An Introduction to Chaos in Nonequilibrium Statistical Mechanics
This book is an introduction to the applications in nonequilibrium statistical mechanics of chaotic dynamics, and also to the use of techniques in statistical mechanics important for an understanding of the chaotic behaviour of fluid systems. The fundamental concepts of dynamical systems theory are reviewed and simple examples are given. Advanced topics including SRB and Gibbs measures, unstable periodic orbit expansions, and applications to billiard-ball systems, are then explained. The text emphasises the connections between transport coefficients, needed to describe macroscopic properties of fluid flows, and quantities, such as Lyapunov exponents and Kolmogorov-Sinai entropies, which describe the microscopic, chaotic behaviour of the fluid. Later chapters consider the roles of the expanding and contracting manifolds of hyperbolic dynamical systems and the large number of particles in macroscopic systems. Exercises, detailed references and suggestions for further reading are included.
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1 Nonequilibrium statistical mechanics
2 The Boltzmann equation
3 Liouvilles equation
4 Boltzmanns ergodic hypothesis
6 The GreenKubo formulae
7 The bakers transformation
8 Lyapunov exponents bakers map and toral automorphisms
12 Transport coefficients and chaos
13 SinaiRuelleBowen SRB and Gibbs measures
14 Fractal forms in GreenKubo relations
15 Unstable periodic orbits
16 Lorentz lattice gases
17 Dynamical foundations of the Boltzmann equation
18 The Boltzmann equation returns
19 Whats next ?
Anosov system Arnold cat map attractor baker’s map baker’s transformation Beijeren Boltzmann equation box-counting dimension Cantor set chaos chaotic Chapter collision compute conﬁguration consider constant-energy surface deﬁned deﬁnition denote density derivation discussion distribution function Dorfman eigenvalues electric ﬁeld ensemble average entropy production equilibrium ergodic hypothesis Ergodic Theory escape rate escape-rate expansion exponential ﬁnd ﬁnite ﬁrst ﬁxed ﬂow ﬂuctuation ﬂuid fractal Frobenius—Perron equation Further reading Gallavotti Gaspard Gibbs measures Green—Kubo formulae Hausdorff dimension hyperbolic systems inﬁnite invariant iterations KS entropy large number lattice Lorentz gas Lyapunov exponents macroscopic Markov partitions mathematical motion moving particle nonequilibrium statistical mechanics number of particles obtain partition function periodic orbits periodic points phase point phase-space positive Lyapunov exponents properties quantum chaos random region repeller Ruelle scatterers set of points Sinai SRB measure tagged particle theorem thermodynamic time-reversal tion topological pressure trajectories transport unit square unstable directions unstable manifolds velocity zero