Statistical Mechanics Fundamentals and Modern Applications

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ISBN: 9780471161653 | 0471161659
Cover: Hardcover
Copyright: 11/6/1997

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Statistical Mechanics reflects the latest techniques and developments in statistical mechanics. Covering a variety of concepts and topics - molecular dynamic methods, renormalization theory, chaos, polymer chain folding, oscillating chemical reactions, and cellular automata. 15 computer programs written in FORTRAN are provided to illustrate the concepts as well as more than 100 chapter-end exercises.

Preface

xiii

PART I ESSENTIALS

(80)

1 Classical Statistical Mechanics

(42)

1.1 Introduction

(1)

1.2 Probability Theory

(8)

1.2.1 The Random Walk Problem

(4)

1.2.2 The Gaussian Distribution

(3)

1.2.3 Poisson Distribution

(1)

1.3 Random Variables and Stochastic Processes

(4)

1.3.1 Introduction

(2)

1.3.2 Moments of Random Variables

(2)

1.4 Equilibrium Statistical Mechanics

(5)

1.4.1 Ergodic Hypothesis

(3)

1.4.2 Ensemble Theory

(1)

1.4.3 Statistical Distribution Function

(1)

1.5 Types of Ensembles

(6)

1.5.1 Microcanonical Ensemble

(2)

1.5.2 Canonical Ensemble

(2)

1.5.3 Grand Canonical Ensemble

(2)

1.6 Thermodynamics

(7)

1.6.1 Canonical and Grand Canonical Partition Functions

(2)

1.6.2 Fluctuations and the Thermodynamic Limit

(2)

1.6.3 Boltzmann Distribution

(1)

1.6.4 Classical Ideal Gas

(2)

1.7 Nonequilibrium Statistical Mechanics

(9)

1.7.1 Liouville's Theorem

(4)

1.7.2 The Joule Expansion

(1)

1.7.3 Boltzmann's H-Theorem

(4)

1.8 Exercises and Problems

(2)

2 Quantum Statistical Mechanics

(36)

2.1 Introduction

(1)

2.2 Essentials of Quantum Mechanics

(11)

2.2.1 Dirac Bra and Ket Vectors

(4)

2.2.2 Schrodinger and Heisenberg Pictures

(1)

2.2.3 Pure and Mixed States

(2)

2.2.4 The Density Matrix

(2)

2.2.5 Quantum Mechanical Liouville's Theorem

(1)

2.2.6 The Fundamental Postulates

(2)

2.3 Quantum Statistics

(10)

2.3.1 Ensembles for Independent Particle Systems

(4)

2.3.2 Quantum Gases of Independent Particles

(3)

2.3.3 Thermal Wavelength and Degeneracy

(1)

2.3.4 Partition Functions

(2)

2.4 Ideal Bose-Einstein and Fermi-Dirac gases

(11)

2.4.1 Weak Degeneracy

(1)

2.4.2 Strong Degeneracy: Bose Gas

(5)

2.4.3 Strong Degeneracy: Fermi Gas

(3)

2.4.4 Example: Black-Body Radiation

(1)

2.4.5 Example: Electrons in Semiconductors

(1)

2.5 Exercises and Problems

(3)

PART II EQUILIBRIUM STATISTICAL MECHANICS

(112)

3 Phase Transitions and Critical Phenomena

(33)

3.1 Introduction

(2)

3.2 Broken Symmetry and Critical Phenomena

(8)

3.2.1 Ising Model: Introduction

(2)

3.2.2 Ising Model: Exact Solution in One Dimension

(2)

3.2.3 Thermodynamic Limit of the Linear Ising Model

(1)

3.2.4 Important Definitions

(3)

3.3 Mean-Field Theories

(10)

3.3.1 Weiss Mean-Field Theory of Ferromagnetism

(1)

3.3.2 Weiss Theory: Critical Behavior

(3)

3.3.3 Problems with the Weiss Approximation

(1)

3.3.4 The van der Waals Theory

(4)

3.3.5 Landau Theory

102

(1)

3.4 Scaling and Universality

103

(7)

3.4.1 Scaling in Mean-Field Theories

103

(2)

3.4.2 Scaling Beyond the Mean-Field Theory

105

(2)

3.4.3 Universality

107

(1)

3.4.4 Symmetry

107

(1)

3.4.5 Dimensionality

108

(1)

3.4.6 Range of Interactions

109

(1)

3.5 Renormalization Group

110

(5)

3.5.1 The Ising Model and the Kadanoff Construction

110

(2)

3.5.2 The Ising Hamiltonian and the RG

112

(3)

3.6 Exercises and Problems

115

(1)

4 The Liquid State

116

(26)

4.1 Introduction

116

(1)

4.2 Distribution Functions

117

(6)

4.2.1 Configurational Partition Function

117

(1)

4.2.2 Reduced Distribution Function

118

(1)

4.2.3 Radial Distribution Function

119

(4)

4.3 Thermodynamic Functions

123

(2)

4.4 The Kirkwood Equation

125

(2)

4.4.1 An Exact Equation for g(r)

125

(1)

4.4.2 Potential of Mean Force

125

(1)

4.4.3 Superposition Approximation

126

(1)

4.5 Takahashi One-Dimensional Fluid Models

127

(9)

4.5.1 Hard-Sphere Potential

127

(4)

4.5.2 Other Potentials

131

(1)

4.5.3 Radial Distribution Function

132

(4)

4.6 Ornstein-Zernike Equation

136

(4)

4.6.1 The Direct Correlation Function

136

(1)

4.6.2 Light Scattering Near Critical Points

137

(1)

4.6.3 Ornstein-Zernike Approach

138

(2)

4.7 Concluding Remarks

140

(1)

4.8 Exercises and Problems

140

(2)

5 Molecular Dynamics Methods

142

(14)

5.1 Introduction

142

(1)

5.2 Early Achievements

142

(2)

5.3 Analytic Methods versus Simulation

144

(2)

5.4 A Typical MD Simulation

146

(8)

5.4.1 Start of the MD Simulation

146

(3)

5.4.2 Predictor-Corrector Method

149

(2)

5.4.3 Equilibration

151

(3)

5.5 Other Techniques

154

(1)

5.6 Exercises and Problems

155

(1)

6 Monte Carlo Methods

156

(19)

6.1 Introduction

156

(1)

6.2 Simple Monte Carlo

156

(2)

6.2.1 Random Experiments

156

(1)

6.2.2 Numerical Integration

157

(1)

6.3 Markovian Chains

158

(3)

6.3.1 Importance Sampling and the Metropolis Method

158

(1)

6.3.2 Transition Probability Matrix

159

(1)

6.3.3 Example: The Random Walk

160

(1)

6.4 Example of Metropolis Monte Carlo

161

(4)

6.4.1 The Ising Model

161

(1)

6.4.2 Metropolis Monte Carlo Applied to the Ising Model

162

(3)

6.5 Non-Boltzmann Sampling

165

(2)

6.5.1 Helmholtz Free Energy

165

(1)

6.5.2 Umbrella Sampling

166

(1)

6.6 Monte Carlo Renormalization Group

167

(5)

6.6.1 The Ising Model

167

(1)

6.6.2 MCRG

167

(5)

6.7 Concluding Remarks

172

(1)

6.8 Exercises and Problems

173

(2)

7 Polymers, Proteins, and Spin Glass Models

175

(18)

7.1 Introduction

175

(1)

7.2 Homopolymers

176

(6)

7.2.1 Self-Avoiding Random Walk and Criticality

176

(3)

7.2.2 The Coil-To-Globule Folding Transition

179

(3)

7.3 Heteropolymers: Proteins

182

(4)

7.3.1 Introduction

182

(1)

7.3.2 HP Lattice Model

183

(2)

7.3.3 Monte Carlo Methods

185

(1)

7.4 Spin Glass Models

186

(5)

7.4.1 Introduction

186

(1)

7.4.2 EA and SK Models

187

(1)

7.4.3 Application of Spin Glasses in Other Fields

188

(1)

7.4.4 Spin Glasses and Polymers

189

(2)

7.5 Exercises and Problems

191

(2)

PART III NONEQUILIBRIUM STATISTICAL MECHANICS

193

(158)

8 The Boltzmann Equation

195

(17)

8.1 Introduction

195

(2)

8.2 Boltzmann Theory

197

(1)

8.3 Binary Collisions

198

(4)

8.3.1 Elastic Collisions

198

(3)

8.3.2 Scattering Cross Sections

201

(1)

8.4 Boltzmann Equation

202

(6)

8.4.1 Derivation of Boltzmann Equation

202

(1)

8.4.2 Collision Term

203

(3)

8.4.3 Equilibrium

206

(2)

8.5 Transport Properties

208

(2)

8.5.1 Hydrodynamic Equations of Change

208

(1)

8.5.2 Transport Equations

209

(1)

8.6 Concluding Remarks

210

(1)

8.7 Exercises and Problems

211

(1)

9 Approaches to Brownian Motion

212

(22)

9.1 Introduction

212

(1)

9.2 Nature of Brownian Motion

213

(1)

9.3 Mathematical Preliminaries

213

(5)

9.3.1 Markovian Processes

213

(2)

9.3.2 Fokker-Planck Equation

215

(3)

9.4 Diffusion of a Brownian Particle

218

(5)

9.4.1 Fick's Laws

218

(1)

9.4.2 Einstein's 1905 Equation

219

(1)

9.4.3 Brownian Particle in a Gravitational Field

220

(3)

9.5 The Langevin Equation

223

(3)

9.6 Brownian Motion as a Time Series

226

(2)

9.7 Microscopic Derivation of the Langevin Equation

228

(4)

9.8 Brownian Motion and Irreversibility

232

(1)

9.9 Exercises and Problems

233

(1)

10 Zwanzig-Mori Formalism

234

(22)

10.1 Introduction

234

(1)

10.2 Linear Response Theory

235

(2)

10.3 Zwanzig Kinetic Equation

237

(5)

10.3.1 Hilbert Space Formulation of Dynamics

237

(2)

10.3.2 Projection Operators

239

(1)

10.3.3 Derivation of the Kinetic Equation

240

(1)

10.3.4 Discussion of Zwanzig's Equation

241

(1)

10.4 Generalized Langevin Equation

242

(4)

10.4.1 Mori's Approach

242

(2)

10.4.2 Correlation Functions

244

(1)

10.4.3 Volterra Equations and the Continued Fraction

245

(1)

10.5 Example: Spectral Line Shape Analysis

246

(4)

10.5.1 Dipolar Absorption of Radiation

246

(1)

10.5.2 Pure Rotational Spectrum

247

(3)

10.6 Example: Hydrodynamics of Flexible Polymers

250

(5)

10.6.1 A Polymer Model

250

(1)

10.6.2 Fluid Dynamics

250

(1)

10.6.3 The Rouse-Zimm Model

251

(2)

10.6.4 Optimized Rouse-Zimm Model

253

(2)

10.7 Exercises and Problems

255

(1)

11 Activated Barrier-Crossing Problem

256

(25)

11.1 Introduction

256

(2)

11.2 Transition State Theory

258

(4)

11.2.1 Arrhenius Law

258

(1)

11.2.2 Assumptions

259

(1)

11.2.3 Derivation of the TST Rate Constant

259

(2)

11.2.4 Discussion of the TST

261

(1)

11.3 Kramers Theory

262

(5)

11.3.1 Introduction

262

(1)

11.3.2 High-Damping Limit

263

(3)

11.3.3 Low-Damping Limit

266

(1)

11.4 Grote-Hynes theory

267

(8)

11.4.1 Introduction

267

(1)

11.4.2 Memory Friction Kernel

268

(1)

11.4.3 The Case N = 1

269

(2)

11.4.4 Derivation of the GH Rate Coefficient

271

(3)

11.4.5 Discussion of the GH Rate Coefficient

274

(1)

11.5 Critical Phenomena in the GH Theory

275

(4)

11.5.1 Singularity in the GH Theory

275

(1)

11.5.2 Scaling and Universality in the GH Theory

276

(3)

11.6 Concluding Remarks

279

(1)

11.7 Exercises and Problems

279

(2)

12 Oscillating Chemical Reactions and Chaos

281

(43)

12.1 Introduction

281

(1)

12.2 History of Oscillating Chemical Reactions

282

(2)

12.2.1 History to 1970

282

(1)

12.2.2 History after 1970

283

(1)

12.3 The Lotka-Volterra Model

284

(8)

12.3.1 Solution of the Lotka-Volterra Equations

284

(6)

12.3.2 Statistical Mechanics and the Lotka-Volterra Model

290

(2)

12.4 The Brusselator Model

292

(11)

12.4.1 Solution of the Brusselator Equations

292

(9)

12.4.2 Statistical Characterization of the Brusselator Model

301

(2)

12.5 The Approach to Chaos in Dynamical Systems

303

(12)

12.5.1 Nonlinear Dynamics

303

(1)

12.5.2 Difference Equations

304

(1)

12.5.3 The Logistic Equation

305

(6)

12.5.4 Characterization of the Logistic Equation

311

(2)

12.5.5 Universality in Chaos

313

(2)

12.6 The Belousov-Zhabotinskii Reaction

315

(7)

12.6.1 The Oregonator

315

(1)

12.6.2 Experimental Observations

315

(6)

12.6.3 Statistical Mechanics of the BZ Reaction

321

(1)

12.7 Exercises and Problems

322

(2)

13 Introduction to Cellular Automation Models

324

(27)

13.1 Introduction

324

(1)

13.2 One-Dimensional Cellular Automata

325

(7)

13.2.1 A Brief History

325

(1)

13.2.2 Elementary Cellular Automata

325

(2)

13.2.3 Three Classes of Cellular Automata

327

(5)

13.3 Two-dimensional Cellular Automata

332

(10)

13.3.1 Neighborhood Structures

332

(1)

13.3.2 The HPP Model

333

(2)

13.3.3 Hydrodynamics of the HPP Model

335

(1)

13.3.4 Reaction Dynamics on a Cellular Automaton

336

(6)

13.4 Fractals

342

(6)

13.4.1 Introduction

342

(1)

13.4.2 Fractal Dimension

343

(2)

13.4.3 Fractal Nature of Cellular Automaton 126

345

(1)

13.4.4 Fractal Nature of Brownian Motion

346

(1)

13.4.5 The Box Dimension

347

(1)

13.4.6 Fractal Nature of Polymers

347

(1)

13.5 Exercises and Problems

348

(3)

Appendix A Resources on the Internet and the World Wide Web

351

(2)

Appendix B Generating Functions

353

(4)

B.1 The Generating Function Concept

353

(1)

B.2 The Moment Generating Function

354

(3)

Appendix C Fermi's Golden Rule

357

(2)

C.1 Transition Probability in Time-Dependent Perturbation Theory

357

(1)

C.2 Use of the Delta Function

358

(1)

Appendix D FORTRAN Programs

359

(22)

D.1 BRNMTN

360

(1)

D.2 BRSLTR

361

(1)

D.3 CELLAUT

362

(1)

D.4 GSSWTN

363

(1)

D.5 ISING

364

(3)

D.6 LJMDFORID

367

(3)

D.7 LOGIS1

370

(1)

D.8 LOGIS2

371

(1)

D.9 LOTKA

372

(1)

D.10 MCINT

373

(1)

D.11 POLYMER

374

(2)

D.12 PWRSPR

376

(1)

D.13 REVCA

377

(2)

D.14 SQRCA

379

(2)

Bibliography

381

(10)

Index

391

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Statistical Mechanics Fundamentals and Modern Applications

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