## Statistical Mechanics of Phase TransitionsThe 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 |

147 | |

### Common terms and phrases

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