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First published in 1974, Dr Croxton's book takes the reader from a consideration of the early ways in which the kinetic theory of gases was modified and applied to the liquid state, through a classical thermodynamic approach, to the modern cluster-diagrammatic quantum and statistical mechanical techniques. He includes chapters on the development and numerical solution of the integral equations relating the atomic structure to the pair potential, on the nature of the liquid surface, on the computer simulation schemes and on transport processes and irreversibility in the liquid phase.
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CAMBRIDGE MONOGRAPHS ON PHYSICS GENERAL EDITORS M . M . WOOLFSON, D.SC. Professor of Theoretical Physics, University of York J. M. ZlMAN, D.PHIL., F.R.S. Professor of Theoretical Physics, University of Bristol
LIQUID STATE PHYSICSA STATISTICAL MECHANICAL INTRODUCTION
LIQUID STATE PHYSICSA STATISTICAL MECHANICAL INTRODUCTION CLIVE A. CROXTON Fellow of Jesus College, Cambridge
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CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521114349 © Cambridge University Press 1974 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1974 This digitally printed version 2009 A catalogue record for this publication is available from the British Library Library of Congress Catalogue Card Number: 72—89803 ISBN 978-0-521-20117-9 hardback ISBN 978-0-521-11434-9 paperback
CONTENTS
page ix
Preface CHAPTER I
Theory of imperfect gases I.I
1.2
1.4 i-5 1.6
Introduction The cluster expansions The Ursell development The cluster expansion of Ree and Hoover Quantum mechanical calculation of the second virial coefficient The virial coefficients
1 2
6 12
16 21
CHAPTER 2
Equilibrium theory of dense fluids: the correlation functions 2.1 Introduction 24 2.2 Generic and specific distributions 25 2.3 A formal relation between g(2)(r) and the pair potential 27 2.4 Equations for the pair correlation function 31 2.5 Differential equations for the pair distribution: the Born-Green-Yvon equation (BGY) 32 2.6 A test of the superposition approximation 38 2.7 The Rice-Lekner (RL) modification of the BGY equation 43 2.8 The Kirkwood equation (K) 47 2.9 The equation of Cole 49 2.10 The equation of Fisher 50 2.11 Abe's series expansion of the BGY equation 52 2.12 Direct and indirect correlation: the OrnsteinZernike equation 53 [v]
VI
2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21
CONTENTS
The HNC and PY approximations: the ^(r) approach page 57 The HNC and PY approximations: the diagrammatic approach 64 71 Solution of the PY equation for hard spheres Functional differentiation 74 Second-order theories 78 Some general comments 81 Non-additive effects 82 Perturbation theories 85 Quantum liquids 92
CHAPTER 3
Numerical solution of the integral equations 3.1 3.2 3.3 3.4 3.5 3.6 3.7
Introduction Low density solutions Liquid-density solutions The hard sphere fluid Realistic fluids Inversion of integral equations Quantum fluids
97 99 111 113 121 133 137
CHAPTER 4
The liquid surface 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10
Introduction A formal theory of the liquid surface The distribution function g&)(z) The distribution function g($(zl9 r) Thermodynamic funct