## Phasetransitions: A Brief Account with Modern ApplicationsThis 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. |

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### Contents

The Ising Model | 13 |

Mean Field Theory | 25 |

Scaling | 37 |

The Renormalization Group | 49 |

Phase Transitions in Quantum Systems | 63 |

Universality | 81 |

Random and Small World Systems | 99 |

SelfOrganized Criticality | 113 |

Bibliography | 129 |

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### Common terms and phrases

2D Ising model atoms average behavior block lattice bonds Bose-Einstein condensation bosons calculations cluster coefficients configurations connected Cooper pair correlation length corresponds critical indices critical point critical temperature Tc defined density dependence described dimensional disordered electrons energy F entropy equation equilibrium exponential ferromagnetic field H finite fixed point fluctuations follows free energy Hamiltonian Heisenberg model high-Tc ij)nn infinite integral interactions Ising model Landau theory leads let us consider low temperatures magnetic field mean field theory nearest neighbors networks non-degenerate number of particles obtained one-dimensional Onsager order parameter order phase transition partition function percolation phase transition occurs phenomena Phys physical power law distribution previous chapter problem properties quantum random graphs renormalization group short-range shown in Fig singularity solution spherical model spin glasses square lattice superconductivity superfluidity symmetry thermodynamic tion universality class values velocity vortices wave function x-y model zero