Chapter 8. Exact Solution of Some Interacting Systems

  1. Prof. Dr. Debashish Chowdhury1 and
  2. Prof. Dr. Dietrich Stauffer2

Published Online: 28 JAN 2005

DOI: 10.1002/3527603158.ch8

Principles of Equilibrium Statistical Mechanics

Principles of Equilibrium Statistical Mechanics

How to Cite

Chowdhury, D. and Stauffer, D. (2005) Exact Solution of Some Interacting Systems, in Principles of Equilibrium Statistical Mechanics, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, FRG. doi: 10.1002/3527603158.ch8

Author Information

  1. 1

    Indian Institute of Technology, Kanpur, India

  2. 2

    University of Cologne, Germany

Publication History

  1. Published Online: 28 JAN 2005
  2. Published Print: 27 JUL 2000

ISBN Information

Print ISBN: 9783527403004

Online ISBN: 9783527603152

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Keywords:

  • statistical mechanics;
  • exact solution of some interacting systems;
  • Ising model;
  • partition function;
  • thermostatics;
  • correlations;
  • phase transitions;
  • fluid models

Summary

This chapter contains sections titled:

  • Introduction

  • Ising Model in d = 1: Partition Function

    • Open Chain in the Absence of External Field

    • Closed Chain: Transfer Matrix Approach

    • Zeroes of the Partition Function

  • Ising Model in d = 1: Thermostatics

    • Spontaneous Magnetization

    • Magnetic Susceptibility

    • Magnetic Specific Heat

  • Ising Model in d = 1: Correlations

    • Open Chain in Zero Field

    • Closed Chain: Transfer Matrix Approach

  • Important Concepts in Phase Transitions

    • Order Parameter

    • Peierls-Griffiths Argument

    • Phase Transitions: A “Mathematical Mechanism”

    • Lower Critical Dimension

  • Critical Exponents

  • Exact Solution of Fluid Models in d = 1

    • Tonks Gas

    • Takahashi Gas

  • Chapter Summary

  • Historical Notes

    • From Ising to Peierls

    • From Kramers to Onsager

    • One-dimensional Models of Fluids

  • Problems

  • Supplementary Notes

    • Peierls-Griffiths Arguments

    • Duality and Star-Triangle Transformations; Exact Tc's

    • Relevance of the Range of the Interaction

    • Exact Solution of the Two-dimensional Ising Model

    • Exact Solution of the n-vector Model

    • Exact Solution of the Spherical Model in d-dimension