Non-equilibrium Thermodynamics and Statistical Mechanics

Name: Non-equilibrium Thermodynamics and Statistical Mechanics | Foundations and Applications
Brand: OUP Oxford
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Foundations and Applications

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Erschienen am 10. September 2012

479 Seiten

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Inhalt

Cover
Contents
1 Prologue
1.1 Entropy and the Second Law
1.2 Time Dependent Systems
1.2.1 The Second Law is Timeless
1.2.2 The Second Entropy
1.3 Nature of Probability
1.3.1 Frequency
1.3.2 Credibility
1.3.3 Measure
1.3.4 Determination of Randomness
1.4 States, Entropy, and Probability
1.4.1 Macrostates and Microstates
1.4.2 Weight and Probability
1.4.3 Entropy
1.4.4 Transitions and the Second Entropy
1.4.5 The Continuum
1.5 Reservoirs
1.5.1 Equilibrium Systems
1.5.2 Non-Equilibrium Steady State
2 Fluctuation Theory
2.1 Gaussian Probability
2.2 Exponential Decay in Markovian Systems
2.3 Small Time Expansion
2.4 Results for Pure Parity Systems
2.4.1 Onsager Regression Hypothesis and Reciprocal Relations
2.4.2 Green-Kubo Expression
2.4.3 Physical Interpretation of the Second Entropy
2.4.4 The Dissipation
2.4.5 Stability Theory
2.4.6 Non-Reversibility of the Trajectory
2.4.7 Third Entropy
2.5 Fluctuations of Mixed Time Parity
2.5.1 Second Entropy and Time Correlation Functions
2.5.2 Small Time Expansion for the General Case
2.5.3 Magnetic Fields and Coriolis Forces
3 Brownian Motion
3.1 Gaussian, Markov Processes
3.2 Free Brownian Particle
3.3 Pinned Brownian Particle
3.4 Diffusion Equation
3.5 Time Correlation Functions
3.6 Non-Equilibrium Probability Distribution
3.6.1 Stationary Trap
3.6.2 UniformlyMoving Trap
3.6.3 Mixed Parity Formulation of the Moving Trap
3.7 Entropy, Probability, and their Evolution
3.7.1 Time Evolution of the Entropy and Probability
3.7.2 Compressibility of the Equations of Motion
3.7.3 The Fokker-Planck Equation
3.7.4 Generalised Equipartition Theorem
3.7.5 Liouville's Theorem
4 Heat Conduction
4.1 Equilibrium System
4.2 First Energy Moment and First Temperature
4.3 Second Entropy
4.4 Thermal Conductivity and Energy Correlations
4.5 Reservoirs
4.5.1 First Entropy
4.5.2 Second Entropy
4.6 Heat and Number Flow
4.7 Heat and Current Flow
5 Second Entropy for Fluctuating Hydrodynamics
5.1 Conservation Laws
5.1.1 Densities, Velocities, and Chemical Reactions
5.1.2 Number Flux
5.1.3 Energy Flux
5.1.4 Linear Momentum
5.2 Entropy Density and its Rate of Change
5.2.1 Sub-system Dissipation
5.2.2 Steady State
5.3 Second Entropy
5.3.1 Variational Principle
5.3.2 Flux Optimisation
5.4 Navier-Stokes and Energy Equations
6 Heat Convection and Non-Equilibrium Phase Transitions
6.1 Hydrodynamic Equations of Convection
6.1.1 Boussinesq Approximation
6.1.2 Conduction
6.1.3 Convection
6.2 Total First Entropy of Convection
6.3 Algorithmfor Ideal Straight Rolls
6.3.1 Hydrodynamic Equations
6.3.2 Fourier Expansion
6.3.3 Nusselt Number
6.4 Algorithmfor the Cross Roll State
6.4.1 Hydrodynamic Equations and Conditions
6.4.2 Fourier Expansion
6.5 Algorithmfor Convective Transitions
6.6 Convection Theory and Experiment
7 Equilibrium Statistical Mechanics
7.1 Hamilton's Equations of Motion
7.1.1 Classical versus Quantum Statistical Mechanics
7.2 Probability Density of an Isolated System
7.2.1 Ergodic Hypothesis
7.2.2 Time, Volume, and Surface Averages
7.2.3 Energy Uniformity
7.2.4 Trajectory Uniformity
7.2.5 Partition Function and Entropy
7.2.6 Internal Entropy of Phase Space Points
7.3 Canonical Equilibrium System
7.3.1 Maxwell-Boltzmann Distribution
7.3.2 Helmholtz Free Energy
7.3.3 Probability Distribution for Other Systems
7.3.4 Equipartition Theorem
7.4 Transition Probability
7.4.1 Stochastic Equations of Motion
7.4.2 Second Entropy
7.4.3 Mixed Parity Derivation of the Second Entropy and the Equations of Motion
7.4.4 Irreversibility and Dissipation
7.4.5 The Fokker-Planck Equation and Stationarity of the Equilibrium Probability
7.5 Evolution in Phase Space
7.5.1 Various Phase Functions
7.5.2 Compressibility
7.5.3 Liouville's Theorem
7.6 Reversibility
7.6.1 Isolated System
7.6.2 Canonical Equilibrium System
7.7 Trajectory Probability and Time Correlation Functions
7.7.1 Trajectory Probability
7.7.2 Equilibrium Averages
7.7.3 Time Correlation Functions
7.7.4 Reversibility
8 Non-Equilibrium Statistical Mechanics
8.1 General Considerations
8.2 Reservoir Entropy
8.2.1 Trajectory Entropy
8.2.2 Reduction to the Point Entropy
8.2.3 Fluctuation Form for the Reservoir Entropy
8.3 Transitions and Motion in Phase Space
8.3.1 Foundations for Time Dependent Weight
8.3.2 Fluctuation Form of the Second Entropy
8.3.3 Time Correlation Function
8.3.4 Stochastic, Dissipative Equations of Motion
8.3.5 Transition Probability and Fokker-Planck Equation
8.3.6 Most Likely Force with Constraints
8.4 Changes in Entropy and Time Derivatives
8.4.1 Change in Entropy
8.4.2 Irreversibility and Dissipation
8.4.3 Various Time Derivatives
8.4.4 Steady State System
8.5 Odd Projection of the Dynamic Reservoir Entropy
8.6 Path Entropy and Transitions
8.6.1 Path Entropy
8.6.2 Fluctuation and Work Theorem
8.7 Path Entropy for Mechanical Work
8.7.1 Evolution of the Reservoir Entropy and Transitions
8.7.2 Transition Theorems
9 Statistical Mechanics of Steady Flow: Heat and Shear
9.1 Thermodynamics of Steady Heat Flow
9.1.1 Canonical Equilibrium System
9.1.2 Fourier's Law of Heat Conduction
9.1.3 Second Entropy for Heat Flow
9.2 Phase Space Probability Density
9.2.1 Explicit Hamiltonian and First Energy Moment
9.2.2 Reservoir Entropy and Probability Density
9.3 Most Likely Trajectory
9.4 Equipartition Theoremfor Heat Flow
9.5 Green-Kubo Expressions for the Thermal Conductivity
9.5.1 Isolated System
9.5.2 Heat Reservoirs
9.5.3 Relation with Odd Projection
9.6 Shear Flow
9.6.1 Second Entropy for Shear Flow
9.6.2 Phase Space Probability Density
9.6.3 Most Likely Trajectory
9.6.4 Equipartition Theorem
10 Generalised Langevin Equation
10.1 Free Brownian Particle
10.1.1 Time Correlation Functions
10.1.2 Mixed ParityDigression
10.1.3 Diffusion Constant
10.1.4 Trajectory Entropy and Correlation
10.2 Langevin and Smoluchowski Equations
10.3 Perturbation Theory
10.3.1 Most Likely Velocity
10.3.2 Alternative Derivation
10.3.3 Most Likely Position
10.3.4 Stochastic Dissipative Equations of Motion
10.3.5 Generalised Langevin Equation for Velocity
10.3.6 Fluctuation Dissipation Theorem
10.3.7 Weiner-Khintchine Theorem
10.3.8 Exponentially Decaying Memory Function
10.4 Adiabatic Linear Response Theory
10.5 Numerical Results for a Brownian Particle in a Moving Trap
10.5.1 Langevin Theory
10.5.2 Smoluchowski Theory
10.5.3 Computer Simulations
10.5.4 Perturbation Algorithm
10.5.5 Relative Amplitude and Phase Lag
10.5.6 Stochastic Trajectory
10.6 Generalised Langevin Equation in the Case of Mixed Parity
10.6.1 Equilibrium System
10.6.2 Regression of Fluctuation
10.6.3 Time Dependent Perturbation
10.6.4 Generalised Langevin Equation
10.7 Projector Operator Formalism
10.8 Harmonic Oscillator Model for the Memory Function
10.8.1 Generalised Langevin Equation
10.8.2 Modified Random Force
10.8.3 Discussion
11 Non-Equilibrium Computer Simulation Algorithms
11.1 Stochastic Molecular Dynamics
11.1.1 Equilibrium Systems
11.1.2 Mechanical Non-Equilibrium System
11.1.3 Driven Brownian Motion
11.1.4 Steady Heat Flow
11.2 Non-Equilibrium Monte Carlo
11.2.1 Equilibrium Systems
11.2.2 Non-Equilibrium Systems
11.2.3 Driven BrownianMotion
11.2.4 Steady Heat Flow
11.3 Brownian Dynamics
11.3.1 Elementary Brownian Dynamics
11.3.2 Perturbative Brownian Dynamics
11.3.3 Stochastic Calculus
References
Index
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