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Ahdelle Aguilera
Ahdelle Aguilera

Honerkamp Stochastic Dynamical Systems Pdf _VERIFIED_ Free


In equilibrium, we can use functions of states --- free energies,thermodynamic potentials --- to determine the most probable state. In fact, wecan even determine the probability of arbitrary states. Out of equilibrium, itwould seem that the natural generalization would be to use a functional of asequence of states, of a trajectory, to determine the probability oftrajectories. In the case of small, linear deviations from equilibrium, theOnsager-Machlup (or Onsager-Rayleigh) "action" gives us such a functional oftrajectories. What works far from equilibrium? In equilibrium, one can linkthe thermodynamic potentials to functions which specify the rate of decayof large deviations, and this is still trueout of equilibrium (see, e.g., Touchette's great review paper), but this ismore of a mathematical result than a "physical" one.Here's an argument for the ubiquity of effective actions. Markov processeshave Gibbs distributions over sequences of states, and Gibbs distributions,just by definition, arise from an effective action. (*) Many nonequilibriumsystems can be described by Markov processes (say, deterministic trajectoryplus noise). But I'd go further and argue that every nonequilbriumsystem can be represented as a Markov process --- that if you haven't foundone, you're not looking hard enough. (That argument's ina separate paper.) So itshould always be possible to find an effective action. But thisdoesn't establish that there should be a common form for these actions acrossdifferent systems, which is what e.g., Keizer and Woo (separately) claim.Are there universal criteria for the stability of non-equilibrium steadystates, or must be actually investigate entire paths? Landauer argued for thelatter, convincingly to my mind, but I need to learn more here.Approach to equilibrium doesn't interest me so much as sustainednon-equilibrium situations, but like everybody else I suppose they're stronglyconnected. Fluctuation-dissipation results are accordingly interesting,especially ones which do not assume nearness to equilibrium. The Evans-Searlesfluctuation theorem, which is well-supported by experiments (see e.g. the Carberry et al. paper) is extremely interesting.I should try to explain some ideas about the role of smooth dynamicalsystems in the statistical mechanics here, but anyone who's geeky enough to beinterested really ought to read Ruelle's review article rather than listen tome, and, after that, Dorfman's book.*: Pedantically, a Gibbs distribution over a collection of variables \( X_t \), index by \( \mathcalT \), collectively \( X \), will take the form \( \mathbbP(X=x) \propto e^-u(x) \), and the "potential" \( u(x) = \sum_A \in 2^\mathcalTu_A(x_A) \), where \( u_A \) is a function only of the values of the random variables at the indices \( A \). (Obvious measure-theoretic complications for continuous index sets are obvious.) The sets \( A \) which actually contribute to the potential are called "cliques". In the particular case of Markov processes in discrete time, the cliques are pairs of successive indices \( \left\ t, t+1 \right\ \), so we can think of \( u(x) \) as being basically a discrete-time effective action. For continuous-time Markov processes, see under Path Integrals and Feynman Diagrams for Classical Stochastic Processes.See alsoFluctuation-Response Relations;Foundatons of Statistical Mechanics;Interacting Particle Systems;Large Deviations;Mori-Zwanzig Formalism;Pattern Formation;Ilya Prigogine;Self-organization;Self-organized Critcality;Statistical Mechanics;Stochastic Processes;Recommended, big picture:S. R. de Groot and P. Mazur, Non-EquilibriumThermodynamicsJ. R. Dorfman, Introduction to Chaos in NonequilibriumStatistical MechanicsDieter Forster, Hydrodynamic Fluctuations, Broken Symmetry,and Correlation Functions [An excellent book which lookshorrible. Bless Donald Knuth for delivering us from type-writenequations!]Josef Honerkamp, Stochastic Dynamical SystemsJoel Keizer, Statistical Thermodynamics of NonequilibriumProcesses [Review: Molecular Fluctuationsfor Fun and Profit]David Ruelle, "Smooth Dynamics and New Theoretical Ideas inNonequilibrium Statistical Mechanics," Journal of StatisticalPhysics 95 (1999): 393--468 = chao-dyn/9812032Geoffrey Sewell, Quantum Mechanics and Its EmergentMacrophysics [Including nonequilibrium quantum statistical mechanics]Eric Smith, "Large-deviation principles, stochastic effectiveactions, path entropies, and the structure and meaning of thermodynamicdescriptions", arxiv:1102.3938Hugo Touchette, "The Large Deviations Approach to StatisticalMechanics", Physics Reports 478 (2009): 1--69, arxiv:0804.0327Recommended, historical:Mark Kac, Probability in Physical Sciences and RelatedTopicsLars Onsager, "Reciprocal relations in irreversible processes",Physical Review37 (1931):405--426 (part I)and 38(1931): 2265--2279 (part II)Lars Onsager and S. Machlup, "Fluctuations and IrreversibleProcesses", Physical Review 91 (1953):1505--1512Recommended, close-ups:D. M. Carberry, J. C. Reid, G. M. Wang, E. M. Sevick, DebraJ. Searles and Denis J. Evans, "Fluctuations and Irreversibility: AnExperimental Demonstration of a Second-Law-Like Theorem Using a ColloidalParticle Held in an Optical Trap", Physical Review Letters92 (2004): 140601 [An extremely good paper,giving a very nice explanation of the fluctuation theorem of Evans and Searles,followed by the neatest imaginable experimental demonstration of its validity.]S. C. Chapman, G. Rowlands and Nick W. Watkins, "The Origin ofUniversal Fluctuations in Correlated Systems: Explicit Calculation for anIntermittent Turbulent Cascade," cond-mat/0302624W. De Roeck, Christian Maes and Karel Netocny, "H-Theorems fromAutonomous Equations", Journal ofStatistical Physics 123 (2006): 571--584, cond-mat/0508089["Iffor a Hamiltonian dynamics for many particles, at all times the presentmacrostate determines the future macrostate, then its entropy is non-decreasingas a consequence of Liouville's theorem. That observation, made since long, ishere rigorously analyzed with special care to reconcile the application ofLiouville's theorem (for a finite number of particles) with the condition ofautonomous macroscopic evolution (sharp only in the limit of infinite scaleseparation); and to evaluate the presumed necessity of a Markov property forthe macroscopic evolution."]S. F. Edwards, "New Kinds of Entropy", Journal ofStatistical Physics 116 (2004): 29--42 [I need tothink about how his last kind of entropy is related to Lloyd-Pagelsthermodynamic depth.]Gregory L. Eyink, "Action principle in nonequilbrium statisticaldynamics," Physical Review E 54 (1996):3419--3435K. H. Fischer and J. A. Hertz, Spin GlassesPierre Gaspard, Chaos, Scattering and StatisticalMechanicsA. Greven, G. Keller and G. Warnecke (eds.), EntropyGiovanni Jona-Lasinio, "From fluctuations in hydrodynamics tononequilibriumthermodynamics", arxiv:1003.4164Rolf Landauer, "Motion Out of NoisyStates," Journal ofStatistical Physics 53 (1988): 233--248 ["Therelative occupation of competing states of local stability is not determinedsolely by the characteristics of the locally favored states, but depends on thenoise along the whole path connecting the competing states. This is not new,but the sophistication of most modern treatments has obscured the simplicity ofthis central point, and here it is argued for in simple physical terms."]Michael Mackey, Time's Arrow: The Origin of ThermodynamicBehavior [This is a very valuable short introduction to theergodic theory of Markov operators, which ishighly relevant to the origins of irreversibility, etc., but I don't think hisapproach works, because he focuses on the relative entropy(Kullback-Leibler divergence from the invariant distribution), rather than theBoltzmann entropy or even the Gibbs entropy.]Mark Millonas (ed.), Fluctuations and Order: The NewSynthesis [Despite the subtitle, no synthesis is in evidence. However,many of the individual papers are very interesting.]Eric Smith, "Thermodynamic dual structure of linear-dissipativedriven systems", PhysicalReview E 72 (2005): 036130Hyung-June Woo, "Statistics of nonequilibrium trajectories andpatternselection", EurophysicsLetters 64 (2003): 627--633R. K. P. Zia, L. B. Shaw, B. Schmittmann and R. J. Astalos,"Contrasts Between Equilibrium and Non-Equilibrium Steady States: ComputerAided Discoveries in Simple Lattice Gases," cond-mat/9906376Modesty forbids me to recommend:CRS and Cristopher Moore, "What Is a Macrostate? SubjectiveMeasurements and Objective Dynamics,"cond-mat/03003625To read:D. Abreu, U. Seifert, "Thermodynamics of genuine non-equilibrium states under feedback control", arxiv:1109.5892Tameem Albash, Daniel A. Lidar, Milad Marvian, and Paolo Zanardi, "Fluctuation theorems for quantum processes", Physical Review E 88 (2013): 032146D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, andA. Petrosyan, "Entropy Production and Time Asymmetry in NonequilibriumFluctuations", Physical ReviewLetters98 (2007): 150601Francis J. Alexander and Gregory L. Eyink, "Rayleigh-RitzCalculation of Effective Potential Far from Equilibrium," Physical ReviewLetters 78 (1997): 1--4Bidhan Chandra Bag"Nonequilibrium stochastic processes: Time dependence ofentropy flux and entropy production," cond-mat/0205500"Upper bound for the time derivative of entropy fornonequilibrium stochastic processes," cond-mat/0201434BCB, Suman Kumar Banik, and Deb Shankar Ray, "The noiseproperties of stochastic processes and entropy production," cond-mat/0104524M. M. Bandi,


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