Phase Transitions. A Brief Account With Modern Applications

Moshe Gitterman, Vivian Haim Halpern9789812389039, 981-238-903-2

This book presents a short, fairly simple course on the basic theory of phase transitions and its modern applications. In physics, these applications include such modern developments as Bose–Einstein condensation of atoms, high temperature superconductivity, and vortices in superconductors, while in other fields they include small world phenomena and scale-free systems (such as stock markets and the Internet). The advantage of treating all these topics together lies in showing their connection with one another and with the general theory of phase transitions.

---

Table of contents :
Contents……Page 6
Preface……Page 10
1. Phases and Phase Transitions……Page 12
1.1 Classification of Phase Transitions……Page 15
1.2 Appearance of a Second Order Phase Transition……Page 18
1.3 Correlations……Page 20
1.4 Conclusion……Page 22
2. The Ising Model……Page 24
2.1 1D Ising model……Page 27
2.2 2D Ising model……Page 28
2.3 3D Ising model……Page 31
2.4 Conclusion……Page 34
3. Mean Field Theory……Page 36
3.1 Landau Mean Field Theory……Page 37
3.2 First Order Phase Transitions in Landau Theory……Page 40
3.3 Landau Theory Supplemented with Fluctuations……Page 41
3.5 Ginzburg Criterion……Page 43
3.6 Wilson’s -Expansion……Page 44
3.7 Conclusion……Page 47
4. Scaling……Page 48
4.1 Relations Between Thermodynamic Critical Indices……Page 50
4.2 Scaling Relations……Page 52
4.3 Dynamic Scaling……Page 56
4.4 Conclusion……Page 58
5.1 Fixed Points of a Map……Page 60
5.2 Basic Idea of the Renormalization Group……Page 62
5.3 RG: 1D Ising Model……Page 64
5.4 RG: 2D Ising Model for the Square Lattice (1)……Page 65
5.5 RG: 2D Ising Model for the Square Lattice (2)……Page 68
5.6 Conclusion……Page 71
6.1 Symmetry of the Wave Function……Page 74
6.2 Exchange Interactions of Fermions……Page 76
6.3 Quantum Statistical Physics……Page 78
6.4 Superfluidity……Page 82
6.5 Bose–Einstein Condensation of Atoms……Page 83
6.6 Superconductivity……Page 84
6.7 High Temperature (High-Tc) Superconductors……Page 89
6.8 Conclusion……Page 91
7.1 Heisenberg Ferromagnet and Related Models……Page 92
7.2 Many-Spin Interactions……Page 96
7.3 Gaussian and Spherical Models……Page 97
7.4 The x–y Model……Page 99
7.5 Vortices……Page 103
7.6 Interactions Between Vortices……Page 104
7.7 Vortices in Superfluids and Superconductors……Page 106
7.8 Conclusion……Page 107
8.1 Percolation……Page 110
8.2 Ising Model with Random Interactions……Page 112
8.3 Spin Glasses……Page 114
8.4 Small World Systems……Page 116
8.5 Evolving Graphs……Page 120
8.6 Phase Transitions in Small World Systems……Page 121
8.7 Conclusion……Page 123
9. Self-Organized Criticality……Page 124
9.1 Power Law Distributions……Page 126
9.2 Sand Piles……Page 128
9.3 Distribution of Links in Networks……Page 129
9.4 Dynamics of Networks……Page 131
9.5 Mean Field Analysis of Networks……Page 135
9.6 Hubs in Scale-Free Networks……Page 137
9.7 Conclusion……Page 139
Bibliography……Page 140
Index……Page 144