Mathematical Theory of Nonequilibrium Steady States: On the Frontier of Probability and Dynamical Systems (Lecture Notes in Mathematics, 1833)

This volume provides a systematic mathematical exposition of the conceptual problems of nonequilibrium statistical physics, such as entropy production, irreversibility, and ordered phenomena. Markov chains, diffusion processes, and hyperbolic dynamical systems are used as mathematical models of physical systems. A measure-theoretic definition of entropy production rate and its formulae in various cases are given. It vanishes if and only if the stationary system is reversible and in equilibrium. Moreover, in the cases of Markov chains and diffusion processes on manifolds, it can be expressed in terms of circulations on directed cycles. Regarding entropy production fluctuations, the Gallavotti-Cohen fluctuation theorem is rigorously proved. Thisvolume provides a systematic mathematical exposition of the conceptual problems of nonequilibrium statistical physics, such as entropy production, irreversibility, and ordered phenomena. Markov chains, diffusion processes, and hyperbolic dynamical systems are used as mathematical models of physical systems. A measure-theoretic definition of entropy production rate and its formulae in various cases are given. It vanishes if and only if the stationary system is reversible and in equilibrium. Moreover, in the cases of Markov chains and diffusion processes on manifolds, it can be expressed in terms of circulations on directed cycles. Regarding entropy production fluctuations, the Gallavotti-Cohen fluctuation theorem is rigorously proved. TOC:Preface.- Introduction.-Circulation Distribution, Entropy Production and Irreversibility of Denumerable Markov Chains.- Circulation Distribution, Entropy Production and Irreversibility of Finite Markov Chains with Continuous Parameter.-General Minimal Diffusion Process: its Construction, Invariant Measure, Entropy Production and Irreversibility.-Measure-theoretic Discussion on Entropy Production of Diffusion Processes and Fluctuation- dissipation Theorem.-Entropy Production, Rotation Numbers and Irreversibility of Diffusion Processes on Manifolds.-On a System of Hyperstable Frequency Locking Persistence under White Noise.- Entropy Production and Information Gain in Axiom A Systems.- Lyapunov Exponents of Hyperbolic Attractors.-Entropy Production, Information Gain and Lyapunov Exponents of Random Hyperbolic Dynamical Systems.- References.- Index Introduction....Pages 1-10 1 Circulation Distribution, Entropy Production and Irreversibility of Denumerable Markov Chains....Pages 11-44 2 Circulation Distribution, Entropy Production and Irreversibility of Finite Markov Chains with Continuous Parameter....Pages 45-66 3 General Minimal Diffusion Process: its Construction, Invariant Measure, Entropy Production and Irreversibility....Pages 67-92 4 Measure-theoretic Discussion on Entropy Production of Diffusion Processes and Fluctuation-dissipation Theorem....Pages 93-120 5 Entropy Production, Rotation Numbers and Irreversibility of Diffusion Processes on Manifolds....Pages 121-148 6 On a System of Hyperstable Frequency Locking Persistence under White Noise....Pages 149-158 7 Entropy Production and Information Gain in Axiom A Systems....Pages 159-188 8 Lyapunov Exponents of Hyperbolic Attractors....Pages 189-214 9 Entropy Production, Information Gain and Lyapunov Exponents of Random Hyperbolic Dynamical Systems....Pages 215-252 References....Pages 253-276