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Cambridge2008Chaos and Coarse Graining in Statistical Mechanics
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1 Basic concepts of dynamical systems theory 1 1.1 Deterministic systems 1 1.2 Unpredictability: systems with many degrees of freedom 7 1.3 Unpredictability: deterministic chaos 12 1.4 Probabilistic aspects of dynamical systems 16 References 23 2 Dynamical indicators for chaotic systems: Lyapunov exponents, entropies and beyond 24 2.1 Dynamical systems approach 25 2.2 Information theory approach 28 2.3 Beyond the Lyapunov exponents and the Kolmogorov–Sinai entropy 44 References 54 3 Coarse graining, entropies and Lyapunov exponents at work 58 3.1 Characterization of the complexity and system modeling 58 3.2 How random is a random number generator? 71 3.3 Lyapunov exponents and complexity in dynamical systems with noise 79 3.4 Conclusions 88 References 89 4 Foundation of statistical mechanics and dynamical systems 92 4.1 The ergodic problem: a brief random walk among an intricate history 93 4.2 Beyond abstract ergodic theory 97 4.3 The connection between analytical mechanics and the ergodic problem 103 4.4 An unexpected result revitalizes interest in the ergodic problem 106 v vi Contents 4.5 Some modern developments 109 4.6 On the role of chaos in statistical mechanics 114 4.7 Some general remarks 116 References 117 5 On the origin of irreversibility 120 5.1 The problem 120 5.2 Toward the solution 122 5.3 Some results 130 5.4 About ensembles, the number of degrees of freedom and chaos 140 References 148 6 The role of chaos in non-equilibrium statistical mechanics 150 6.1 On the connection between the Kolmogorov–Sinai entropy and production rate of the coarse-grained Gibbs entropy 152 6.2 Gibbs and Boltzmann entropies: the role of chaos, interaction and coarse graining 159 6.3 Fluctuation-response relation and chaos 167 6.4 Chaos and pseudochaos for diffusion and conduction 174 6.5 Remarks and perspectives 180 References 182 7 Coarse-graining equations in complex systems 185 7.1 A short parenthesis: secular terms and multiscale analysis 187 7.2 From molecular level to Brownian motion 189 7.3 Diffusion at large scale and eddy diffusivity 198 7.4 The adiabatic piston: a system between the microscopic and macroscopic realms 203 7.5 Remarks and perspectives 210 References 215 8 Renormalization-group approaches 217 8.1 Renormalization group(s): a brief overview 218 8.2 Renormalization groups to cure singular perturbation expansions 220 8.3 A multiscale and constructive approach to capture critical behavior 225 8.4 Renormalization groups: a multiscale approach for asymptotic analysis 236 8.5 Probabilistic viewpoint on renormalization groups 243 8.6 Conclusions and perspectives 262 References 264 Index |
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