Nonequilibrium statistical mechanics / Gene F. Mazenko.

Format:

Book

Author/Creator:

Mazenko, G. (Gene)

Contributor:

Emma Louise McClellan Fund.

Language:

English

Subjects (All):

Nonequilibrium statistical mechanics.

Physical Description:

xiii, 478 pages : illustrations ; 24 cm

Place of Publication:

Weinheim : Wiley-VCH ; Chichester : John Wiley [distributor], [2006]

Summary:

Nonequilibrium Statistical Mechanics offers a graduate level treatment of time dependent phenomena in condensed matter physics. Conventional ideas of linear response theory and kinetic theory are treated in detail. The general emphasis is placed on the development of generalized Langevin equations for treating nonlinear behaviour in a wide variety of systems. A complete introduction to the dynamic renormalization group is presented.

This is the third volume of a set of texts by the same author. Two books of this set have already been published (Fluctuations, Order, and Defects; Equilibrium Statistical Mechanics). The present volume is self-contained and can stand alone from the preceding volumes. From the contents: Systems out of Equilibrium, Time Dependent Phenomena in Condensed Matter Systems: Relationship of Theory and Experiment, Dynamic Critical Phenomena and Broken Symmetry, Nonlinear Systems, Unstable Growth.

Contents:

1 Systems Out of Equilibrium 1

1.1 Problems of Interest 1

1.2 Brownian Motion 6

1.2.1 Fluctuations in Equilibrium 6

1.2.2 Response to Applied Forces 13

1.4 Problems for Chapter 1 16

2 Time-Dependent Phenomena in Condensed-Matter Systems 19

2.1 Linear Response Theory 19

2.1.2 Linear Response Formalism 19

2.1.3 Time-Translational Invariance 27

2.1.4 Vector Operators 29

2.1.5 Example: The Electrical Conductivity 29

2.1.6 Example: Magnetic Resonance 32

2.1.7 Example: Relaxation From Constrained Equilibrium 37

2.1.8 Field Operators 40

2.1.9 Identification of Couplings 41

2.2 Scattering Experiments 42

2.2.1 Inelastic Neutron Scattering from a Fluid 42

2.2.2 Electron Scattering 49

2.2.3 Neutron Scattering: A More Careful Analysis 50

2.2.4 Magnetic Neutron Scattering 52

2.2.5 X-Ray and Light Scattering 55

2.4 Problems for Chapter 2 59

3 General Properties of Time-Correlation Functions 63

3.1 Fluctuation-Dissipation Theorem 63

3.2 Symmetry Properties of Correlation Functions 67

3.3 Analytic Properties of Response Functions 70

3.4 Symmetries of the Complex Response Function 73

3.5 The Harmonic Oscillator 75

3.6 The Relaxation Function 77

3.8 The Classical Limit 82

3.9 Example: The Electrical Conductivity 83

3.10 Nyquist Theorem 85

3.11 Dissipation 87

3.12 Static Susceptibility (Again) 89

3.13 Sum Rules 91

3.15 Problems for Chapter 3 96

4 Charged Transport 101

4.2 The Equilibrium Situation 101

4.3 The Nonequilibrium Case 104

4.3.1 Setting up the Problem 104

4.3.2 Linear Response 106

4.4 The Macroscopic Maxwell Equations 113

4.5 The Drude Model 116

4.5.1 Basis for Model 116

4.5.2 Conductivity and Dielectric Function 118

4.5.3 The Current Correlation Function 119

4.7 Problems for Chapter 4 121

5 Linearized Langevin and Hydrodynamical Description of Time-Correlation Functions 123

5.2 Spin Diffusion in Itinerant Paramagnets 124

5.2.1 Continuity Equation 124

5.2.2 Constitutive Relation 126

5.2.3 Hydrodynamic Form for Correlation Functions 128

5.2.4 Green-Kubo Formula 130

5.3 Langevin Equation Approach to the Theory of Irreversible Processes 134

5.3.1 Choice of Variables 134

5.3.2 Equations of Motion 134

5.3.3 Example: Heisenberg Ferromagnet 136

5.3.4 Example: Classical Fluid 138

5.3.6 Generalized Langevin Equation 142

5.3.7 Memory-Function Formalism 144

5.3.8 Memory-Function Formalism: Summary 147

5.3.9 Second Fluctuation-Dissipation Theorem 147

5.4 Example: The Harmonic Oscillator 149

5.5 Theorem Satisfied by the Static Part of the Memory Function 154

5.6 Separation of Time Scales: The Markoff Approximation 155

5.7 Example: Brownian Motion 156

5.8 The Plateau-Value Problem 158

5.9 Example: Hydrodynamic Behavior; Spin-Diffusion Revisited 161

5.10 Estimating the Spin-Diffusion Coefficient 165

5.12 Problems for Chapter 5 171

6 Hydrodynamic Spectrum of Normal Fluids 175

6.2 Selection of Slow Variables 175

6.3 Static Structure Factor 177

6.4 Static Part of the Memory Function 182

6.5 Spectrum of Fluctuations with No Damping 188

6.6 Dynamic Part of the Memory Function 191

6.7 Transverse Modes 192

6.8 Longitudinal Modes 194

6.9 Fluctuation Spectrum Including Damping 196

6.11 Problems for Chapter 6 203

7 Kinetic Theory 205

7.2 Boltzmann Equation 206

7.2.1 Ideal Gas Law 206

7.2.2 Mean-Free Path 210

7.2.3 Boltzmann Equation: Kinematics 213

7.2.4 Boltzmann Collision Integral 215

7.2.5 Collisional Invariants 219

7.2.6 Approach to Equilibrium 221

7.2.7 Linearized Boltzmann Collision Integral 223

7.2.8 Kinetic Models 225

7.2.9 Single-Relaxation-Time Approximation 228

7.2.10 Steady-State Solutions 231

7.3 Traditional Transport Theory 233

7.3.1 Steady-State Currents 233

7.3.2 Thermal Gradients 237

7.3.3 Shear Viscosity 241

7.3.4 Hall Effect 244

7.4 Modern Kinetic Theory 246

7.4.1 Collisionless Theory 250

7.4.2 Noninteracting Gas 252

7.4.3 Vlasov Approximation 253

7.4.4 Dynamic Part of Memory Function 256

7.4.5 Approximations 257

7.4.6 Transport Coefficients 260

7.6 Problems for Chapter 7 266

8 Critical Phenomena and Broken Symmetry 271

8.1 Dynamic Critical Phenomena 271

8.1.1 Order Parameter as a Slow Variable 271

8.1.2 Examples of Order Parameters 273

8.1.3 Critical Indices and Universality 277

8.1.4 The Scaling Hypothesis 277

8.1.5 Conventional Approximation 279

8.2 More on Slow Variables 283

8.3 Spontaneous Symmetry Breaking and Nambu-Goldstone Modes 285

8.4 The Isotropic Ferromagnet 286

8.5 Isotropic Antiferromagnet 290

8.6 Summary 295

8.8 Problems for Chapter 8 296

9 Nonlinear Systems 299

9.1 Historical Background 299

9.2 Motivation 301

9.3 Coarse-Grained Variables and Effective Hamiltonians 302

9.4 Nonlinear Coarse-Grained Equations of Motion 306

9.4.1 Generalization of Langevin Equation 306

9.4.2 Streaming Velocity 307

9.4.3 Damping Matrix 310

9.4.4 Generalized Fokker-Planck Equation 311

9.4.5 Nonlinear Langevin Equation 312

9.5 Discussion of the Noise 314

9.5.2 Gaussian Noise 314

9.5.3 Second Fluctuation-Dissipation Theorem 315

9.7 Examples of Nonlinear Models 317

9.7.1 TDGL Models 317

9.7.2 Isotropic Magnets 320

9.7.3 Fluids 322

9.8 Determination of Correlation Functions 326

9.8.1 Formal Arrangements 326

9.8.2 Linearized Theory 329

9.8.3 Mode-Coupling Approximation 329

9.8.4 Long-Time Tails in Fluids 330

9.9 Mode Coupling and the Glass Transition 335

9.10 Mode Coupling and Dynamic Critical Phenomena 336

9.12 Problems for Chapter 9 338

10 Perturbation Theory and the Dynamic Renormalization Group 343

10.1 Perturbation Theory 343

10.1.1 TDGL Model 343

10.1.2 Zeroth-Order Theory 344

10.1.3 Bare Perturbation Theory 345

10.1.4 Fluctuation-Dissipation Theorem 349

10.1.5 Static Limit 352

10.1.6 Temperature Renormalization 356

10.1.7 Self-Consistent Hartree Approximation 360

10.1.8 Dynamic Renormalization 361

10.2 Perturbation Theory for the Isotropic Ferromagnet 369

10.2.1 Equation of Motion 369

10.2.2 Graphical Expansion 370

10.2.3 Second Order in Perturbation Theory 374

10.3 The Dynamic Renormalization Group 379

10.3.1 Group Structure 379

10.3.2 TDGL Case 380

10.3.3 Scaling Results 388

10.3.4 Wilson Matching 390

10.3.5 Isotropic Ferromagnet 392

10.6 Problems for Chapter 10 400

11 Unstable Growth 403

11.2 Langevin Equation Description 407

11.3 Off-Critical Quenches 411

11.4 Nucleation 413

11.5 Observables of Interest in Phase-Ordering Systems 416

11.6 Consequences of Sharp Interfaces 418

11.7 Interfacial motion 420

11.8 Scaling 423

11.9 Theoretical Developments 425

11.9.1 Linear Theory 425

11.9.2 Mean-Field Theory 426

11.9.3 Auxiliary Field Methods 428

11.9.4 Auxiliary Field Dynamics 432

11.9.5 The Order Parameter Correlation Function 435

11.9.6 Extension to n-Vector Model 437

11.10 Defect Dynamics 439

11.11 Pattern Forming Systems 445

11.13 Problems for Chapter 11 450

A Time-Reversal Symmetry 455

B Fluid Poisson Bracket Relations 461

C Equilibrium Average of the Phase-Space Density 463

D Magnetic Poisson Bracket Relations 465

E Noise and the Nonlinear Langevin Equation 467.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Emma Louise McClellan Fund.

ISBN:

3527406484

9783527406487

OCLC:

65768372

Publisher Number:

9783527406487