Non-Equilibrium Statistical Mechanics

✍ Scribed by James H. Luscombe

• Publisher: CRC Press
• Year: 2025
• Tongue: English
• Leaves: 257
• Category: Library

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
CHAPTER 1: Irreversibility, entropy, and fluctuations
1.1. ENTROPY AND IRREVERSIBILITY, CLAUSIUS INEQUALITY
1.2. EQUILIBRIUM, GLOBAL AND LOCAL
1.3. BALANCE EQUATIONS: FLUXES AND SOURCES
1.4. HYDRODYNAMIC CONSERVATION LAWS
1.4.1. Mass conservation, convective derivative
1.4.2. Momentum balance, stress tensor
1.4.3. Energy conservation, heat flux
1.5. ENTROPY SOURCES: FLUXES AND THERMO-FORCES
1.6. LINEAR FORCE-FLUX RELATIONS, KINETIC COEFFICIENTS
1.7. FORCES AND FLUXES LINKED THOUGH FLUCTUATIONS
1.8. THERMOELECTRICITY, KELVIN RELATION
1.9. STEADY STATES AND MINIMUM ENTROPY PRODUCTION
CHAPTER 2: Fluctuations as stochastic processes
2.1. EINSTEIN FLUCTUATION THEORY
2.1.1. Characteristic functions and moments
2.1.2. Force-fluctuation correlations
2.2. MICROSCOPIC REVERSIBILITY AND ONSAGER RECIPROCITY
2.3. STOCHASTIC PROCESSES: ADDING TIME TO PROBABILITY
2.3.1. The consistency conditions
2.3.2. Stochastic calculus
2.4. STATIONARY, INDEPENDENT, AND MARKOV PROCESSES
2.4.1. Stationary processes: No unique origin of time
2.4.2. Independent processes: Present is independent of the past
2.4.3. Markov processes: Present depends on the immediate past
2.5. ERGODIC AND SPECTRAL PROPERTIES
2.5.1. Ergodicity
2.5.2. Spectral analysis, Wiener-Khinchin theorem
2.6. THERMAL NOISE, NYQUIST THEOREM
2.7. THE RANDOM WALK IN ONE DIMENSION, DIFFUSION
2.7.1. Asymptotic form
2.7.2. Passing to the continuous limit, Einstein diffusion equation
2.7.3. Wiener-Levy process
2.8. THE MASTER EQUATION
2.8.1. Derivation
2.8.2. Detailed balance
2.8.3. Matrix form
2.8.4. Eigenfunction expansion
2.9. GAUSSIAN PROCESSES
2.9.1. Gaussian distributions and cumulants
2.9.2. Multivariate Gaussian distributions
2.9.3. Many-time cumulants
2.9.4. Gaussian stochastic processes, Doob’s theorem
CHAPTER 3: Brownian motion and stochastic dynamics
3.1. LANGEVIN EQUATION, EINSTEIN RELATION
3.1.1. Thermal noise
3.1.2. Brownian motion of free particles; the random force
3.1.3. Uniform external force; drift speed
3.2. LANGEVIN EQUATION AS A STOCHASTIC PROCESS
3.3. FOKKER-PLANCK EQUATION FOR ONE RANDOM VARIABLE
3.3.1. Derivation
3.3.2. Free Brownian particles in spatially homogeneous systems
3.3.3. Ornstein-Uhlenbeck process
3.4. BROWNIAN PARTICLES IN EXTERNAL FORCE FIELDS
3.4.1. The stationary solution
3.4.2. Passage over potential barriers: The Kramers escape problem
3.4.3. The field-free case with spatially dependent initial conditions
3.4.4. Strong damping regime, Smoluchowski equation
3.4.5. Uniform field
CHAPTER 4: Kinetic theory: Boltzmann’s approach to irreversibility
4.1. FROM MECHANICS TO STATISTICAL MECHANICS
4.2. REDUCED PROBABILITY DISTRIBUTIONS
4.3. DYNAMICS OF REDUCED DISTRIBUTIONS: THE HIERARCHY
4.4. HYDRODYNAMICS AND THE HIERARCHY
4.4.1. Densities macroscopic and microscopic
4.4.2. Balance equations from the hierarchy
4.4.3. Microscopic balance equations
4.4.4. Hydrodynamics: Adding constitutive relations to conservation laws
4.4.5. Linearized hydrodynamics, normal modes
4.5. THE BOLTZMANN EQUATION, MOLECULAR CHAOS
4.5.1. Derivation
4.5.2. Comments
4.6. BOLTZMANN’S H-THEOREM, CONNECTION WITH ENTROPY
4.6.1. Proof of the H-theorem
4.6.2. Equilibrium, collisional invariants, Maxwell-Boltzmann distribution
4.6.3. Connection with the second law, Gibbs and Boltzmann entropies
4.6.4. Coarse graining and loss of information
4.6.5. The reversibility paradox, not
4.7. COLLISION FREQUENCY, MEAN FREE PATH
4.8. NORMAL SOLUTIONS OF THE BOLTZMANN EQUATION
4.8.1. Hilbert’s theorem, hydrodynamic regime
4.8.2. Chapman-Enskog method
4.9. CHAPMAN-ENSKOG THEORY OF TRANSPORT COEFFICIENTS
4.9.1. The heat and momentum fluxes associated with f(1)
4.9.2. Calculating A and B
CHAPTER 5: Weakly coupled systems: Landau-Vlasov theories
5.1. HOMOGENEOUS WEAK COUPLING: THE LANDAU EQUATION
5.2. CONNECTION WITH THE FOKKER-PLANCK EQUATION
5.3. INHOMOGENEOUS PLASMAS: THE VLASOV EQUATION
5.4. LANDAU DAMPING
5.4.1. The linearized Vlasov equation
5.4.2. Initial value problem
CHAPTER 6: Dissipation, fluctuations, and correlations
6.1. FLUCTUATIONS AND THEIR CORRELATIONS
6.2. LINEAR RESPONSE THEORY
6.2.1. Response functions: General derivations, classical and quantum
6.2.2. The generalized susceptibility χ(ω) and its analytic properties
6.2.3. Identification of χ′′(ω) with dissipation
6.2.4. Explicit formulae for the quantum response
6.2.5. Brownian motion of a harmonically bound classical particle
6.2.6. The relaxation function
6.3. FLUCTUATION-DISSIPATION THEOREM
6.4. GREEN-KUBO THEORY OF TRANSPORT COEFFICIENTS
6.4.1. Mechanical transport processes
6.4.2. Thermal transport processes
6.4.3. Einstein relation
6.4.4. Comments
6.5. GENERALIZED LANGEVIN EQUATION
6.5.1. Derivation
6.5.2. Recovering the Langevin equation for Brownian motion
6.6. MEMORY FUNCTIONS
APPENDIX A: Statistical mechanics
A.1. CLASSICAL ENSEMBLES: PROBABILITY DENSITY FUNCTIONS
A.1.1. Classical dynamics of many particle systems; Γ-space
A.1.2. The transition to statistical mechanics
A.1.3. Canonical ensemble, closed systems
A.1.4. Grand canonical ensemble, open systems
A.2. QUANTUM ENSEMBLES: PROBABILITY DENSITY OPERATORS
A.2.1. Quantum indeterminacy
A.2.2. Many-particle wave functions
A.2.3. The density operator
A.2.4. Canonical ensemble
A.2.5. Grand canonical ensemble
A.3. ENTROPY
A.4. THE WEYL CORRESPONDENCE PRINCIPLE
APPENDIX B: Probability theory
B.1. EVENTS, SAMPLE SPACE, AND PROBABILITY
B.2. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
B.2.1. Probability distributions on discrete sample spaces; joint distributions
B.2.2. Probability densities on continuous sample spaces; joint densities
B.2.3. Moments of distributions
APPENDIX C: Elastic collisions
APPENDIX D: Integral equations and resolvents
APPENDIX E: Dynamical representations in quantum mechanics
APPENDIX F: Causality and analyticity
Bibliography
Index