Cover Page -- Title Page -- Copyright Page -- Preface -- Contents -- Detailed Contents -- Chapter 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 -- Chapter 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 -- Chapter 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 Uniformly Moving 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
Chapter 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 -- Chapter 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 -- Chapter 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 Algorithm for Ideal Straight Rolls -- 6.3.1 Hydrodynamic Equations -- 6.3.2 Fourier Expansion -- 6.3.3 Nusselt Number -- 6.4 Algorithm for the Cross Roll State -- 6.4.1 Hydrodynamic Equations and Conditions -- 6.4.2 Fourier Expansion -- 6.5 Algorithm for Convective Transitions -- 6.6 Convection Theory and Experiment -- Chapter 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 -- Chapter 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 -- Chapter 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 Theorem for 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 -- Chapter 10 Generalised Langevin Equation -- 10.1 Free Brownian Particle -- 10.1.1 Time Correlation Functions -- 10.1.2 Mixed Parity Digression -- 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 -- Chapter 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 Brownian Motion -- 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 -- Footnotes

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Context of Non-equilibrium thermodynamics and statistical mechanics : foundations and applications, Phil Attard

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