.i:6)

qP

N 2 where ||f (r)|| = i=1 f (ri ) is the norm of a function f defined over a mesh of N points. We assume that the norm of the perturbation depends exponentially on the number of iterations

||δn || = ||δ0 ||2λn

,

(6.i:7)

where λ is the Lyapunov exponent related to the fictitious dynamics. Then one can write the average exponential stretching of initially nearby points as ! n−1 Y 1 λ = lim log2 . (6.i:8) Si n→∞ n i=0

Malescio et. al. have calculated the dependence of λ on the density for various simple three dimensional liquids (and various closures): hard spheres [84], Yukawa, inverse power and

CHAPTER 6. STABILITY OF THE ITERATIVE SOLUTIONS OF INTEGRAL EQUATIONS AS ONE PHASE FREEZING CRITERION II. TECHNICAL DETAILS 75

Lennard-Jones potentials [85]. For all these systems they found that λ increases with the density and the density at which λ becomes positive, ρinst , falls close to the freezing density ρf of the fluid system. This occurrence lead them to propose this kind of analysis as a one-phase criterion to predict the freezing transition of a dense fluid and to estimate ρf . However, we think that there are some practical and conceptual difficulties with such one-phase criterion. First of all, it does not depend only on the closure adopted but also on the kind of algorithm used to solve the integral equation. Indeed, different algorithms give different ρinst and Malescio et. al. choose to use as instability threshold for their criterion the one obtained using Picard algorithm, thus giving to it a special status. However, it is hard to understand why the particular algorithm adopted in the solution of the integral equation should be directly related to a phase boundary. Moreover, one would expect that the estimate of ρinst would improve in connection with improved closures. This is not the case, at least in the one component hard sphere fluid. Even a more serious doubt about the validity of the proposed criterion comes from its behavior in one dimensional systems. In this chapter we present the same Lyapunov exponent analysis on a system of hard rods in one dimension treated using either the Percus-Yevick (PY) or the hypernetted chain (HNC) approximations. What we find is that the Lyapunov exponent as a function of density has the same behavior as that for the three dimensional system (hard spheres): it becomes positive beyond a certain ρinst . Since it is known [10] that a one dimensional fluid of hard rods does not have a phase transition, our result sheds some doubts on the validity of the proposed criterion.

ii

Technical details

As numerical scheme to calculate the fixed point we used Zerah’ s algorithm [75] for the three dimensional systems and a modified iterative method for the hard rods in one dimension. In the modified iterative method input and output are mixed at each iteration γn+1 = Amix γn = αAγn + (1 − α)γn

,

(6.ii:1)

where α is a real parameter 0 < α < 1. Note that while for a non expansive operator A the Picard iterative method (6.i:3) needs not converge, one can prove convergence results on an Hilbert space for the modified iterative method with fixed α (see proposition 10.16 in [87]). In all the computations we used a uniform grid of N = 1024 points with a spacing δr = 0.025. Generally, we observed a marginal increase of ρinst by lowering N . A method to find a Lyapunov exponent, equivalent but more accurate than the one of Malescio et. al. (6.i:8), goes through the diagonalization of the Floquet matrix. Note that in general this matrix is non symmetric, thus yielding complex eigenvalues. A Lyapunov exponent can then be defined as [88] ′



λ = log max i

q

eri2

+ ei2i



,

(6.ii:2)

where eri and eii are respectively the real and imaginary part of the i-th eigenvalue. In our numerical computations we always used recipe (6.ii:2) to calculate the Lyapunov exponents since it is explicitly independent from the choice of an initial perturbation. We constructed the Floquet matrix in the following way [89]. In a Picard iteration we start from γ(r) we calculate c(r) from the closure approximation, we calculate its Fourier transform

CHAPTER 6. STABILITY OF THE ITERATIVE SOLUTIONS OF INTEGRAL EQUATIONS AS ONE PHASE FREEZING CRITERION III. NUMERICAL RESULTS 76

c˜(k), we calculate γ˜ (k) from the OZ equation, and finally we anti transform γ˜ to get γ ′ (r). For example for a three dimensional system a PY iteration in discrete form can be written as follows   ci = (1 + γi ) e−βφi − 1 , (6.ii:3) c˜j

=

N −1 4πδr X ri sin(kj ri )ci kj

,

(6.ii:4)

i=1

γ˜j

= ρ˜ c2j /(1 − ρ˜ cj )

γi′ =

N −1 δk X kj sin(kj ri )˜ γj 2π 2 ri

(6.ii:5) ,

(6.ii:6)

j=1

where ri = iδr are the N mesh points in r space, kj = jδk are the N mesh points in k space, with δk = π/(N δr), ci = c(ri ), γi = γ(ri ), c˜j = c˜(kj ), γ˜j = γ˜ (kj ), and φi = φ(ri ) is the interparticle potential calculated on the grid points. The Floquet matrix will then be Mij

= =

N −1

X ∂γ ′ ∂˜ cm ∂cj ∂γi′ i γm ∂˜ = ∂γj ∂˜ γm ∂˜ cm ∂cj ∂γj m=1    δrδk rj e−βφj − 1 (Di−j − Di+j ) , π ri

(6.ii:7)

where Dl =

N −1 X m=1

"

2ρ˜ cm cos(km rl ) + 1 − ρ˜ cm



ρ˜ cm 1 − ρ˜ cm

2 #

.

(6.ii:8)

The HNC case can be obtained replacing in (6.ii:7) [exp(−βφj )− 1] with [exp(−βφj + γj )− 1] and p the Martynov p Sarkisov (MS) [62] closure can be implemented replacing it with [exp(−βφj + 1 + 2γj − 1)/ 1 + 2γj − 1]. To derive the expression for the Floquet matrix valid for the one dimensional system and consistent with a trapezoidal discretization of the integrals, we need to replace (6.ii:4) and (6.ii:6) with ! N −1 X 1 , (6.ii:9) cos(kj ri )ci + c0 c˜j = 2δr 2 i=1 ! N −1 1 δk X ′ cos(kj ri )˜ γj + γ˜0 . (6.ii:10) γi = π 2 i=1

iii iii.1

Numerical results Three dimensional systems

We have calculated the Lyapunov exponent (6.ii:2) as a function of the density for a three dimensional hard spheres fluid and a Lennard-Jones fluid at a reduced temperature T ∗ = 2.74, using both the PY and the HNC closures. For the hard spheres we have also used the MS closure. The results of the calculations are shown in figure 6.1 and 6.2 respectively. In good agreement with the results of Malescio et. al. [84, 85], we can see how the slope of the curves starts high at low densities and decreases rapidly with ρ. At high densities the Lyapunov exponent becomes

CHAPTER 6. STABILITY OF THE ITERATIVE SOLUTIONS OF INTEGRAL EQUATIONS AS ONE PHASE FREEZING CRITERION III. NUMERICAL RESULTS 77

0.6

Lyapunov exponent

PY 0.4 HNC MS 0.2 0 -0.2

0.5 0.45 0.4

-0.4

0.35 0.3

-0.6

0.25 0.2 0.15

-0.8

0.1 0.05 0

-1 -1.2 0.05

-0.05 0.42

0.1

0.15

0.2

0.25

0.3 η

0.44

0.46

0.35

0.48

0.4

0.5

0.52

0.45

0.54

0.5

0.55

Figure 6.1: We show the Lyapunov exponent as a function of the packing fraction η = ρπd3 /6 for a system of three dimensional hard spheres of diameter d as calculated using the PY, the HNC, and the MS closures.

zero at ρinst . Before reaching the instability threshold the curves show a rapid change in their slope at ρc < ρinst . The insets show a magnification of the region around ρc from which we are lead to conclude that, within the numerical accuracy of the calculations, the slope of the curves dλ′ /dρ undergoes a jump at ρc . For the hard spheres fluid we found ηinst = ρinst πd3 /6 of about 0.445 in the PY approximation, around 0.461 in the HNC approximation, and around 0.543 in the MS approximation. For the Lennard-Jones fluid our results were indistinguishable from those of Malescio et. al. [85]. We found a reduced instability density ρ∗inst around 1.09 in the PY approximation and around 1.06 in the HNC approximation.

iii.2

The one dimensional hard spheres

We have calculated the Lyapunov exponent (6.ii:2) as a function of the density for a one dimensional hard spheres fluid using both PY and HNC closures. The results of the calculation are shown in figure 6.3. The curves show the same qualitative behavior as the ones for the three dimensional fluids.

iii.3

The Floquet matrix

In figure 6.4 we show a surface plot of the non-zero region of the Floquet matrix (6.ii:7) as calculated for the three dimensional hard spheres fluid in HNC approximation at η = 0.3. As we approach the critical density the peaks near i = 1 accentuate themselves. This suggests that the trace of the transition of operator A from k contractive to nonexpansive can be found in a local change of the Floquet matrix.

CHAPTER 6. STABILITY OF THE ITERATIVE SOLUTIONS OF INTEGRAL EQUATIONS AS ONE PHASE FREEZING CRITERION IV. CONCLUSIONS 78

1

PY HNC

Lyapunov exponent

0.5 0 0.06

-0.5

0.05 0.04 0.03 0.02

-1

0.01 0 -0.01 -0.02

-1.5

-0.03 -0.04 1

1.02

1.04

1.06

1.08

1.1

-2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

*

ρ

Figure 6.2: We show the Lyapunov exponent as a function of the reduced density for a Lennard-Jones fluid at a reduced temperature T ∗ = 2.74 as calculated using the PY and the HNC closures.

iv

Conclusions

The fictitious dynamics associated to the iterative solution of an integral equation can signal the transition of the map of the integral equation from k contractive to non expansive. If the Lyapunov exponent is negative the map is k contractive, if it is positive the map is non expansive. Since it is possible to modify in an arbitrary way the fictitious dynamics keeping the same fixed point, it is difficult to understand a deep direct connection between the stability properties of the map and a one-phase criterion for a thermodynamic transition. Admittedly the correlations shown by Malescio et al. are striking. We calculated the Lyapunov exponent as a function of the density for various fluids (hard spheres in one and three dimensions and three dimensional Lennard-Jones fluid) both in the HNC and PY approximations. For the three dimensional fluids the instability density falls close to the freezing density ρf . For example, the Lennard-Jones fluid studied with HNC should undergo a freezing transition at ρ∗ ≃ 1.06 or at ρ∗ ≃ 1.09, if studied with PY , rather close to the freezing density ρ∗f ≃ 1.113. For hard spheres ρ∗inst is about 10% smaller than ρ∗f ∼ 0.948. The Hansen-Verlet “rule” states that a simple fluid freezes when the maximum of the structure factor is about 2.85 [9]. According to this rule the three dimensional hard spheres fluid studied with HNC should undergo a freezing transition at ρ ≃ 1.01 while when studied with PY the transition should be at ρ ≃ 0.936. The corresponding estimates obtained through ρ∗inst , 0.879 (HNC) and 0.850 (PY) are poorer and, more important, are not consistent with the well known better performance of PY in the case of hard spheres. In one dimension, a fluid of hard spheres (hard rods), cannot undergo a phase transition [10]. From Fig. 6.3 we see that the system still becomes structurally unstable. This can be explained by observing that the structural stability as defined by Malescio et. al. is a property of the map A and in particular of the algorithm used to get solution of the integral equation under study. As such, it is not directly related to the thermodynamic properties even at the approximate level

CHAPTER 6. STABILITY OF THE ITERATIVE SOLUTIONS OF INTEGRAL EQUATIONS AS ONE PHASE FREEZING CRITERION IV. CONCLUSIONS 79

1

PY HNC

Lyapunov exponent

0.5 0 0.35

-0.5

0.3 0.25 0.2

-1

0.15 0.1 0.05 0

-1.5

-0.05 -0.1 0.6

0.62

0.64

0.66

0.68

0.7

-2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

*

ρ

Figure 6.3: We show the Lyapunov exponent as a function of the reduced density for a one dimensional fluid of hard spheres as calculated using the PY and the HNC closures.

Mij 1 0.8 0.6 0.4 0.2 0 -0.2

25 20 15

0 5

10

10 i

j

5

15 20

0

Figure 6.4: We show a surface plot of the Floquet matrix (6.ii:7) calculated for three dimensional hard spheres in HNC approximation at η = 0.3. The matrix was generated using N = 256 grid points but only the region significantly different from zero is shown.

of the theory (there is no direct relation between the contractiveness properties of A and the thermodynamics). It looks more reasonable that the increase of the correlations would be the common origin of the numerical instability of the Picard iteration and, whenever it is possible, of thermodynamic phase transitions.

Chapter 7

Direct correlation functions of the Widom-Rowlinson model In this chapter we calculate, through Monte Carlo numerical simulations, the partial total and direct correlation functions of the three dimensional symmetric Widom-Rowlinson mixture. We find that the differences between the partial direct correlation functions from simulation and from the Percus-Yevick approximation (calculated analytically by Ahn and Lebowitz) are well fitted by Gaussians. We provide an analytical expression for the fit parameters as function of the density. We also present Monte Carlo simulation data for the direct correlation functions of a couple of non additive hard sphere systems to discuss the modification induced by finite like diameters.

i

Introduction

Fluid binary mixtures may exhibit the phenomenon of phase separation. The simplest system able to undergo a demixing phase transition is the model introduced by Widom and Rowlinson some years ago [90]. Consider a binary mixture of non-additive hard spheres. This is a fluid made of hard spheres of specie 1 of diameter R11 and number density ρ1 and hard spheres of specie 2 of diameter R22 and number density ρ2 , with a pair interaction potential between species i and j that can be written as follows vij (r) =



∞ r < Rij 0 r > Rij

,

(7.i:1)

where R12 = (R11 + R22 )/2 + α. The Widom-Rowlinson (WR) model is obtained choosing the diameters of the spheres equal to 0, R11 = R22 = 0 ,

(7.i:2)

so that there is no interaction between like spheres and there is a hard core repulsion of diameter α between unlike spheres. The symmetry of the system induces the symmetry of the unlike correlations [h12 (r) = h21 (r), c12 (r) = c21 (r)]. The WR model has been studied in the past by exact [91] and approximate [54, 92–94] methods and it has been shown that it exhibits a phase transition at high density. More recently, additional studies have appeared and theoretical predictions have been confirmed by Monte Carlo (MC) computer simulations [95–100] 80

CHAPTER 7. DIRECT CORRELATION FUNCTIONS OF THE WIDOM-ROWLINSON MODEL II. MONTE CARLO SIMULATION AND PY SOLUTION 81

In this chapter we will study the three dimensional symmetric Widom-Rowlinson mixture for which ρ1 = ρ2 = ρ/2, where ρ is the total number density of the fluid, and h11 (r) = h22 (r) ,

(7.i:3)

c11 (r) = c22 (r) .

(7.i:4)

Moreover we know from (7.i:1) that the partial pair correlation function gij = hij + 1 must obey gij (r) = 0

for r < Rij .

(7.i:5)

Our main goal is to focus on the direct correlation functions (dcf) of the WR model as a simplified prototype of non-additive hard spheres (NAHS) systems. The reasons to focus on the dcf’s is twofold: on the one hand, they are easier functions to model and fit. On the other hand, they play a central role in approximate theories like the Percus-Yevick approximation or mean spherical approximation (MSA) [12]. We hope that a better understanding of the dcf’s properties in the WR model, could help in developing accurate analytical theories for the NAHS systems. (M C)

(M C)

We calculate through Monte Carlo simulations the like g11 (r) and unlike g12 (r) pair distribution functions for a system large enough to allow a meaningful determination of the (M C) (M C) correspondent partial direct correlation functions c11 (r) and c12 (r), using the OrnsteinZernike equation [12]. We compare the results for the unlike direct correlation function with the results of the Percus-Yevick (PY) analytic solution found by Ahn and Lebowitz [54, 92]. In the same spirit as the work of Grundke and Henderson for a mixture of additive hard spheres [101], (M C) (P Y ) (M C) we propose a fit for the functions ∆c11 (r) = c11 (r) and ∆c12 (r) = c12 (r) − c12 (r). At the end of the chapter we also show the results from two Monte Carlo simulations on a mixture of non-additive hard spheres with equal diameter spheres R11 = R22 = R12 /2 and on one with different diameter spheres R11 = 0 and R22 = R12 to study the effect of non zero like diameters on the WR dcf’s.

ii

Monte Carlo simulation and PY solution

The Monte Carlo simulation was performed with a standard NVT Metropolis algorithm [14] using N = 4000 particles. Linked lists [14] have been used to reduce the computational cost. We generally used 5.2 × 108 Monte Carlo steps where one step corresponds to the attempt to move a single particle. The typical CPU time for each density was around 20 hours (runs at higher densities took longer than runs at smaller densities) on a Compaq AlphaServer 4100 5/533. We run the simulation of WR model at 6 different densities ρ¯ = ρα3 = 0.28748, 0.4, 0.45, 0.5, 0.575, and 0.65. Notice that the most recent computer simulation calculations [97, 98] give consistent estimates of the critical density around 0.75. Our data at the highest density (0.65) are consistent with a one phase system. The Monte Carlo simulation returned the gij (r) over a range not less than 9.175α for the densest system. In all the studied cases the pair distribution functions attained their asymptotic value well inside the maximum distance they were evaluated. Thus, it has been possible to obtain accurate Fourier transforms of the correlation functions [hij (k)]. To obtain the cij (r) we used

CHAPTER 7. DIRECT CORRELATION FUNCTIONS OF THE WIDOM-ROWLINSON MODEL II. MONTE CARLO SIMULATION AND PY SOLUTION 82

Ornstein-Zernike equation as follows c11 (k) = c12 (k) =

  h11 (k) 1 + ρ2 h11 (k) − ρ2 h212 (k)  2  2 1 + 2ρ h11 (k) − ρ2 h12 (k) h12 (k)  2  2 ρ 1 + 2 h11 (k) − ρ2 h12 (k)

(7.ii:1) (7.ii:2)

From the hij (k) and cij (k) we get the difference γij (k) = hij (k) − cij (k) which is the Fourier transform of a continuous function in real space. So it is safe to transform back in real space [to get γij (r)]. Finally, the dcf’s are obtained from the differences hij (r) − γij (r). While for a system of non-additive hard sphere in three dimensions a closed form solution to the PY approximation is still lacking, Ahn and Lebowitz have found an analytic solution of this approximation for the WR model (in one and three dimensions). The PY approximation consists of the assumption that cij (r) does not extend beyond the range of the potential cij (r) = 0

for r > Rij .

(7.ii:3)

Combining this with the exact relation (7.i:5) and using the Ornstein-Zernike equation we are left with a set of equations for cij (r) and gij (r) which have been solved analytically by Ahn and Lebowitz. Their solution is parameterized by a parameter z0 . They introduce the following two functions of z0 (which can be written in terms of elliptic integrals of the first and third kind) Z ∞ dz p , (7.ii:4) I1 ≡ 3 z0 z z + 4z/z0 − 4 Z ∞ dz p , (7.ii:5) I2 ≡ 3 z + 4z/z0 − 4 z0

and define z0 in terms of the partial densities ρ1 and ρ2 as follows √ (I2 /2)3 η ≡ 2π ρ1 ρ2 = . cos I1

(7.ii:6)

They then define the following functions (note that in the last equality of equation (3.76) in [54] there is a misprint) s 1+Y 2 c¯12 (k) ≡ − √ 3 3 ρ1 ρ2 z0 Y + 4Y + 4 # " q Z ∞ dz 1 p , (7.ii:7) z03 Y 3 + 4Y + 4 × sin 2 (z + Y ) z03 z 3 + 4z − 4 1 ¯ 12 (k) ≡ c¯12 (k)[1 − ρ1 ρ2 c¯2 (k)]−1 , h (7.ii:8) 12

where Y ≡ (2k/I2 )2 . We also realized that some other misprint should be present in the Ahn and Lebowitz paper since we have found empirically that the PY solution (with k in units of α) should be given by c12 (k) = c¯12 (ks) ,

(7.ii:9)

CHAPTER 7. DIRECT CORRELATION FUNCTIONS OF THE WIDOM-ROWLINSON MODEL II. MONTE CARLO SIMULATION AND PY SOLUTION 83

where s is a scale parameter to be determined as follows ¯ 12 (r = 0)]1/3 . s = −[h

(7.ii:10)

Notice that for the symmetric case ρ1 = ρ2 = ρ/2 and η = πρ = 0.90316 . . . we find z0 = 1 and s = 1. In Figs. 7.1, 7.2, and 7.3 we show three cases corresponding to the extreme and one intermediate density. In the figures, we compare the MC simulation data with the PY solution for

Figure 7.1: Top panel: partial direct correlation functions obtained from the Monte Carlo (P Y ) simulation (points) with the c12 (r) obtained from the PY approximation (line) at a density 3 ρα = 0.28748. Bottom panel: partial pair distribution functions obtained from the Monte Carlo simulation compared with the ones obtained from the PY approximation at the same density. The open circles and the dashed line: the like correlation functions. Closed circles and the continuous line: the unlike correlation functions.

the partial pair distribution functions and the partial direct correlation functions. Our results for the partial pair distribution functions at ρα3 = 0.65 are in good agreement with the ones of Shew and Yethiraj [97]. The figures show how the like correlation functions obtained from the PY approximation are the ones that differ most from the MC simulation data. The difference is more marked in a neighborhood of r = 0 and becomes more pronounced as the density increases. In Fig. 7.4 we also show the results for the partial direct correlation function in k space at a density ρα3 = 0.28748.

CHAPTER 7. DIRECT CORRELATION FUNCTIONS OF THE WIDOM-ROWLINSON MODEL III. FIT OF THE DATA 84

Figure 7.2: Same as in Fig. 7.1 at a density ρα3 = 0.4.

iii

Fit of the data (M C)

From the simulations we found that c12 (r) < 8 × 10−3 for r > α at all the densities studied. This allows us to say that ∆c12 (r) ≃ 0 for r > α. Moreover we found that both ∆c12 (r) for r < α, and ∆c11 (r) are very well fitted by Gaussians ∆c11 (r) ≃ b11 exp[−a11 (r + d11 )2 ]

∆c12 (r) ≃ b12 exp[−a12 r 2 ]

for all r > 0,

(7.iii:1)

for 0 < r < α,

(7.iii:2)

In Figs. 7.5 and 7.6 we show the behaviors of the parameters of the best fit (7.iii:1) and (7.iii:2), with density. In order to check the quality of the fit, we did not use the data at ρ¯ = 0.45 in the determination of the parameters. The points for a12 and b12 are well fitted by a straight line or a parabola. As shown in Fig. 7.5 the best parabolae are a12 (¯ ρ) = 0.839 + 0.096¯ ρ − 1.287¯ ρ2 , 2

b12 (¯ ρ) = −0.155 + 0.759¯ ρ − 0.159¯ ρ

(7.iii:3) .

(7.iii:4)

Fig. 7.6 shows how the parameters for ∆c11 (r) are much more scattered and hard to fit. The quartic polynomial going through the five points, for each coefficient, are a11 (¯ ρ) = −55.25 + 504.8¯ ρ − 1659.¯ ρ2 + 2364.¯ ρ3 − 1236.¯ ρ4 ,

(7.iii:5)

2

3

4

,

(7.iii:6)

2

3

4

,

(7.iii:7)

b11 (¯ ρ) = 171.4 − 1556.¯ ρ + 5166.¯ ρ − 7421.¯ ρ + 3906.¯ ρ

d11 (¯ ρ) = 128.9 − 1144.¯ ρ + 3747.¯ ρ − 5328.¯ ρ + 2782.¯ ρ

CHAPTER 7. DIRECT CORRELATION FUNCTIONS OF THE WIDOM-ROWLINSON MODEL III. FIT OF THE DATA 85

Figure 7.3: Same as in Fig. 7.1 at a density ρα3 = 0.65.

1.0

c11(k), c12(k)

0.0 -1.0 -2.0

(MC)

c11

(MC)

-3.0 -4.0 0.0

c12 (PY) c12 5.0

10.0

15.0 kα

20.0

25.0

30.0

Figure 7.4: We compare the partial direct correlation function in k space obtained from the Monte Carlo simulation (with superscript MC) with the one obtained from the PY approximation (with superscript PY) at a density ρα3 = 0.28748.

The difficulty in finding a good fit for these parameters may be twofold: first we are fitting ∆c11 (r) with a three (instead of two) parameters curve and second the partial pair distribution

CHAPTER 7. DIRECT CORRELATION FUNCTIONS OF THE WIDOM-ROWLINSON MODEL IV. FROM WR TO NON ADDITIVE HARD SPHERES 86

Figure 7.5: We plot, for five different values of the density, the parameters a12 (diagonal crosses) and b12 (starred crosses) of the best Gaussian fit (7.iii:2) to ∆c12 (r) for r < α, and fit them with parabolae (lines). The parameters at ρα3 = 0.45 where not used for the parabolic fit and give an indication of the quality of the fit.

functions obtained from the Monte Carlo simulation are less accurate in a neighborhood of the origin (due to the reduced statistics there). This inaccuracy is amplified in the process of finding the partial direct correlation functions. Such inaccuracy will not affect significantly ∆c12 (r) which has a derivative very close or equal to zero near the origin, but it will significantly affect ∆c11 (r) which is very steep near the origin. In order to estimate the quality of the fit we have used the simulation data at ρ¯ = 0.45. From Fig. 7.5 we can see how the parabolic fit is a very good one. In Fig. 7.6 the point at ρ¯ = 0.45 gives an indication of the accuracy of the quartic fit. We have also compared the pair distribution and direct correlation functions obtained from the fit with those from MC: both the like and unlike distribution functions are well reproduced while there is a visible discrepancy in the dcf from the origin up to r = 0.5α. However we expect that moving on the high density or low density regions (where the quartic polynomial becomes more steep) the quality of the fit will get worst. In particular the predicted negative values for a11 , in those regions, are completely unphysical and the fit should not be used to extrapolate beyond the range 0.28 < ρ¯ < 0.65.

iv

From WR to non additive hard spheres

In order to see how the structure, and in particular the dcf’s of the Widom-Rowlinson model change as one switches on the spheres diameters we have made two additional Monte Carlo 3 and R simulations. In the first one we chose ρ1 = ρ2 = 0.65/R12 11 = R22 = R12 /2. The resulting partial pair distribution functions and partial direct correlation functions are shown in Fig. 7.7. From a comparison with Fig. 7.3 we see how in this case the switching on of the like diameters causes both c12 (r) for r < R12 and g12 (r) for r > R12 to approach r = R12 with a slope close to zero. 3 and R In the second simulation we chose ρ1 = ρ2 = 0.65/R12 11 = 0, R22 = R12 . The

CHAPTER 7. DIRECT CORRELATION FUNCTIONS OF THE WIDOM-ROWLINSON MODEL V. CONCLUSIONS 87

Figure 7.6: We plot, for five different values of the density, the parameters a11 , b11 and d11 (stars) of the best Gaussian fit (7.iii:1) to ∆c11 (r), and draw the quartic polynomial (lines) through them. The parameters at ρα3 = 0.45 where not used to determine the quartic polynomial and give an indication of the quality of the fit.

resulting partial pair distribution functions and partial direct correlation functions are shown in Fig. 7.8. From a comparison with Fig. 7.3 we see how in this case the switching on of the like diameters causes both g11 (0) and c11 (0) to increase, and c12 (r) to lose the nearly zero slope at r = 0. As in the previous case g12 (r) for r > R12 approaches r = R12 with a slope close to zero. The like 22 correlation functions for r > R12 vary over a range comparable to the one over which vary the like 11 correlation functions of the WR model. For both these cases there is no analytic solution of the PY approximation available and a better understanding of the behavior of the direct correlation functions may help in finding approximate expressions [53].

v

Conclusions

In this chapter we have evaluated the direct correlation functions of a Widom-Rowlinson mixture at different densities through Monte Carlo simulation and we have studied the possibility of fitting the difference between MC data and the PY dcf’s. We found a very good parameterization of c12 (r) for r < α [see equations (7.iii:2) and (7.iii:3)-(7.iii:4)] and a poorer one for c11 (r) [see equations (7.iii:1) and (7.iii:5)-(7.iii:7)]. The difficulty in this last case probably arises from the necessity of using three parameters [instead of just two needed for parameterizing c12 (r)], although it cannot be completely excluded some effect of the decreasing precision of the

CHAPTER 7. DIRECT CORRELATION FUNCTIONS OF THE WIDOM-ROWLINSON MODEL V. CONCLUSIONS 88

3 Figure 7.7: Monte Carlo results at a density ρ = ρ1 = ρ2 = 0.65/R12 for the partial direct correlation function (on top) and the partial pair distribution function (below) of a mixture of non additive hard spheres with R11 = R22 = R12 /2. The open circles denote the like correlation functions. The closed circles denote the unlike correlation functions.

simulation data near the origin. In the last part of the chapter we have illustrated with additional Monte Carlo data the changes induced in the WR dcf’s by a finite size of the excluded volume of like correlations. These results are meant to provide a guide in the search of a manageable, simple analytical parameterization of the structure of mixtures of non additive hard spheres which is still not available although highly desirable.

CHAPTER 7. DIRECT CORRELATION FUNCTIONS OF THE WIDOM-ROWLINSON MODEL V. CONCLUSIONS 89

3 Figure 7.8: Monte Carlo results at a density ρ = ρ1 = ρ2 = 0.65/R12 for the partial direct correlation function (on top) and the partial pair distribution function (below) of a mixture of non additive hard spheres with R11 = 0 and R22 = R12 . The open circles denote the like 11 correlation functions. The open triangles denote the like 22 correlation functions. The closed circles denote the unlike correlation functions.

Chapter 8

Pressures for a One-Component Plasma on a pseudosphere The classical (i.e. non-quantum) equilibrium statistical mechanics of a two dimensional one component plasma (a system of charged point-particles embedded in a neutralizing background) living on a pseudosphere (an infinite surface of constant negative curvature) is considered. In the case of a flat space, it is known that, for a one-component plasma, there are several reasonable definitions of the pressure, and that some of them are not equivalent to each other. In the present chapter, this problem is revisited in the case of a pseudosphere. General relations between the different pressures are given. At one special temperature, the model is exactly solvable in the grand canonical ensemble. The grand potential and the one-body density are calculated in a disk, and the thermodynamic limit is investigated. The general relations between the different pressures are checked on the solvable model. We study the ergodicity of a classical finite dynamical system moving in a connected and compact domain of a pseudosphere. In particular we derive a condition on its potential and kinetic energy sufficient for the system to be ergodic. We discuss the existence and uniqueness of the grand canonical Gibbs distribution as the limit distribution for the system with an infinite number of particles. We consider the special case of the one component Coulomb plasma on a pseudosphere and prove the ergodicity of the system obtained by switching off the mutual interaction amongst the particles. We also derive an equation of state for the one component Coulomb plasma on a pseudosphere using a field theoretical argument, and argue that the same equation of state holds for the same system on a large class of Riemannian surfaces.

i

Introduction

This chapter is divided into four parts: in the first part (from subsection i.1 to subsection iii.1) we study a two dimensional one component Coulomb plasma (2D OCP) as a dynamical system moving in a connected and compact domain of a pseudosphere. In the second part (from subsection iii.2 to subsection iii.6) we compare four different definitions for the pressure of this system and derive some general sum rules. In the third part (section iv) we study a swarm of free particles moving on a pseudosphere and coupled to a massive scalar field, a Yukawa field. This is a field theoretical description of a system of particles interacting through a screened Coulomb potential of the Debye-Yukawa form. When the Yukawa interaction tends to the Coulomb interaction the system reduces to a one component Coulomb plasma. In the last part (section 90

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE I. INTRODUCTION 91

v) we solve exactly the 2D OCP on a finite disk of the pseudosphere, in the grand canonical ensemble, at a special temperature. The thermodynamic limit is also investigated and the sum rules among the different definitions for the pressure are checked. Coulomb systems such as plasmas or electrolytes are made of charged particles interacting through Coulomb’ s law. The simplest model of a Coulomb system is the one component plasma, also called Jellium: an assembly of identical point charges, embedded in a neutralizing uniform background of the opposite sign. Here we consider the classical (i.e. non quantum) equilibrium statistical mechanics of the one component plasma. Although many features of more realistic systems are correctly reproduced, this model has the peculiarity that there are several reasonable definitions of its pressure, and some of these definitions are not equivalent to each other [102, 103]. The two-dimensional version of the one component plasma has been much studied. Provided that the Coulomb potential due to a point-charge is defined as the solution of the Poisson equation in a two-dimensional world (i.e. is a logarithmic function − ln r of the distance r to that point-charge), the two dimensional one component plasma mimics many generic properties of the three-dimensional Coulomb systems. Of course, this toy logarithmic model does not describe real charged particles, such as electrons, confined on a surface, which nevertheless interact through the three dimensional Coulomb potential 1/r (with the electric field lines coming out of the surface). One motivation for studying the two dimensional one component plasma is that its equilibrium statistical mechanics is exactly solvable at one special temperature: both the thermodynamical quantities and the correlation functions are available [104]. How the properties of a system are affected by the curvature of the space in which the system lives is a question which arises in general relativity. This is an incentive for studying simple models. Thus, the problem of a two dimensional one component plasma on a pseudosphere has been considered [105]. For this two dimensional one component plasma on a pseudosphere, the problem of studying and comparing the different possible definitions of the pressure also arises. This is the subject of the present chapter. A pseudosphere is a non compact Riemannian surface of constant negative curvature. Unlike the sphere it has an infinite area and it is not embeddable in the three dimensional Euclidean space. The property of having an infinite area makes it interesting from the point of view of Statistical Physics because one can take the thermodynamic limit on it. Riemannian surfaces of negative curvature play a special role in the theory of dynamical systems [106]. Hadamard study of the geodesic flow of a point particle on a such surface [107] has been of great importance for the future development of ergodic theory and of modern chaos theory. In 1924 the mathematician Emil Artin [108] studied the dynamics of a free point particle of mass m on a pseudosphere closed at infinity by a reflective boundary (a billiard). Artin’ s billiard belongs to the class of the so called Anosov systems. All Anosov systems are ergodic and posses the mixing property [109]. Sinai [110] translated the problem of the Boltzmann-Gibbs gas into a study of the by now famous “Sinai’ s billiard”, which in turn could relate to Hadamard’ s model of 1898. Recently, smooth experimental versions of Sinai’ s billiard have been fabricated at semiconductor interfaces as arrays of nanometer potential wells and have opened the new field of mesoscopic physics [111]. The following important theorem holds for Anosov systems [112],[113]: Theorem i.1 Let M be a connected, compact, orientable analytic surface which serves as the configurational manifold of a dynamical system whose Hamiltonian is H = T + U . Let the dynamical system be closed and its total energy be h. Consider the manifold M defined by

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE I. INTRODUCTION 92

the Maupertuis Riemannian metric (ds)2 = 2(h − U )T (dt)2 on M . If the curvature of M is negative everywhere then the dynamical system is an Anosov system and in particular is ergodic on Mh = {h = H}. If the dynamical system is composed of N particles, the same conclusions hold, we need only require that the curvature be negative when we keep the coordinates of all the particles but anyone constant. A simple example for the application of this theorem is given by the free symmetrical top. In this case the configurational manifold M is given by the Euler angles ~q = (q θ , q ϕ , q ψ ) = (θ, ϕ, ψ). Since the potential energy U is zero the Maupertuis Riemannian metric is ds2 /(2h) = T dt2 = gµ,ν dq µ dq ν where the kinetic energy is T =

1 ˙2 ˙ 3 cos θ} , {θ I1 + ϕ˙ 2 [I3 + (I2 − I3 ) sin2 θ] + ψ˙ 2 I3 + ϕ˙ ψ2I 2

(8.i:1)

with I1 = I2 , and I3 the three moments of inertia. The manifold M is then SO(3) with the following metric tensor  gθθ = 12 I1 ,    gϕϕ = 12 [I3 + (I2 − I3 ) sin2 θ] , 1   gψψ = 2 I3 ,  1 gϕψ = 2 I3 cos θ . If we calculate the scalar curvature of M, this is what we find   1 I3 R= 4− . I1 I2

(8.i:2)

We conclude that when I3 > 4I2 the system is ergodic. This is also shown at the end of section 37 of [114] where it is said that “. . . the top does not at any time return exactly at its original position”. The chapter has the following structure. In subsection i.1 we give a brief description of the pseudosphere. In subsection i.2 we introduce the one component Coulomb plasma as a dynamical system confined in a connected and compact domain of the pseudosphere. We discuss the ergodicity of the system in section ii: we calculate the curvature of M for a general dynamical system with potential energy U . Requiring the curvature to be negative we find a disequality containing T and partial derivatives of U whose fulfillment we are able to prove for the one component Coulomb plasma with the Coulomb interaction amongst the particles switched off. In subsection ii.3 we discuss the thermodynamic limit from the point of view of ergodic theory. In section iii we compare four different definitions of pressure for the one component plasma on the pseudosphere. In subsection iii.1 we use the virial theorem to derive an expression for the virial pressure of the finite or infinite one component plasma in terms of the one and two particle correlation functions. It is known that, due to the presence of an inert background without kinetic energy, the thermal pressure of a flat one component plasma is negative for particular values of the temperature [115, 116] (this pathology occurs also in three dimensions). A pressure that is always positive is the kinetic pressure which is defined [102, 103] as one would define the pressure in the kinetic theory of gases. In subsection iii.2 we show the equivalence between the virial pressure and the kinetic pressure for the one component plasma on the pseudosphere. In subsection iii.3 we derive a relationship between the thermal pressure and the kinetic pressure in the thermodynamic limit (although for usual fluids the thermal pressure

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE I. INTRODUCTION 93

and the kinetic pressure are equivalent, in the presence of a background they are different). In subsection iii.4 we extend a “contact theorem” proved by Totsuji [117] for the flat one component plasma, to the plasma on the pseudosphere. In subsection iii.5 we treat the non neutral one component plasma on the pseudosphere: we review the previous definitions of pressure, define the mechanical pressure [102], and determine the relationship between the mechanical pressure and the kinetic pressure in the thermodynamic limit. In subsection iii.6 we derive expressions of the thermal and mechanical pressures appropriate for the grand canonical ensemble. These will be used in the last section. On a pseudosphere since the area of a large domain is of the same order as the area of the neighborhood of the boundary, all the above definitions of pressure depend on the boundary conditions. In section iv we show that a bulk pressure independent of the boundary conditions can be defined from the Maxwell stress tensor [118, 119] at some point well inside the fluid. We derive an equation of state for this Maxwell tensor pressure and show that it holds for the one component plasma on a large class of Riemannian surfaces including the plane, the sphere, and the pseudosphere. In the last section v, we illustrate the general properties of the one component plasma on the pseudosphere at the special value of the Coulombic coupling constant at which all properties can be explicitly and exactly calculated. The grand potential and the one particle density are calculated in a disk, and the thermodynamic limit is investigated. The general relations between the different pressures are checked on the solvable model.

i.1

The pseudosphere

There are at least three commonly known sets of coordinates to describe a pseudosphere S. The one which render explicit the “similarity” with the sphere is ~q = (q 1 , q 2 ) = (q τ , q ϕ ) = (τ, ϕ) with τ ∈ [0, ∞[ and ϕ ∈ [0, 2π) the metric being, ds2 = gµν dq µ dq ν = a2 (dτ 2 + sinh2 τ dϕ2 )

.

(8.i:3)

Another set of coordinates often used is (r, ϕ) with r = tanh(τ /2). They are the polar coordinates of the unitary disk, D = {ω ∈

C | |ω| < 1}

.

(8.i:4)

.

(8.i:5)

The metric in terms of this new coordinates is, ds2 = 4a2

dr 2 + r 2 dϕ2 (1 − r 2 )2

The unitary disk with such a metric is called the Poincar´e disk 1 . A third set of coordinates used is (x, y) obtained from (r, ϕ) through the Cayley transformation, z ≡ x + iy =

ω+i 1 + iω

.

(8.i:6)

which establishes a bijective transformation between the unitary disk and the complex half plane, H = {z = x + iy | x ∈ 1

R, y > 0}

.

(8.i:7)

Notice that in this chapter, instead of working with a dimensionless r, we preferred to work with r = 2a tanh(τ /2), so that at small r, the geodesic distance (8.i:9) of a point (r, ϕ) from the origin would simply have r as its leading term.

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE I. INTRODUCTION 94

The center of the unitary disk corresponds to the point zo = i, “the center of the plane”. The metric becomes, ds2 = a2

dx2 + dy 2 y2

.

(8.i:8)

The complex half plane with such a metric is called the hyperbolic plane, and the metric the Poincar´e’ s metric. Cayley transformation is a particular M¨obius transformation. Poincar´e metric is invariant under M¨obius transformations. And any transformation that preserves Poincar´e metric is a M¨obius transformation. The geodesic distance d01 between any two points q0 = (τ0 , ϕ0 ) and q1 = (τ1 , ϕ1 ) on S is given by, cosh(d01 /a) = cosh τ1 cosh τ0 − sinh τ1 sinh τ0 cos(ϕ1 − ϕ0 ) .

(8.i:9)

Given the set of points Ωd at a geodesic distance from the origin less or equal to d, Ωd = {(τ, ϕ) | τ a ≤ d, ϕ ∈ [0, 2π)}

,

(8.i:10)

that we shall call a disk of radius d, we can determine its circumference, Z q τ˙ 2 + sinh2 τ ϕ˙ 2 dt C = L(∂Ωd ) = a τ =d/a   d = 2π a sinh π a ed/a , a d→∞



and its area, Z



Z

d/a

dτ a2 sinh τ dϕ 0  0 d π a2 ed/a = 4π a2 sinh2 2a d → ∞

A = V(Ωd ) =

(8.i:11)



.

The Laplace-Beltrami operator on S is,   √ µν ∂ 1 ∂ ∆ = √ gg g ∂q µ ∂q ν   1 ∂2 1 ∂ ∂ 1 = sinh τ + a2 sinh τ ∂τ ∂τ sinh2 τ ∂ϕ2

(8.i:12)

,

(8.i:13)

where g is the determinant of the metric tensor g = det||gµν ||. The curvature is expressed in terms of the Riemannian tensor which for a surface has 22 (22 − 1)/12 = 1 independent components. For a pseudosphere if we choose the coordinates (τ, ϕ), the metric tensor is,  2  a 0 ||gµν || = . (8.i:14) 0 a2 sinh2 τ The characteristic component of the Riemann tensor is, Rτϕτ ϕ = − sinh2 τ

.

(8.i:15)

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE I. INTRODUCTION 95

The Gaussian curvature is given by Rτ ϕ τ ϕ = g ϕϕ Rτ ϕτ ϕ = −

1 a2

.

(8.i:16)

Contraction gives the components of the Ricci tensor, Rτ τ = Rϕ ϕ = −

1 , a2

Rτ ϕ = 0 ,

(8.i:17)

and further contraction gives the scalar curvature, R=−

i.2

2 . a2

(8.i:18)

The one component Coulomb plasma

The one component Coulomb plasma is an ensemble of N identical pointwise particles of mass m and charge q, constrained to move in a connected and compact domain Ω ⊂ S by an infinite potential barrier on the boundary of the domain ∂Ω. The total charge of the system is neutralized by a background surface charge distribution uniformly smeared on Ω with density ρb = −nq (ρb is 0 outside of Ω), where n = N/V(Ω) is the particle number density. The pair Coulomb potential between two unit charges a geodesic distance d apart, satisfies Poisson equation on S, ∆v(d) = −2πδ(2) (d)

,

(8.i:19)

√ where δ(2) (d01 ) = δ(~ q0 − ~ q1 )/ g is the Dirac delta function on the curved manifold. Poisson equation admits a solution vanishing at infinity,    dij . (8.i:20) v(dij ) = − ln tanh 2a The electrostatic potential of the background w(~q) satisfies, ∆w(~q) = −2πρb

.

(8.i:21)

If we choose Ω = Ωaτ0 , the electrostatic potential of the background inside Ω can be chosen to be just a function of τ (see appendix J),     1 − tanh2 (τ0 /2) 2 2 w(τ ) = 2πa qn ln + sinh (τ0 /2) ln[tanh (τ0 /2)] 1 − tanh2 (τ /2) 2

.

(8.i:22)

The self energy of the background is (see equation (J.:9)), 1 v0 = − (2πa2 qn)2 {1 − cosh τ0 + 4 ln[cosh(τ0 /2)] + 2 2 sinh4 (τ0 /2) ln[tanh2 (τ0 /2)]} .

(8.i:23)

The total potential energy of the system is then, U = v0 + vpb + vpp

,

(8.i:24)

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE II. ERGODICITY 96

where vpp is the potential energy due to the interactions amongst the particles, vpp =

N 1X 2 q v(dij ) , 2 i,j=1

(8.i:25)

i6=j

and vpb is the potential energy due to the interaction between the particles and the background, vpb =

N X

q w(τi ) .

(8.i:26)

i=1

This expression can be rewritten as follows, vpb = v1 + v¯pb

,

(8.i:27)

where, v1 = N 2πa2 q 2 n {ln[1 − tanh2 (τ0 /2)] + sinh2 (τ0 /2) ln[tanh2 (τ0 /2)]}

,

(8.i:28)

is a constant and, v¯pb =

N X

q w(τ ¯ i) ,

(8.i:29)

i=1

with, w(τ ¯ ) = −2πa2 qn ln[1 − tanh2 (τ /2)] ≥ 0

∀ τ

(8.i:30)

Since the interaction between the particles is repulsive we conclude that, up to an additive constant (v0 + v1 ), the potential U is a positive function of the coordinates of the particles.

ii

Ergodicity

Consider a closed one component Coulomb plasma of N charges and total energy h, confined in the domain Ωaτ0 ⊂ S. Let the coordinates of particle i be ~qi = q(i) α~eα = (q(i) 1 , q(i) 2 ) ∈ Ωaτ0 , where ~eα = ∂/∂q α (α = 1, 2) is a coordinate basis for S. The trajectory of the dynamical system, Tt0 = {q N (t) ≡ (~ q1 , . . . , ~qN ) | t ∈ [0, t0 ]}

,

(8.ii:1)

is a geodesic on the 2N dimensional manifold M defined by the metric, Gαβ = (h − U )gµν (~qi ) ⊗ · · · ⊗ gµν (~qN ) ,

(8.ii:2)

on S N . Since v¯pb and vpp are positive on Ωaτ0 we have, ′ = (h − v0 − v1 )gµν (~qi ) ⊗ · · · ⊗ gµν (~qN ) Gαβ < Gαβ

,

(8.ii:3)

where G ′ has a negative curvature along the coordinates of any given particle. In the next subsection we will calculate the curvature of G along the coordinates of one particle. According to the theorem stated in the introduction we will require the curvature to be negative everywhere

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE II. ERGODICITY 97

on S N . This will determine a condition on the kinetic and potential energy of the system, sufficient for its ergodicity to hold on Mh . ˜ α are the 1-forms of the dual Let p˜i = p(i) α ω ˜ α be the momentum of charge i, where ω ˜ α = dq coordinate basis, and define pN (t) ≡ (˜ p1 , . . . , p˜N ). The ergodicity of the system tells us that N N given any dynamical quantity A(q , p ), its time average, Z 1 T A(q N , pN ) dt , (8.ii:4) hAit = lim T →∞ T 0 coincides with its microcanonical phase space average, R N N 4N µ ps Mps A(q , p ) δ(h − H) d R hAih = 4N µ ps Mps δ(h − H) d

,

(8.ii:5)

where the phase space of the system is,

Mps = {(q N , pN ) | ~qi ∈ S i = 1, . . . , N ;

p(i) α ∈ [−∞, ∞] i = 1, . . . , N, α = 1, 2}

,

(8.ii:6)

the phase space measure is, 4N

d

µps =

2 Y

α=1

dq(1) α · · · dq(N ) α dp(1) α · · · dp(N ) α

,

(8.ii:7)

and δ is the Dirac delta function.

ii.1

Calculation of the curvature of M

We calculate the curvature of M along particle 1 using Cartan structure equations. Let T = h − U (τ, ϕ) be the kinetic energy of the N particle system of total energy h, as a function of the coordinates of particle 1 (all the other particles having fixed coordinates). We choose an orthonormal basis, √  τˆ ˜ ω ˜ = a T dτ √ (8.ii:8) ϕ ˆ ˜ ω ˜ = a sinh(τ ) T dϕ By Cartan second theorem we know that the connection 1-form satisfies ω ˜ αˆ βˆ + ω ˜ βˆαˆ = 0. Then we must have, ( ˜ ϕˆϕˆ = 0 ω ˜ τˆτˆ = ω (8.ii:9) ω ˜ τˆϕˆ = −˜ ωϕˆτˆ = −˜ ω ϕˆτˆ We use Cartan first theorem to calculate ω ˜ τˆϕˆ , ˜ω τˆ = −˜ ˜ ϕˆ d˜ ω τˆϕˆ ∧ ω √ ˜ ˜ ) = d(a T dτ

(8.ii:10)

1 ˜ ∧ dτ ˜ =0 , = a T 2 ,ϕ dϕ

where in the last equality we used the fact that the pair interaction is a function of ϕi − ϕj and that the interaction with the background is a function of τ only (being the system confined in a

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE II. ERGODICITY 98

domain which is symmetric under translations of ϕ). We must then conclude that ω ˜ τˆϕˆ is either ϕ ˆ zero or proportional to ω ˜ . We proceed then calculating, ˜ω ϕˆ = −˜ d˜ ω ϕˆτˆ ∧ ω ˜ τˆ √ ˜ sinh(τ ) T dϕ) ˜ = d(a

(8.ii:11)

1

˜ ∧ dϕ ˜ , = a(sinh(τ )T 2 ),τ dτ which tells us that indeed, 1

ω ˜ ϕˆτˆ =

(sinh(τ )T 2 ),τ ϕˆ ω ˜ a sinh(τ )T

.

(8.ii:12)

˜ω αˆ + ω Next we calculate the characteristic component of the curvature 2-form Rαˆβˆ = d˜ ˜ αˆγˆ ∧ βˆ

ω ˜ γˆ ˆ, β

˜ω τˆ Rτˆϕˆ = d˜ ϕ ˆ

1 1 ˜ ˜ = d[−(sinh(τ )T 2 ),τ T − 2 dϕ] 1

1

= −

[(sinh(τ )T 2 ),τ T − 2 ],τ τˆ ω ˜ ∧ω ˜ ϕˆ a2 sinh(τ )T

.

(8.ii:13)

and use Cartan third theorem to read off the characteristic component of the Riemann tensor, 1

Rτˆϕˆ ˆτ ϕ ˆ

1

[(sinh(τ )T 2 ),τ T − 2 ],τ =− a2 sinh(τ )T

.

(8.ii:14)

We find then for the scalar curvature, ˆ

R = Rαˆ β

α ˆ βˆ

= 2Rτˆϕˆτˆϕˆ ) ( 1 1 2 [(sinh(τ ) T 2 ),τ T − 2 ],τ = − 2 a sinh(τ ) T

which can be rewritten in terms of the Laplacian as follows,    (U,τ )2 U,ϕϕ 2 1 2 − R=− 2 1+ −a ∆U + a T 2T T sinh2 τ

,

(8.ii:15)

.

(8.ii:16)

For finite values of h, the condition for R to be negative on all the accessible region of S N is then, 2πa2 q 2 n −

ii.2

(U,τ )2 U,ϕϕ < 2T + T sinh2 τ

.

(8.ii:17)

Ergodicity of the semi-ideal Coulomb plasma

Consider a one component Coulomb plasma where we switch off the mutual interactions between the particles, leaving unchanged the interaction between the particles and the neutralizing background (U = v0 + vpb ). We will call it the “semi-ideal” system. Define, Ω(h, τ0 ) = {q N |~ qi ∈ Ωaτ0 ∀i, h − U (q N ) ≥ 0}

,

(8.ii:18)

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE II. ERGODICITY 99

¯ = h − v0 − v1 and and call h

f (N ) = −N ln[1 − tanh2 (τ0 /2)] = N ln[1 + sinh2 (τ0 /2)]   N . = N ln 1 + 4πa2 n

(8.ii:19)

We will have (α = 2πa2 nq 2 ) r=

inf

2

q N ∈Ω(h,τ0 )

2T =



¯ − αf (N )]2 h ¯ > αf (N ) 2[h ¯h ≤ αf (N ) 0

,

Notice that for large N , at constant n, we have     N N α 1 + ln 1 + −2 −v0 /α = + + O(1/N ) q2 4πa2 n 4πa2 n 2 α −v1 /α = f (N ) + N − 2 + O(1/N ) . q

(8.ii:20)

,

(8.ii:21) (8.ii:22)

Using the extensive property of the energy we may assume that h = N h0 , where h0 is the total energy per particle. Then for large N we will have    2   N 1 α ¯ ln 1 + − + O(1/N ) > αf (N ) , (8.ii:23) h = N h0 + αf (N ) + q 4πa2 n 2 if h0 ≥ 0. ¯ > αf (N ) we have On the other hand for h l =

sup q N ∈Ω(h,τ0 )

[αT + (U,τ )2 ] ≤

¯ + α2 tanh2 (τ0 /2) = l+ = αh

sup

[αT ] +

q N ∈Ω(h,τ0 )

sup

[(U,τ )2 ]

q N ∈Ω(h,τ0 )

,

(8.ii:24)

Condition (8.ii:17) is always satisfied if l < r. Then the semi-ideal system is ergodic if,   q ¯>h ¯ + = αf (N ) + α 1 + 1 + 8f (N ) + 8 tanh2 (τ0 /2) h , (8.ii:25) 4

¯ + is the largest root of the equation l+ = r. Recalling that tanh2 (τ0 /2) → 1 at lare N , where h one can verify that, given equation (8.ii:23), equation (8.ii:25) must be satisfied at large N if h0 > 0. We conclude that the semi ideal system is certainly ergodic if the total enery is extensive and the total energy per particle is positive.

ii.3

The thermodynamic limit

From the point of view of ergodic theory, given the microcanonical phase space probability distribution for the N particle dynamical system, PN (q N , pN ) = R

δ(hN − HN ) 4N µ ps Mps δ(hN − HN ) d

,

(8.ii:26)

it is natural to assume the existence of an asymptotic probability distribution P (γ) with γ = (N, q N , pN ) as the number of particles tends to infinity [120]. One usually has, PN n=

−→ N →∞

N V(Ω)

constant

P

,

(8.ii:27)

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 100

where in the limit process one has to take into account the extensive property of hN . This freedom in taking the limit translate itself in the existence of an whole family of limit distribution {Pβ } parameterized by the temperature 1/β of the infinite system. By the theorem of equivalence of ensembles [121] we know that Pβ indeed exists and is the grand-canonical Gibbs distribution, which is well defined for the one component Coulomb plasma [105]. The uniqueness of the limit distribution is discussed in [122]. Given the existence and uniqueness of the limit distribution one can reach the averages of the infinite system using the following procedure, Z Z A P dγ , (8.ii:28) A P dγ −→ hAi = hAiΩ = MΩ

MS

Ω→S

where A is any given dynamical variable and, MΩ = {(N, q N , pN ) | N ∈

N; ~qi ∈ Ω

i = 1, . . . , N ;

p(i) α ∈ [−∞, ∞] i = 1, . . . , N, α = 1, 2}

iii

,

(8.ii:29)

Pressures of the one component Coulomb plasma

For a one component Coulomb plasma several different definitions of the pressure are possible [102]. In this section we review four of them. We treat the neutral system in the first four subsections and the non neutral system in the last subsection.

iii.1

The virial theorem

The Hamiltonian of our dynamical system of N particles is, ¯ (q N ) , H(q N , pN ) = T (q N , pN ) + U

(8.iii:1)

where we are assuming the particles confined in Ωaτ0 (we will omit the subscript aτ0 unless ¯ = U + confining potential. The kinetic energy is, explicitly needed), with U N

T =

1 X αβ g (~qi )p(i) α p(i) β 2m

.

(8.iii:2)

i=1

The Roman indices label the particles, and the lower or upper Greek indices denote covariant or contravariant components respectively. A sum over repeated Greek indices is tacitly assumed. The equations of motion for particle i are,  ∂H 1   q(i) ˙ α= = gαβ (~qi )p(i) β   ∂p(i) α m 

  1 µν ∂H  ¯,α (~qi )  =− g ,α (~qi )p(i) µ p(i) ν − U ˙ α=−  p(i) α ∂q(i) 2m

,

(8.iii:3)

where the comma stands for partial differentiation and the dot for total derivative with respect to time.

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 101

P

If we take the time derivative of N

d X q(i) τ p(i) τ dt

=

i=1

i q(i)

τp

(i) τ

we find 2 ,

N

N

i=1 N X

i=1

1 X 1 X τν g (~qi )p(i) τ p(i) ν − q(i) τ gµν ,τ (~qi )p(i) µ p(i) ν m 2m −

¯,τ (~qi ) q(i) τ U

,

(8.iii:4)

i=1

where the last term is called the virial of the system. Since the system is confined in Ω the coordinates q(i) τ (t) and their canonically conjugated momenta p(i) τ (t) remain finite at all times. We then must have, N

h

d X q(i) τ p(i) τ it = 0 . dt

(8.iii:5)

i=1

(v)

We define the virial pressure PΩ of the system as minus the time average of the force per unit length exerted by the confining potential on the particles. By the ergodic hypothesis we have,

(v) PΩ

I

τ 1

q d Στ

∂Ω

=

N N 1 X 1 X τν g (~qi )p(i) τ p(i) ν it − q(i) τ gµν ,τ (~qi )p(i) µ p(i) ν it h h m 2m i=1 N X

−h

i=1

q(i) τ U,τ (~qi )it

,

(8.iii:6)

i=1

√ Q where d1 Σα = g 2β=1;β6=α dq β , is the elementary “surface” element, on the pseudosphere, orthogonal to the direction α. The line integral is, I

∂Ωaτ0

q τ d1 Στ =

Z

2π 0

τ0 a2 sinh τ0 dϕ = τ0 aL(∂Ωaτ0 )

.

(8.iii:7)

Moreover the ergodic hypothesis allows us to replace the time averages with microcanonical phase space averages. To reach the thermodynamic limit we further replace the microcanonical averages with grand-canonical phase space averages over MΩ and let Ω → S. We call P (v) the virial pressure of the system in the thermodynamic limit, P (v) =

lim

Ω→S

(v)

PΩ

.

(8.iii:8)

n constant

P One may be tempted to start with the time derivative of i ~ qi · p ~i . Note however that this quantity does not remain finite at all times. This is because, when you follow the motion of a particle colliding with the boundary, it may go around the origin indefinitely, and ϕ (which must be defined as a continuous variable, without any 2π jumps) may increase indefinitely. Thus the time average of the time derivative of this quantity does not vanish. 2

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 102

We calculate next the three terms contributing to the pressure. The first one is, N 1 X τν g (~ qi )p(i) τ p(i) ν iΩ = h m i=1 Z ∞ N X X zN −β(T +U ) 1 gτ ν (~qi )p(i) τ p(i) ν d4N µps e h2N N ! 2m i 2 N =0 = Z ∞ N X z −β(T +U ) 4N e d µps h2N N ! N =0 Z ∞ X zN e−βU dS1 · · · dSN N 2N N ! λ Ω n 1 N =0 hN i = V(Ω) , = Z ∞ β X zN β β −βU e dS · · · dS 1 N λ2N N ! Ω

(8.iii:9)

N =0

p where z is theQfugacity of the system, λ = 2πβ~2 /m is the de Broglie thermal wavelength, √ and dS = g 2α=1 dq α = a2 sinh τ dϕ dτ is the elementary area element on the pseudosphere. In the following we will introduce a generalized fugacity ζ = z/λ2 . Since V(Ωaτ0 ) diverges less rapidly than τ0 L(∂Ωaτ0 ) as τ0 → ∞, this term does not contribute to the pressure. The second term is, N 1 X − q(i) τ gµν ,τ (~qi )p(i) µ p(i) ν iΩ = h 2m i=1 N X

1 h 2m



i=1 N X

1 h 2m

τi

i=1 N X

2h

τi g ϕϕ ,τ (~qi )[p(i) ϕ ]2 iΩ = gϕϕ,τ (~qi ) [p ]2 iΩ = [gϕϕ (~qi )]2 (i) ϕ

(p(i) ϕ )2 τi iΩ = tanh τi 2ma2 sinh2 τi

i=1 N X

τi 1 h iΩ = β tanh τi i=1 Z τ1 1 (1) q1 ) nΩ (~ dS1 β Ω tanh τ1

,

(8.iii:10)

(1)

where nΩ is the one particle correlation function, Z ∞ X ζN e−βU dS2 · · · dSN N N ! Ω (1) q1 ) = N =1 nΩ (~ Z ∞ X ζN e−βU dS1 · · · dSN N! Ω

.

(8.iii:11)

N =0

Since for the infinite system U does not depend on the choice of the pseudosphere origin we (1) (1) must have nS (~q1 ) = n. In the event that nΩaτ (τ ) ∼ n near the boundary (see appendix I), 0

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 103

Rτ comparing the asymptotic behaviors of 2πa2 0 0 τ cosh τ dτ and τ0 L(∂Ωaτ0 ) as τ0 → ∞ we find that the second term gives a contribution n/β to the pressure. The third term is, Z N X (1) ¯,τ (τ1 ) dS1 nΩ (~q1 ) τ1 w q(i) τ U,τ (~ qi )iΩ = q −h

(8.iii:12)



i=1

q2 − 2

Z



(2)

nΩ (~q1 , ~q2 ) (q(1) τ v,τ1 (d12 ) + q(2) τ v,τ2 (d12 )) dS1 dS2 ,

(2)

where nΩ is the two particle correlation function, Z ∞ X ζN e−βU dS3 · · · dSN N (N − 1) N ! Ω (2) q1 , ~ q2 ) = N =2 ∞ nΩ (~ Z X ζN e−βU dS1 · · · dSN N! Ω

,

(8.iii:13)

N =0

(2)

which for the infinite system can be rewritten as nS (~q1 , ~q2 ) = n2 g(d12 ), where g is the usual pair correlation function. Notice that since the charges are indistinguishable we must have (2) (2) q2 , ~ q1 ). Then equation (8.iii:12) can be rewritten as follows, q2 ) = nΩ (~ nΩ (~q1 , ~ Z N X (1) τ ¯,τ (τ1 ) dS1 nΩ (~q1 ) τ1 w q(i) U,τ (~ qi )iΩ = q −h Ω

i=1

−q

2

Z



(2)

nΩ (~q1 , ~q2 ) q(1) τ v,τ1 (d12 ) dS1 dS2

,

(8.iii:14)

(1)

When nΩaτ (τ ) ∼ n near the boundary (see appendix I), we find the contribution of the back0 ground to the pressure comparing the asymptotic behaviors of,   Z Z τ0 2 2 τw ¯,τ (τ ) dS = 2πa nq −2πa τ tanh(τ /2) sinh τ dτ , (8.iii:15) S

0

and τ0 L(∂Ωaτ0 ) as τ0 → ∞. So doing we find that the background contributes to the pressure a term −2πa2 (nq)2 . We then reach the following expression for the pressure in terms of the one and two particle correlation functions, " Z 1 1 τ1 (1) P (v) = lim dS1 + (8.iii:16) nΩaτ (~q1 ) 0 τ0 →∞ aτ0 L(∂Ωaτ0 ) β Ω tanh τ1 aτ0 # Z Z (1) (2) 2 q 1 ) τ1 w ¯,τ (τ1 ) dS1 − q nΩaτ (~ nΩaτ (~q1 , ~q2 ) τ1 v,τ1 (d12 ) dS1 dS2 . q Ωaτ0

iii.2

0

0

Ωaτ0

Equivalence of virial and kinetic pressures

The average force exerted by the particles on a perimeter element ds = a sinh τ0 dϕ of the (1) (1) boundary ∂Ωaτ0 , is [nΩ (τ0 )/β]ds where nΩ (τ ) is the one particle density at a distance aτ from the origin. Therefore the kinetic pressure is, (k)

(1)

PΩ = nΩ (τ0 )/β

.

(8.iii:17)

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 104

We assume that this quantity has a limit when τ0 → ∞. It will now be shown that the virial (k) (v) pressure PΩ , is the same as PΩ . Replacing the virial pressure with the kinetic pressure in the left hand side of equation (8.iii:6) we have, I I 1 (1) (k) τ 1 q d Στ = (8.iii:18) nΩ (~q)q τ d1 Στ . PΩ β ∂Ω ∂Ω Using Gauss theorem we find, (k) PΩ

I

τ

1

q d Στ

=

∂Ω

Z p ( g(~q1 )q(1) τ ),τ1 (1) 1 p nΩ (~q1 ) dS1 + β Ω g(~q1 ) Z 1 (1) (~q1 ) dS1 . q τn β Ω (1) Ω ,τ1

The first term on the right hand side of this equation can be further developed into, Z √ ( g),τ1 hN i 1 (1) + q τ n (~q1 ) dS1 . √ β β Ω g (1) Ω

(8.iii:19)

(8.iii:20)

We see then that we recover the term (8.iii:9) plus the term (8.iii:10) of the virial pressure. In the second term on the right hand side of equation (8.iii:19) we can replace the gradient of the one particle correlation function with its expression in terms of one and two particle correlation functions. We know that the equilibrium states of the finite system contained in the domain Ω are described by correlation functions which are solutions of the BGY hierarchy, 1 (m) n (~ q1 , . . . , ~ qm ) = βq 2 Ω ,α1   Z m X (m) −n Fα1 (d1j ) nΩ (~q1 , . . . , ~qm ) Fα1 (d10 ) dS0 + Ω

+

Z



j=2

(m+1)

Fα1 (d10 )nΩ

(~q1 , . . . , ~qm , ~q0 ) dS0

,

where Fα1 (d10 ) = −v,α1 (d10 ). For m = 1 we have, Z 1 (1) (1) (2) Fα1 (d10 )[nΩ (~q1 , ~q0 ) − nnΩ (~q1 )] dS0 nΩ ,α1 (~ q1 ) = 2 βq Ω

,

(8.iii:21)

(8.iii:22)

which when inserted into the second term on the right hand side of equation (8.iii:19) gives the term (8.iii:14) of the virial pressure. We then find that, (k)

(v)

PΩ = PΩ

iii.3

.

(8.iii:23)

The thermal pressure in the Canonical ensemble

The thermal pressure is defined as the partial derivative with respect to the area of the Helmholtz free energy F (β, A, N ) keeping the number of particles N , the background charge, and the temperature T constants,   ∂F (t) . (8.iii:24) PΩ = − ∂A β,N

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 105

The free energy is related to the logarithm of the partition function Q(β, A, N ) as follows, F =−

1 ln Q β

.

(8.iii:25)

and the partition function is defined as, Q(N, A, T ) = =

Z N N h−2N e−βH(q ,p ) d4N µps N! Z λ−2N N e−βU (q ) dS1 · · · dSN N! Ω

,

(8.iii:26)

p where λ = 2πβ~2 /m is the de Broglie thermal wavelength. We calculate the thermal pressure using the dilatation method. We make the following change of variables in the definition of the partition function,  ϕi → ϕi ϕi ∈ [0, 2π) (8.iii:27) τi → τ0 ti ti ∈ [0, 1] This enables us to calculate the partial derivative with respect to A = 2πa2 (cosh τ0 − 1) through a partial derivative with respect to τ0 ,   dτ0 ∂ 1 (t) PΩ = ln Q(τ0 ) dA ∂τ0 β 1 1 ∂Q(τ0 ) 1 , (8.iii:28) = 2 2πa sinh τ0 β Q(τ0 ) ∂τ0 where, Q(τ0 ) =

λ−2N N!

Z



e−βUτ0 a2N τ0N

N Y

sinh(τ0 ti ) dti dϕi

,

(8.iii:29)

i=1

with, Uτ0

= q2

 1 X 2

v([dij ]τ0 )

i6=j

Z NX v([dip ]τ0 )a2 sinh(τ0 tp )τ0 dtp dϕp − A i )  2 Z 1 N 4 2 v([dpq ]τ0 )a sinh(τ0 tp ) sinh(τ0 tq )τ0 dtp dϕp dtq dϕq 2 A

(8.iii:30) ,

and, cosh([dij ]τ0 /a) = cosh(τ0 ti ) cosh(τ0 tj ) − sinh(τ0 ti ) sinh(τ0 tj ) cos(ϕi − ϕj ) . At the end of the calculation we undo the change of variables going back to (τi , ϕi ). We find then, ( ) 1 1 N 1 X β ∂Uτ0 τ 0 ti (t) PΩ = . + h iΩ − hτ0 iΩ 2πa2 sinh τ0 β τ0 τ0 tanh(τ0 ti ) τ0 ∂τ0 i

(8.iii:31)

(8.iii:32)

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 106

Recalling that aτ0 L(∂Ω) = τ0 2πa2 sinh τ0 we see that for the ideal gas (U = 0) the thermal pressure coincides with the virial pressure [see terms (8.iii:9) and (8.iii:10)]. It remains to calculate the excess thermal pressure, (t)

PΩ,exc = −

∂Uτ0 1 hτ0 iΩ aτ0 L(∂Ω) ∂τ0 (t)

(t)

(t)

= PΩ,pp + PΩ,pb + PΩ,bb

,

(8.iii:33)

which is made up of three contributions: the one from the particle-particle interactions, the one from the particle-background interactions, and the one from the background-background interaction. (t) Let us calculate PΩ,pp . Since, ∂v([dij ]τ0 ) = τ0 ∂τ0



∂ ∂ ti + tj ∂ti ∂tj



v([dij ]τ0 )

,

we find, (t)

PΩ,pp = −

X q2 h τi v,τi (dij )iΩ aτ0 L(∂Ω)

,

(8.iii:34)

i6=j

which coincides with the second term on the right hand side of equation (8.iii:14). (t)

Let us calculate next PΩ,pb , (t) PΩ,pb

q2 =− aτ0 L(∂Ω)

(

XZ [τi v,τi (dip ) + τp v,τp (dip )] dSp iΩ −nh Ω

i

XZ −nh v(dip ) dSp iΩ i



τp dSp iΩ tanh τp Ω i ) Z aτ0 L(∂Ω) X +n v(dip ) dSp iΩ h A Ω i ( XZ nh τi v,τi (dip ) dSp iΩ XZ −nh v(dip )

=

q2 aτ0 L(∂Ω)



i

XI +nh i

∂Ω

τp v(dip ) d1 Σpτ iΩ

Z aτ0 L(∂Ω) X v(dip ) dSp iΩ −n h A Ω i

)

.

(8.iii:35)

We see that the first term on the right hand side coincides with the first term on the right hand side of equation (8.iii:14).

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 107

(t)

Let us calculate in the end PΩ,bb , (t) PΩ,bb

q2 =− aτ0 L(∂Ω)



n2 2

+n2

Z Z

v(dpq ) dSp dSq   Z Z τp τq n2 + v(dpq ) + dSp dSq 2 Ω Ω tanh τp tanh τq  Z Z aτ0 L(∂Ω) v(dpq ) dSp dSq − n2 A Ω Ω  Z I n2 τp v(dpq ) d1 Σpτ dSq Ω ∂Ω  Z Z aτ0 L(∂Ω) . v(dpq ) dSp dSq −n2 A Ω Ω Ω

=−

q2 aτ0 L(∂Ω)

[τp v,τp (dpq ) + τq v,τq (dpq )] dSp dSq

Z ΩZ Ω Ω

(8.iii:36)

We find then the following relationship between the thermal and the virial pressure, Z Z n (v) (t) (1) − q2 PΩ = PΩ v(dpq )[nΩ (τp ) − n] dSp dSq A Ω Ω Z I n (1) (8.iii:37) τp v(dpq )[nΩ (τq ) − n] d1 Σpτ dSq . + q2 aτ0 L(∂Ω) Ω ∂Ω The second integral on the right hand side of this equation is zero: the electric potential at ~qp created by the total charge distribution (particles plus background) is given by the quantity R (1) q Ω v(dpq )[nΩ (τq ) − n] dSq . Since the total charge is zero, by Newton’s theorem the above potential vanishes on the Rdisk’s boundary τp = τ0 . In the first integral on the right hand side of equation (8.iii:37) −qn Ω v(dpq ) dSq is the electric potential created by the background at ~ qp . We then have, Z 1 (1) (v) (t) (8.iii:38) w(τp )[nΩ (τp ) − n] dSp . PΩ = PΩ + q A Ω We want to find an expression for the difference between the thermal and the virial pressure (1) in the thermodynamic limit. Since nΩ (τp ) − n is localized near the boundary (see appendix I) we change the integration variable from τ to σ = τ0 − τ and take the limit τ0 → ∞. We have that the electric potential of the background behaves as, w(σ)



τ0 →∞

2πa2 qn(−σ − 1)

,

(8.iii:39)

Then using the normalization condition for the one particle correlation function we find, Z ∞ (1) (t) (v) 2 2 P − P = −2πa nq (8.iii:40) [nS (σ) − n]σe−σ dσ . 0

This latter formula is the same as in the case of a flat system in the thermodynamic limit (half-space), except for the factor exp(−σ) (see [102] section 5.1.2. The flat system expression is recovered taking the limit a → ∞, σ → 0, aσ = x).

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 108

iii.4

Difference between thermal and kinetic pressure

In a flat half space (thermodynamic limit of a disc) the difference between the thermal and the kinetic pressure P (t) − P (k) is related to the potential difference between the surface and the bulk [117]. We want to give the analog of this relation in the case of a pseudosphere. In this subsection we will omit the subscripts on the correlation functions. Let us first review the flat case. In reference [102] the last equation of section 5.1.2. becomes in our notation, Z ∞ (t) (k) 2 x[n(1) (x) − n] dx , P − P = −2πq n 0

where x is the distance from the boundary. The electric potential φ(x) obeys the Poisson equation, d2 φ(x) = −2πq [n(1) (x) − n] . dx2 After an integration by parts, and taking into account that xdφ/dx vanishes at x = 0 (because of the overall neutrality) and at x = ∞ (because the electric field goes fast to zero in the bulk), one finds, P (t) − P (k) = −qn [φbulk − φsurface ] .

(8.iii:41)

This equation (8.iii:41) can be found in [117], equations (2.18) and (2.20). Let us now give another, more pictorial, proof of equation (8.iii:41). We consider a large disk filled with a one component Coulomb plasma, of area A. For compressing it a little, changing the area by dA < 0, we must provide the reversible work δW = P (t) |dA|. We may achieve that compression in two steps. First, one compresses the particles only, leaving the background behind; the corresponding work is δW (1) = P (k) |dA|, since P (k) is the force per unit length exerted on the wall by the particles alone. Then, one compresses the background, i.e. brings the charge −qnb |dA| from a region where the potential is φsurface R = 0 into the plasma, spreading it uniformly; the corresponding work is δW (2) = −[qn|dA|/A] φ(r)dS, where φ(r) is the potential at distance r from the center. Therefore, Z qn (t) (k) φ(r) dS . (8.iii:42) P =P − A Since φ(r) differs from φbulk only in the neighborhood of the boundary circle, in the large disc limit, Z 1 φ(r) dS ∼ φbulk , (8.iii:43) A and (8.iii:42) becomes (8.iii:41). Let us now follow the same steps on a pseudosphere (see figure 8.1). We again get (8.iii:42), with φ(τ ) instead of φ(r). But now, the neighborhood of the boundary circle has an area of the same order of magnitude as the whole area A, and (8.iii:43) is no longer valid. In the large disc limit, we rather have, Z Z τ0 1 −τ0 φ(τ )eτ dτ . (8.iii:44) φ(τ ) dS ∼ e A 0

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 109

large disk of area A

charge −qn b |dA|

00000000000000000000000 11111111111111111111111 11111111111111111111111 00000000000000000000000 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111

2a

Poincare disk

φ (r) electrostatic potential at distance r from the origin

Figure 8.1: Shows the Poincar´e disk, the disk of area A initially containing the 2D OCP, and the disk of area A − dA containing the compressed 2D OCP. Since we are working at constant β, N, Nb , the leftover background charge qnb dA must be spreaded uniformly within the compressed disk.

Using (8.iii:44) in (8.iii:42), we recover, after some manipulation, equation (8.iii:40), P

(t)

−P

(k)

2

∼ −2πa nq

2

Z

τ0 0

[n(1) (τ ) − n](τ0 − τ )eτ −τ0 dτ

.

(8.iii:45)

Indeed, in (8.iii:45) [n(1) (τ ) − n] can be expressed in terms of φ(τ ) through the Poisson equation, ∆φ(τ ) = −2πq[n(1) (τ ) − n]. Since the charge density is localized at large τ , we can use for the Laplacian ∆ ∼ a−2 [d2 /dτ 2 + d/dτ ]. After integrations by parts, (8.iii:45) becomes, P

(t)

−P

(k)

−τ0

∼ −qn e

Z

τ0

φ(τ )eτ dτ

,

(8.iii:46)

0

which is the same as (8.iii:42) in the large τ0 limit. In conclusion, (8.iii:41) valid for a large flat disc generalizes into (8.iii:46) on a pseudosphere

iii.5

Non neutral system and the mechanical pressure

In this subsection we want to revisit the various definitions of pressure and the relations between them for a non neutral one component Coulomb plasma, i.e. a system with ρb = −nb q with nb 6= n. It is convenient to introduce the number of elementary charges in the background: Nb = nb A In this case we find for the virial and kinetic pressure of the finite system, (k) PΩ

=

(v) PΩ

=

 Z hN i τ1 1 1 (1) + nΩ (τ1 ) dS1 aτ0 L(∂Ω) β β Ω tanh τ1  Z Z (1) (2) 2 −q τ1 v,τ1 (d10 )[nΩ (~q1 , ~q2 ) − nb nΩ (τ1 )] dS0 dS1 Ω



(8.iii:47) .

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 110

The thermal pressure becomes equation (8.iii:37) with n replaced by nb , Z Z (1) (v) (t) 2 nb v(dpq )[nΩ (τp ) − nb ] dSp dSq − q PΩ = PΩ A Ω Ω Z I nb (1) 2 + q τp v(dpq )[nΩ (τq ) − nb ] d1 Σpτ dSq aτ0 L(∂Ω) Ω ∂Ω

.

(8.iii:48)

For the non neutral system we can introduce a fourth type of pressure: the mechanical pressure, or partial pressure due to the particles. For a fluid parameterized by (β, A, N, nb ) the mechanical pressure is defined as follows,   ∂F (m) . (8.iii:49) PΩ = − ∂A β,N,nb Using the dilatation method again, we find, Z I nb (v) (m) (1) 2 PΩ = PΩ + q τp v(dpq )[nΩ (τq ) − nb ] d1 Στp dSq aτ0 L(∂Ω) Ω ∂Ω

.

Using Newton’s theorem this expression can be simplified as follows, I nb (v) (m) 2 = q (N − Nb ) PΩ − PΩ τp v(dp0 )d1 Στp aτ0 L(∂Ω) ∂Ω = q 2 nb (N − Nb )v(aτ0 ) .

(8.iii:50)

(8.iii:51)

The difference between the thermal and the mechanical pressure can be rewritten as, Z q (1) (m) (t) (8.iii:52) w(τ )[nΩ (τ ) − nb ] dS . PΩ − PΩ = A Ω In the thermodynamic limit we find, P (m) − P (v) = 2πa2 q 2 nb (n − nb ) .

(8.iii:53)

(1)

In equation (8.iii:52) [nΩ (τ ) − nb ] differs from zero just in a neighborhood of the disk boundary (the system tends to be electrically neutral in the bulk). Then changing variables from τ to the distance from the boundary σ = τ0 − τ we find, Z 2q τ0 (1) (t) (m) (8.iii:54) w(σ)[nS (σ) − nb ] sinh(τ0 − σ) dσ . P −P = τ0 e 0 Using the asymptotic expansion (8.iii:39) for the background potential we have, Z ∞ (1) (t) (m) 2πa2 qnb (−σ − 1)[nS (σ) − nb ]e−σ dσ P −P = q 0

= −2πa2 q 2 nb (n − nb ) Z ∞ (1) −2πa2 nb q 2 [nS (σ) − nb ]σe−σ dσ

,

(8.iii:55)

0

where we used the asymptotic form for the normalization condition of the one particle correlation function. We also have, Z ∞ (1) (t) (v) 2 2 P − P = −2πa nb q (8.iii:56) [nS (σ) − nb ]σe−σ dσ . 0

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE III. PRESSURES OF THE ONE COMPONENT COULOMB PLASMA 111

large disk of area A

charge −qn b |dA|

111111111111111111111111 000000000000000000000000 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 (1) 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 1 2 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111

2a

φ(q ) =

q[n

Poincare disk

( τ ) − n b ] v(d 12 ) dS 1

Figure 8.2: Shows the Poincar´e disk, the disk of area A initially containing the 2D OCP, and the disk of area A − dA containing the compressed 2D OCP. Since we are working at constant β, N, nb , the leftover background charge qnb dA must be sent to infinity.

The difference P (m) − P (k) can be obtained by a slight change in the argument of subsection iii.4 (see figure 8.2). Again we consider a large disk filled with a one component Coulomb plasma of area A, and we compress it infinitesimally, changing its area by dA < 0, now at constant β, N, nb , providing the reversible work δW = −P (m) dA, in two steps. First one compresses the particles only, leaving the background behind, and the corresponding work is δW (1) = −P (k) dA. Then one must withdraw the leftover background charge qnb dA, bringing it from the surface, where the potential is φsurface to infinity, where the potential vanishes. The corresponding work is δW (2) = −qnb dAφsurface . Therefore one finds, P (m) − P (k) = qnb φsurface . In the thermodynamic limit on the pseudosphere, φsurface → 2πa2 q(n − nb ).

iii.6

Thermal and mechanical pressures in the Grand Canonical ensemble

In the following we shall also need an expression of the thermal and mechanical pressures appropriate for the grand canonical ensemble. It should be remembered that, for a one component plasma, the grand canonical partition function must be defined [123] as an ensemble of systems with any number N of particles in a fixed area and with a fixed background charge density −qnb (using am ensemble of neutral systems, i.e. varying nb together with N does not give a well behaved grand partition function). Thus the grand partition function Ξ and the corresponding grand potential Ω = − ln Ξ/β are functions of β, A, ζ, nb , where ζ is the fugacity. We assume that even on a pseudosphere, the grand potential is extensive, i.e. of the form Ω = Aω(β, ζ, nb ). The usual Legendre transformation from F to Ω and from N to ζ changes (8.iii:24) into,   ∂Ω (t) . (8.iii:57) P =− ∂A β,ζ,Nb Since ω depends on A through nb = Nb /A, (8.iii:57) becomes, P (t) = −ω + nb

∂ω ∂nb

.

(8.iii:58)

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE IV. THE YUKAWA FLUID AND THE MAXWELL TENSOR PRESSURE 112

Note the difference with an ordinary fluid, without a background, for which P (t) = −ω. The mechanical pressure (8.iii:49) is changed by the Legendre transformation into,   ∂Ω (m) = −ω (8.iii:59) P =− ∂A β,ζ,nb

iv

The Yukawa fluid and the Maxwell tensor pressure

In the previous sections we have described a method for calculating the virial and kinetic pressures exerted by the Coulomb plasma on the “surface” of its container comparing them with the thermal pressure. In this section, following closely a derivation due to Jancovici [119], we will calculate the Maxwell tensor pressure P (θ) of the plasma on the Riemannian surface, using a field theoretical argument. This bulk pressure has been shown to be equivalent to the thermal pressure as defined by Choquard, for the flat plasma. On a pseudosphere, since the surface of a large domain is of the same order of magnitude as the volume, the thermal pressure would depend on the boundary conditions and probably would be different from the Maxwell tensor pressure. A field theoretical description of a one component Coulomb plasma on a Riemannian manifold R can be obtained as follows. Consider a system of particles of mass m and charge q living on the whole manifold with a number density n, immersed in a uniform neutralizing background charge distribution of charge density ρb = −nq, and coupled to a scalar field φ of “mass” α. This we will call a Yukawa fluid. In the limit of a vanishing α the Yukawa fluid reduces to the one component Coulomb plasma. Let us introduce the mass density of particle i, √ (8.iv:1) ρi = m δ(~ q − ~qi )/ g , where ~qi is the position of particle i, and the total charge density as, ρ = ρp + ρb , X √ ρp = q δ(~q − ~qi )/ g

.

(8.iv:2)

i

The Hamiltonian of the fluid can be written as, H = Hp + Hφ + Hpφ

,

(8.iv:3)

where Hp is the kinetic energy of the particles, Z √ Hp = Hp g d~ q , Hp =

1X ρi gµν (~qi )(p(i) µ /m)(p(i) ν /m) 2

,

(8.iv:4)

i

and Hφ + Hpφ is their total “electrostatic” potential energy. If the dimension of the manifold is d we have, Z √ Hφ = Hφ g d~ q , Hφ = −

1 µν [g φ,µ φ,ν + α2 φ2 ] , 2ǫd

(8.iv:5)

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE IV. THE YUKAWA FLUID AND THE MAXWELL TENSOR PRESSURE 113

where ǫ1 = 2, ǫ2 = 2π, ǫ3 = 4π, and, Hpφ =

Z

√ q , Hpφ g d~

Hpφ = ρφ . √ √ The Lagrangian density L g = (Lp + Lφ ) g has a particle contribution, Lp =

1X ˙ µ q(i) ˙ν ρi gµν (~qi )q(i) 2

,

(8.iv:6)

(8.iv:7)

i

and a field contribution, Lφ = −(Hφ + Hpφ ) . The field equation of motion is,  √  √ ∂[Lφ g] ∂ ∂(Lφ g) = ∂q µ ∂φ,µ ∂φ

(8.iv:8)

,

(8.iv:9)

which reduces to Helmholtz equation, (−∆ + α2 )φ = ǫd ρ

,

(8.iv:10)

whose solution may be written in terms of its Green function G, as follows, Z √ q1 . φ(~ q0 ) = G(d01 )ρ(~q1 ) g d~

(8.iv:11)

In appendix H we give a collection of Green functions for the Euclidean spaces of dimension d ≤ 3 and for some simple Riemannian manifolds of dimension d = 2 and d = 3. Performing an integration by parts and neglecting the “surface” contribution of the field at infinity (when such “surface” does not reduce to a “point” as in finite manifolds) we can rewrite the energy density of the Yukawa fluid as, √ (Hφ + Hpφ ) g =

√ 1 µν [g φ,µ φ,ν + α2 φ2 ] g 2ǫd 1 √ ρφ g . 2

(8.iv:12)

The total stress tensor is, Tµν

√ 2 δ[L g] = −√ = (Tp )µν + (Tφ )µν g δgµν

.

(8.iv:13)

It has a particle contribution, (Tp )µν = −

X i

ρi

p(i) µ p(i) m

m

ν

,

(8.iv:14)

and a field contribution, (Tφ )µν

δLφ + gµν Lφ δgµν 1 1 [φ,µ φ,ν − gµν (φγ, φ,γ + α2 φ2 )] . ǫd 2

= −2 =

(8.iv:15)

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE IV. THE YUKAWA FLUID AND THE MAXWELL TENSOR PRESSURE 114

The mass current density of particle i is, Jiµ = ρi

p(i) µ m

.

(8.iv:16)

From the conservation of the mass of a particle (Jiµ ; iµ = 0, where the semicolon stands for covariant derivative), the equation of motion of the particles, and the equation of motion of the field, follows the conservation of the stress tensor (Tµν; ν = 0). The Maxwell tensor pressure of the Yukawa fluid is given by, P (θ) = −hT 11 iR 1 = − hT µµ iR d

(8.iv:17) ,

(8.iv:18)

where as in the previous sections h. . .iR stands for the grand-canonical average and fluid isotropy was used in the last equality. The particle contribution to the pressure is, Pp(θ) = −h(Tp )11 iR ˆˆ

= −h(Tp )11 iRˆ   2 X ρˆi β 2 = h (p ) iRˆ β m 2m (i) ˆ1

,

i

where in the second equality we changed coordinate basis on RN : from the coordinate basis {~eiµ } to the non-coordinate orthonormal reference frame {~eiµˆ }, defined by, ~eiµˆ = Liµ iµˆ ~eiµ

(8.iv:19)

where the transition matrix satisfies, giµ iν Liµˆ iµ Liνˆ iν = giµˆ iνˆ

,

with giµˆ iνˆ = δiµˆ iνˆ . We denoted with h. . .iRˆ the grand-canonical average using the new phase space coordinates, ( q(i) µˆ = Liµˆ iµ q(i) µ , p(i) µˆ = Liµ iµˆ p(i) µ and with ρˆi the mass density in the local orthonormal frame. Carrying out the integration over the momenta we find, Pp(θ) =

1 X ρˆi h iˆ β m R

.

i

Switching back to the original coordinate basis yields, Pp(θ) = =

1 X √ h δ(~q − ~qi )/ giR β i n . β

(8.iv:20)

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE IV. THE YUKAWA FLUID AND THE MAXWELL TENSOR PRESSURE 115

We see then that the field contribution to the pressure turns out to be the excess pressure over the one of the ideal fluid. Making use of the rotational symmetry of the fluid, the excess pressure may be written as, (θ)



1 = − h(Tφ )µµ iR d 1 = − h(Tφ )µˆµˆ iRˆ d d 1 1 hφ,ˆµ φ,ˆµ − (φ,ˆµ φ,ˆµ + α2 φ2 )iRˆ = − d ǫd 2 1 = hφ,µ φ,µ (1/2 − 1/d) + α2 φ2 /2)iR . ǫd

(8.iv:21)

In particular for a two dimensional manifold we have, (θ)

Pφ =

α2 2 hφ iR 4π

.

(8.iv:22)

Using equation (8.iv:11) we find, (θ)

Pφ =

X α2 X h[q G(d0i ) − nqIG ][q G(d0j ) − nqIG ]iR 4π i

,

(8.iv:23)

j

R

where IG = G(d01 ) dS1 . The homogeneity and isotropy of the fluid allows us to rewrite the excess pressure as sum of two terms, (θ)

(θ)

(θ)

Pφ = Pself + Pnon−self

,

(8.iv:24)

where the self term is, (θ)

(θ)

(θ)

¯ + P (d)] ¯ Pself = lim [P0 (d) , 1

(8.iv:25)

¯ d→0

(θ)

¯ = regularized term (see next subsection) P0 (d) Z α2 2 (θ) ¯ P1 (d) = nq G2 (d01 ) dS1 , 4π ¯ d01 >d

,

(8.iv:26) (8.iv:27)

and the non-self term is, (θ) Pnon−self

α2 2 2 n q = 4π

Z

G(d01 )G(d02 )h(d12 ) dS1 dS2

.

(8.iv:28)

where h = g − 1 is the pair correlation function.

iv.1

Calculation of the self part of the excess pressure

In the calculation of the self part of the excess pressure care is needed in neglecting the force that each particle exerts on itself [118]. Such contribution is responsible for the divergence of the self part of the excess pressure at small geodesic distances from the origin when one calculates it, for example, on the q 1 = 0 “surface” (without taking advantage of the rotational symmetry of the fluid, i.e. from equation (8.iv:17) instead of equation (8.iv:18)). In order to cure such (θ) divergence one can employ the prescription described in equation (8.iv:25): split Pself into the

CHAPTER 8. PRESSURES FOR A ONE-COMPONENT PLASMA ON A PSEUDOSPHERE IV. THE YUKAWA FLUID AND THE MAXWELL TENSOR PRESSURE 116

(θ)

(θ)

contributions P0 of geodesic distance from the origin d01 < d¯ and P1 of geodesic distance ¯ and let d¯ → 0 in the end. Now P (θ) is convergent and can be computed from the origin d01 > d, 1 using the rotational symmetry as shown in equation (8.iv:27). For a pseudosphere one finds [124], (θ)

P1

(θ)

¯ = = lim P1 (d) ¯ d→0

=

Z ∞ α2 2 nq 2πa2 Q2ν (y) dy 4π 1 (aα)2 2 ψ ′ (ν + 1) nq , 2 2ν + 1

(8.iv:29) (θ)

where ψ is the psi function (the logarithmic derivative of the gamma function). P0 must be regularized by the prescription that no particle sits on the q 1 = 0 “surface”. This can be realized by removing from the integration domain a thin slab |q 1 | < ε and taking the limit ε → 0 afterwards,   Z nq 2 1 (θ) 2 2 2 P0 = − lim G,1 (d01 ) − [G,µ (d01 )G,µ (d01 ) + α G (d01 )] ε Z nq 2 2 2 lim = − , (8.iv:30) ε where in the last equality we kept just the divergent part of the integrand. Since d¯ can be taken (θ) arbitrarily small, the regularized P0 can be computed using the small d form of G(d), which is just the Coulomb potential in 2 , p d −→ r = x2 + y 2 ,

R

G(d) −→ G(r) = − ln r + constant

.

(θ)

We then find for P0 , (θ) ¯ P0 (d)

nq 2 lim =− 2ǫ2 ε→0

Z √d¯2 −x2 x2 − y 2 nq 2 dy 2 dx √ = − (x + y 2 )2 4 − d¯2 −x2 ε 0, it follows as R → ∞ when r < 0 since [1 − Q(k)] → 0 as y → ∞. Since Q from Cauchy’ s theorem that Q(r) = 0

r −ǫ we take ˆ Q(k) Hto be the analytic continuation found in property (4). Then the limit (B.:24) ensures us that γ2 . . . → 0 as R → ∞ when r ≥ 0. It again follows from Cauchy’ s theorem that, Q(r) = 0

r≥d

.

(B.:30)

We can then write ˆ Q(k) = 1 − 2πρ

Z

d 0

eikr Q(r) dr

.

(B.:31)

APPENDIX B. THE WIENER-HOPF FACTORIZATION

149

y +ε R γ1

x

−ε The analytic continuation ^ of Q(k) is analytic for y0 ,

(B.:44)

where the prime denotes differentiation. Since dS/dr = −rc(r) from equation (B.:36) we find   Z d dQ(s − r) ds Q(s) rc(r) = −Q′ (r) + 2πρ −Q(r)Q(0) − ds r   Z d ′ ′ d Q (s)Q(s − r) ds = −Q (r) + 2πρ −Q(r)Q(0) − [Q(s)Q(s − r)]r + r   Z d ′ ′ Q (s)Q(s − r) ds . (B.:45) = −Q (r) + 2πρ −Q(d)Q(d − r) + r

Since Q(d) = 0 (see equation (B.:30)) we have that ′

rc(r) = −Q (r) + 2πρ

Z

d r

Q′ (s)Q(s − r) ds

0