Statistical Mechanics of Phase Transitions

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Clarendon Press, May 7, 1992 - 164 pages
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The book provides an introduction to the physics which underlies phase transitions and to the theoretical techniques currently at our disposal for understanding them. It will be useful for advanced undergraduates, for post-graduate students undertaking research in related fields, and for established researchers in experimental physics, chemistry, and metallurgy as an exposition of current theoretical understanding. - ;Recent developments have led to a good understanding of universality; why phase transitions in systems as diverse as magnets, fluids, liquid crystals, and superconductors can be brought under the same theoretical umbrella and well described by simple models. This book describes the physics underlying universality and then lays out the theoretical approaches now available for studying phase transitions. Traditional techniques, mean-field theory, series expansions, and the transfer matrix, are described; the Monte Carlo method is covered, and two chapters are devoted to the renormalization group, which led to a break-through in the field. The book will be useful as a textbook for a course in `Phase Transitions', as an introduction for graduate students undertaking research in related fields, and as an overview for scientists in other disciplines who work with phase transitions but who are not aware of the current tools in the armoury of the theoretical physicist. - ;Introduction; Statistical mechanics and thermodynamics; Models; Mean-field theories; The transfer matrix; Series expansions; Monte Carlo simulations; The renormalization group; Implementations of the renormalization group. -
 

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Contents

Introduction
1
11 Phase transitions in other systems
4
112 Surfactants in solution
7
12 A microscopic model
8
121 A renormalization group
13
Statistical mechanics and thermodynamics
15
22 Thermodynamics
16
23 Convexity properties of the free energy
19
544 The correlation length
74
Series expansions
79
61 High temperature series expansions
80
62 Low temperature series expansions
85
63 The onedimensional Ising model
86
64 Analysis of series expansions
88
65 Problems
92
Monte Carlo simulations
95

24 Correlation functions
20
25 Firstorder and continuous phase transitions
21
26 Critical point exponents
25
261 Universality
27
262 Exponent inequalities
29
27 Problems
31
Models
33
31 The spin12 Ising model
35
311 Orderdisorder transitions in binary alloys
36
312 Lattice gas models
39
32 The spin1 Ising model
41
34 XY and Heisenberg models
43
35 Universality revisited
45
36 Discussion
47
37 Problems
48
Meanfield theories
50
411 Meanfield critical exponents
53
42 Landau theory
54
421 Meanfield critical exponents revisited
56
43 The correlation function
57
44 Classical meanfield theories
59
441 Van der Waals theory of fluids
60
442 Weiss theory of magnetism
61
46 Problems
63
The transfer matrix
67
52 The free energy
69
53 The correlation function
70
54 Results for the Ising model
72
541 The free energy
73
72 Practical details
97
73 Considerations in the data analysis
100
732 Statistical errors
101
733 Finitesize corrections
102
742 More complicated systems
103
75 Problem
104
The renormalization group
105
81 Definition of a renormalization group transformation
106
82 Flows in parameter space
108
83 Universality
112
84 An example
113
85 Scaling and critical exponents
115
86 Scaled variables
118
87 Conformal invariance
120
88 Problems
121
Implementations of the renormalization group
124
911 Derivation of the recursion equations
125
912 Fixed points
127
913 Fixed points and scaling
129
914 The free energy
130
92 Higher dimensions
132
93 The gstate Potts model
136
94 The Monte Carlo renormalization group
139
95 The eexpansion
140
96 Problems
141
Further reading
145
Index
147
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