# Mathematical Physics in One Dimension

## 1st Edition

### Exactly Soluble Models of Interacting Particles

**Authors:**Elliott H. Lieb Daniel C. Mattis

**eBook ISBN:**9781483218564

**Imprint:**Academic Press

**Published Date:**1st January 1966

**Page Count:**580

## Description

Mathematical Physics in One Dimension: Exactly Soluble Models of Interacting Particles covers problems of mathematical physics with one-dimensional analogs. The book discusses classical statistical mechanics and phase transitions; the disordered chain of harmonic oscillators; and electron energy bands in ordered and disordered crystals. The text also describes the many-fermion problem; the theory of the interacting boson gas; the theory of the antiferromagnetic linear chains; and the time-dependent phenomena of many-body systems (i.e., classical or quantum-mechanical dynamics). Physicists and mathematicians will find the book invaluable.

## Table of Contents

Foreword

• Chapter 1 · Classical Statistical Mechanics

Introduction

The Takahashi Nearest-Neighbor Gas

The Kac-Baker Model

The Coulomb Gas and the Statistical Theory of Energy Levels

Bibliography

Reprinted papers

A Simple Method for Treating the Statistical Mechanics of One-Dimensional Substances

Sur l'Intégrale de Configuration pour les Systèmes de Particules à Une Dimension

Molecular Distribution Functions in a One-Dimensional Fluid

Certain General Order-Disorder Models in the Limit of Long-Range Interactions

On the van der Waals Theory of the Vapor-Liquid Equilibrium

On the van der Waals Theory of the Vapor-Liquid Equilibrium

One-Dimensional Phase Transition in the Spherical Model of a Gas

Exact Statistical Mechanics of a One-Dimensional System with Coulomb Forces

One-Dimensional Plasma Model at Thermodynamic Equilibrium

Statistical Mechanics of a One-Dimensional Coulomb System with a Uniform Charge Background

• Chapter 2 · The Disordered Chain of Harmonic Oscillators

Introduction

Dyson's Method

Other Rigorous Solutions

Special Theorems

Bibliography

Reprinted papers

The Dynamics of a Disordered Linear Chain

Disordered One-Dimensional Crystals

Special Frequencies in the Vibrational Spectra of Disordered Chains

Some Exact Results for the Vibrational Spectrum of a Disordered Chain

Structure of the Spectra of Disordered Systems

The Vibrational Spectrum of a Disordered Linear System

Vibrations of Glass-Like Disordered Chains

Theory of Normal Vibrations of Chain Molecules with Finite Length

• Chapter 3 · Electron Energy Bands in Ordered and Disordered Crystals

General Considerations about Periodic Structures

Special Periodic Structures

Slight Deviations from Periodicity

Large Deviations from Periodicity

Bibliography

Reprinted papers

he Approximate Solution of One-Dimensional Wave Equations

Quantum Mechanics of Electrons in Crystal Lattices

New Soluble Energy Band Problem

Energy Bands and Wave Functions in Periodic Potentials

Wave Propagation in One-Dimensional Structures

Analytic Properties of Bloch Waves and Wannier Functions

A Theory of the Electrical Breakdown of Solid Dielectrics

Electron Levels in a One-Dimensional Random Lattice

Existence of Energy Gaps in One-Dimensional Liquids

The Electronic Structure of a One-Dimensional Random Alloy

Existence of Energy Gaps in the Spectrum of a One-Dimensional Atomic Chain

• Chapter 4 · The Many-Fermion Problem

Introduction

Tomonaga's Model

Exactly Soluble Model

Is Perturbation Theory Valid

Delta-Function Model

Lack of Ferromagnetism in One Dimension

General

Field Theory

Bibliography

Reprinted papers

Remarks on Bloch's Method of Sound Waves Applied to Many-Fermion Problems

Exact Solution of a Many-Fermîon System and Its Associated Boson Field

Theory of Ferromagnetism and the Ordering of Electronic Energy Levels

• Chapter 5 · The Bose Gas

Introduction

Bibliography

Reprinted papers

Exact Analysis of an Interacting Bose Gas

I. The General Solution and the Ground State

Exact Analysis of an Interacting Bose Gas

II. The Excitation Spectrum

Momentum Distribution in the Ground State of the One-Dimensional System of Impenetrable Bosons

Study of Exactly Soluble One-Dimensional N-Body Problems

• Chapter 6 · Magnetism

Origins of the Magnetic Force

Algebra of Spin Operators

Ferromagnetic Heisenberg Chain

Antiferromagnetic Heisenberg Chain

Some Exactly Soluble Models

Applications of One-Dimensional Theories

Bibliography

Reprinted papers

Two Soluble Models of an Antiferromagnetic Chain

Linear Magnetic Chains with Anisotropic Coupling

Spin-Wave Spectrum of the Antiferromagnetic Linear Chain

Magnetization Curve at Zero Temperature for the Antiferromagnetic Heisenberg Linear Chain

• Chapter 7 · Time-Dependent Phenomena

Introduction

Bibliography

Reprinted papers

Statistical Mechanics of Assemblies of Coupled Oscillators

Dynamics of a Simple Many-Body System of Hard Rods

Equipartition of Energy for Nonlinear Systems

Computer Studies of Energy Sharing and Ergodicity for

Nonlinear Oscillator Systems

## Details

- No. of pages:
- 580

- Language:
- English

- Copyright:
- © Academic Press 1966

- Published:
- 1st January 1966

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483218564