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Non-equilibrium thermodynamics and statistical mechanics
'Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications' builds from basic principles to advanced techniques, and covers the major phenomena, methods, and results of time-dependent systems. It is a pedagogic introduction, a comprehensive reference manual, and an original research monograph. Uniquely, the book treats time-dependent systems by close analogy with their static counterparts, with most of the familiar results of equilibrium thermodynamicsand statistical mechanics being generalized and applied to the non-equilibrium case. The book is notable for its unified treatment of thermodynamics, hydrodynamics, stochastic processes, and statistical mechanics, for its self-contained, coherent derivation of a variety of non-equilibrium theorems, and for its quantitative tests against experimental measurements and computer simulations. Systems that evolve in time are more common than static systems, and yet until recently they lacked any over-arching theory. 'Non-equilibrium Thermodynamics and Statistical Mechanics' is unique in its unified presentation of the theory of non-equilibrium systems, which has now reached the stage of quantitative experimental and computational verification. The novel perspective and deep understanding that this book brings offers the opportunity for new direction and growth in the study oftime-dependent phenomena.'Non-equilibrium Thermodynamics and Statistical Mechanics' is an invaluable reference manual for experts already working in the field. Research scientists from different disciplines will find the overview of time-dependent systems stimulating and thought-provoking. Lecturers in physics and chemistry will be excited by many fresh ideas and topics, insightful explanations, and new approaches. Graduate students will benefit from its lucid reasoning and its coherent approach, as well as from thechem12physof mathematical techniques, derivations, and computer algorithms.
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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Includes bibliographical references and index.
Oxford : Oxford Univiversity Press, 2012