Nonequilibrium and Irreversibility (eBook)

Equilibrium.- Many particles systems: kinematics, timing.- Birth of kinetic theory.- Heat theorem and Ergodic hypothesis.- Least action and heat theorem.- Heat Theorem and Ensembles.- Boltzmann’s equation, entropy, Loschmidt’s paradox.- Conclusion.- Stationary Nonequilibrium.- Thermostats and infinite models.- Finite thermostats.- Examples of nonequilibrium problems.- Initial data.- Finite or infinite thermostats? Equivalence?.- SRB distributions.- Chaotic Hypothesis.- Phase space contraction in continuous time.- Phase space contraction in timed observations.- Conclusions.- Discrete phase space.- Recurrence.- Hyperbolicity: stable & unstable manifolds.- Geometric aspects of hyperbolicity. Rectangles.- Symbolic dynamics and chaos.- Examples of hyperbolic symbolic dynamics.- Coarse graining and discrete phase space.- Coarse cells, phase space points and simulations.- The SRB distribution: its physical meaning.- Other stationary distributions.- Phase space cells and entropy.- Counting phase space cells out of equilibrium.- kB logN entropy or Lyapunov function?.- Fluctuations.- SRB potentials.- Chaos and Markov processes.- Symmetries and time reversal.- Pairing rule and Axiom C.- Large deviations.- Time reversal and fluctuation theorem.- Fluctuation patterns.- Onsager reciprocity, Green-Kubo formula, fluctuation theorem.- Local fluctuations: an example.- Local fluctuations: generalities.- Quantum systems, thermostats and non equilibrium.- Quantum adiabatic approximation and alternatives.- Applications.- Equivalent thermostats.- Granular materials and friction.- Neglecting granular friction: the relevant time scales.- Simulations for granular materials.- Fluids.- Developed turbulence.- Intermittency.- Stochastic evolutions.- Very large fluctuations.- Thermometry.- Processes time scale and irreversibility.- Historical comments.- Proof of the second fundamental theorem.- Collision analysis and equipartition.- Dense orbits: an example.- Clausius’ version of recurrence and periodicity.- Clausius’ mechanical proof of the heat theorem.- Priority discussion of Boltzmann (vs. Clausius ).- Priority discussion: Clausius’ reply.- On the ergodic hypothesis (Trilogy: #1).- Canonical ensemble and ergodic hypothesis (Trilogy: #2).- Heat theorem without dynamics (Trilogy: #3).- Irreversibility: Loschmidt and “Boltzmann’s sea”.- Discrete phase space, count of its points and entropy.- Monocyclic and orthodic systems. Ensembles.- Maxwell 1866.- Appendices.- A Appendix: Heat theorem (Clausius version).- B Appendix: Aperiodic Motions as Periodic with Infinite Period!.- C Appendix: The heat theorem without dynamics.- D Appendix: Keplerian motion and heat theorem.- E Appendix: Gauss’ least constraint principle.- F Appendix: Non smoothness of stable/unstable manifolds.- G Appendix: Markovian partitions construction.- H Appendix: Axiom C.- I Appendix: Pairing theory.- J Appendix: Gaussian fluid equations.- K Appendix: Jarzinsky’s formula.- L Appendix: Evans-Searles’ formula.- M Appendix: Forced pendulum with noise.- N Appendix: Solution Eq.(eM.10).- O Appendix: Iteration for Eq.(eM.10).- P Appendix: Bounds for the theorem in Appendix M.- Q Appendix: Hard spheres, BBGKY hierarchy.- R Appendix: Interpretation of BBGKY equations.- S Appendix: BGGKY; an exact solution (?).- T Appendix: Comments on BGGKY and stationarity.- References.- Index.