Nonequilibrium and Irreversibility

ISBN-13: 9783319383279 / Angielski / Miękka / 2016 / 248 str.

Giovanni Gallavotti

Opis

This book concentrates on the properties of the stationary states in chaotic systems of particles or fluids, leaving aside the theory of the way they can be reached. The stationary states of particles or of fluids (understood as probability distributions on microscopic configurations or on the fields describing continua) have received important new ideas and data from numerical simulations and reviews are needed. The starting point is to find out which time invariant distributions come into play in physics. A special feature of this book is the historical approach. To identify the problems the author analyzes the papers of the founding fathers Boltzmann, Clausius and Maxwell including translations of the relevant (parts of) historical documents. He also establishes a close link between treatment of irreversible phenomena in statistical mechanics and the theory of chaotic systems at and beyond the onset of turbulence as developed by Sinai, Ruelle, Bowen (SRB) and others: the author gives arguments intending to support strongly the viewpoint that stationary states in or out of equilibrium can be described in a unified way. In this book it is the "chaotic hypothesis," which can be seen as an extension of the classical ergodic hypothesis to non equilibrium phenomena, that plays the central role. It is shown that SRB - often considered as a kind of mathematical playground with no impact on physical reality - has indeed a sound physical interpretation; an observation which to many might be new and a very welcome insight. Following this, many consequences of the chaotic hypothesis are analyzed in chapter 3 - 4 and in chapter 5 a few applications are proposed. Chapter 6 is historical: carefully analyzing the old literature on the subject, especially ergodic theory and its relevance for statistical mechanics; an approach which gives the book a very personal touch. The book contains an extensive coverage of current research (partly from the authors and his coauthors publications) presented in enough detail so that advanced students may get the flavor of a direction of research in a field which is still very much alive and progressing. Proofs of theorems are usually limited to heuristic sketches privileging the presentation of the ideas and providing references that the reader can follow, so that in this way an overload of this text with technical details could be avoided.

Spis treści

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.