Table Of ContentApplications of Field Theory Methods in
Statistical Physics of
Nonequilibrium Systems
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Applications of Field Theory Methods in
Statistical Physics of
Nonequilibrium Systems
Bohdan Lev
National Academy of Science of Ukraine, Ukraine
Anatoly Zagorodny
National Academy of Science of Ukraine, Ukraine
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Lev, Bohdan, author. | Zagorodny, A., author.
Title: Applications of field theory methods in statistical physics of nonequilibrium systems /
Bohdan Lev, National Academy of Science of Ukraine, Ukraine, Anatoly Zagorodny,
National Academy of Science of Ukraine, Ukraine.
Description: New Jersey : World Scientific, [2021] | Includes bibliographical references and index.
Identifiers: LCCN 2021005226 (print) | LCCN 2021005227 (ebook) |
ISBN 9789811229978 (hardcover) | ISBN 9789811229985 (ebook)
Subjects: LCSH: Statistical physics. | Field theory (Physics)
Classification: LCC QC174.7 .L47 2021 (print) | LCC QC174.7 (ebook) | DDC 530.13/3--dc23
LC record available at https://lccn.loc.gov/2021005226
LC ebook record available at https://lccn.loc.gov/2021005227
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd.
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Preface
In this book, we formulate a unified approach to the description
of interacting many-particle systems combining the methods of
statistical physics and quantum field theory. The strong point
of this approach is that it may be applied to the description of
phasetransitions with theformation of newspatially inhomogeneous
phases.Anotheradvantageconcernsthepossibilitytodescribequasi-
equilibrium systems with spatially inhomogeneous particle distribu-
tions and metastable states.
The validity of the methods of statistical description of many-
particle systems and models (theory of phase transitions included) is
discussed and compared. The idea of using the quantum field theory
approach and related topics (path integration, saddle-point and
stationary-phase methods, Hubbard–Stratonovich transformation,
mean-field theory, and functional integrals) is also described in
sufficient detail for understanding as well as for further applications.
To some extent, the book could be treated as a brief encyclopedia
of methods applicable to the statistical description of spatially
inhomogeneous equilibrium and metastable particle distributions.
Theadditionalstrengthofthebookisthatwenotonlyformulate
the general approach, but also apply it to solve various practical
and important problems (gravitating gas, Coulomb-like systems,
dusty plasmas, thermodynamics of cellular structures, nonuniform
dynamics of gravitating systems, etc).
The book could be useful not only for students, but also
for researchers dealing with various methods of equilibrium and
nonequilibrium statistical mechanics.
v
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Contents
Preface v
Introduction 1
1. Statistical Physics of Interacting
Particle Systems 13
1.1 Systems of Particles with Interaction . . . . . . . . . 13
1.2 Models of Statistical Physics . . . . . . . . . . . . . . . 20
1.3 The Model of Hard Spheres with Attractive
Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4 Nonideal Gas at Low Temperatures . . . . . . . . . . 38
2. Statistical Description of Phase Transitions 41
2.1 Theory of the Second-Order Phase Transitions . . . 41
2.2 Unification of the Theories of Phase
Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 First-Order Phase Transitions . . . . . . . . . . . . . . 61
2.4 Dynamics of Metastable States . . . . . . . . . . . . . 73
3. Path Integration and Field Theory 79
3.1 Classical and Quantum Systems . . . . . . . . . . . . 79
3.2 Saddle-Point Method or Stationary-Phase
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3 Construction of the Field Theory . . . . . . . . . . . . 93
vii
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viii Statistical Physics of Nonequilibrium Systems
3.4 Hubbard–Stratonovich Transformation . . . . . . . . 96
3.5 The Mean-Field Theory and the Functional
Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4. Peculiarity of Calculation of Some Models 102
4.1 Special Cases of the Calculation of Path
Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2 Harmonic Lattice Model . . . . . . . . . . . . . . . . . 110
4.3 The n-Vector Model . . . . . . . . . . . . . . . . . . . . 112
4.4 Potts Model . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5 Villain and Gauss Lattice Models . . . . . . . . . . . 116
4.6 Two-Dimensional Coulomb-Gas Models . . . . . . . 119
5. Statistical Description of Condensed Matter 123
5.1 Partition Function for Model Systems . . . . . . . . . 123
5.2 Ideal Classical and Quantum Gases . . . . . . . . . . 128
5.3 Hard Spheres Model . . . . . . . . . . . . . . . . . . . . 134
5.4 Two Exactly Solvable Models of Statistical
Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.5 Gravitating Gas Model . . . . . . . . . . . . . . . . . . 148
5.6 Coulomb-like Systems . . . . . . . . . . . . . . . . . . . 152
6. Inhomogeneous Distribution in Systems
of Particles 163
6.1 Microcanonical Description of Gravitating
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.2 Spatial Distribution Function . . . . . . . . . . . . . . 174
6.3 Inhomogeneity of Self-Gravitating Systems . . . . . 184
6.4 Conditions for the Gravothermal Catastrophe . . . 189
6.5 Models with Attraction and Repulsion . . . . . . . . 209
7. Cellular Structures in Condensed Matter 216
7.1 Cellular Structures and Selection of States . . . . . . 216
7.2 Thermodynamic of Cellular Structures . . . . . . . . 218
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Contents ix
7.3 Cellular Structures in Colloids. . . . . . . . . . . . . . 223
7.4 Geometry of the Distribution of Interacting
Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
8. Statistical Description of Nonequilibrium
Systems 259
8.1 Nonequilibrium Gravitating Systems . . . . . . . . . 259
8.2 Systems with Repulsive Interaction . . . . . . . . . . 274
8.3 Saddle States of Nonequilibrium Systems . . . . . . 282
8.4 Nonequilibrium Dynamics of Universe
Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Conclusions 305
Bibliography 313
Index 323
