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Probability, Typicality and Emergence in Statistical Mechanics Probability, Typicality and Emergence in Statistical Mechanics Angelo VULPIANI Dept. of Physics, University Sapienza, Rome, Italy <[email protected]> Seminario delle Meccaniche, Roma, 11 Maggio 2020 (online talk) Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 1 / 41 Acknowledgments THANKS to: Marco Baldovin, Lorenzo Caprini, Patrizia Castiglione, Fabio Cecconi, Massimo Cencini, Luca Cerino, Sergio Chibbaro, Massimo Falcioni, Annick Lesne, Andrea Puglisi and Lamberto Rondoni. * M. Baldovin, L. Caprini and A. Vulpiani Irreversibility and typicality: a simple analytical result for the Ehrenfest model Physica A, 524, 422 (2019). * P. Castiglione, M. Falcioni, A. Lesne and A. Vulpiani, Chaos and coarse graining in statistical mechanics, (Cambridge University Press, 2008). * L. Cerino, F. Cecconi, M. Cencini, and A. Vulpiani, The role of the number of degrees of freedom and chaos in macroscopic irreversibility Physica A 442, 486 (2016). * S. Chibbaro, L. Rondoni, and A. Vulpiani, On the Foundations of Statistical Mechanics: Ergodicity, Many Degrees of Freedom and Inference Communications in Theoretical Physics 62, 469 (2014). Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 2 / 41 Why a talk on foundations of statistical mechanics? a) Interest for itself, both for scientists and philosophers. b) Today the development of statistical mechanics has reached the point of delving into its origin, this was already stressed by S.K. Ma in his book. For instance a challenging frontier in modern statistical physics is concerned with systems with a small number of degrees of freedom, far from the thermodynamic limit, and non Hamiltonian systems (granular material, active matter etc). c) Pedagogical motivations: discuss in an explicit way some difficulties in the treatment of basic topics. Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 3 / 41 The conceptual problem The relevance of the probability theory for the statistical mechanics is obvious and cannot be underestimated. At physical, as well conceptual, level one has to face a problem: what is the link between the probabilistic computations (i.e. the averages over an ensemble) and the actual results obtained in laboratory experiments which, a fortiori, are conducted on a single realization (or sample) of the system under investigation? In my opinion the most important role of the probability theory is to explain typicality, i.e. the (experimental) fact that a macroscopic observable in a single system is close to its average, this is an emergent property. Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 4 / 41 A proposal for he link between probability and physics: the Cournot's principle Antoine Cournot (1801-1877) proposed the principle of impossibility: An event with very small probability will not happen. Such a principle dates back to Jakob Bernoulli (Ars Conjectandi 1713) Something is morally certain if its probability is so closed to certainty that shortfall is imperceptible Eminent mathematicians (e.g. P.Levy, J.Hadamard, A.N.Kolmogorov) considered the Cournot's principle as the only sensible connection between probability and empirical world. The non abstract character of the probability: Probability is a physical property just like lenght and weight (Paul Levy) Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 5 / 41 The status of the understanding of typicality There are (few) rigorous results in the limit N 1: * In the equilibrium problem (ergodicity problem) for weakly interacting systems and a proper class of observables (Khinchin, Mazur and van der Linden). * For the non equilibrium problem (H theorem), for diluted gases in special regime, Grad- Boltzmann limit (Lanford). Several numerical computations and philosophical debates. In the following I discuss a) the Ehrenfest model: in such a stochastic system it is possible to show in esplicit rigorous way that, in the limit N ! 1, the irreversibility is present for almost all the realizations, b) some simple numerical experiments of systems with many particles, showing irreversibility in a single macroscopic body. Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 6 / 41 The Great Vision Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 7 / 41 Probability and real world The two great ideas of Boltzmann: 1 Introduction of probability to compute physical quantities 2 A link between the microscopic dynamics (Hamilton equations) and the macroscopic world (thermodynamics) The issue 2 is summarized by the celebrated relation S = k ln W : Where S is the entropy, W is the "volume of the phase space" of the macroscopic configuration. 1 Z W (E; V ; N) = δ(H(Q; P) − E)d3N Qd3N P : N!h3N Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 8 / 41 The ergodic hypothesis Consider a macroscopic system with N interacting particles, the microscopic state is described by a X 2 R6N whose components are the positions and the momenta of the particles. When an instrument measures some quantity, e.g. the pressure, it performs a time average, on a measurement time T , of some function of X. The instrument computes 1 Z T A¯T = A(X(t))dt : T 0 Of course for the computation of A¯T one needs the knowledge of the initial condition X(0) and the ability to determine the time evolution X(t). This is surely beyond the human possibilities. Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 9 / 41 The ingenious Boltzmann's idea is to replace the time average with a suitable average on the the phase space. He assumed that 1 Z T Z lim A(X(t))dt = A(X)ρmicro(X)dX (1) T !1 T 0 where ρmicro(X) is the microcanonical probability density. This is the ERGODIC HYPOTHESIS Once the ergodic hypothesis is assumed to be valid in the limit of N 1 it is easy to derive, for a system which exchanges energy with an external environment, the Boltzmann-Gibbs distribution for the canonical ensemble. Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 10 / 41 Beyond mathematics: physical ergodicity Now, after many clear numerical evidences, starting from FPUT (Fermi, Pasta, Ulam and Tsingou) work (1955) on chain of non linear oscillators, and the KAM (Kolmogorov, Arnold and Moser) theorem on non integrable systems (1954-1963) we know that for sure the ergodic hypothesis, strictly speaking, cannot be valid. * Why, in spite of this, is the Boltzmann-Gibbs distribution valid? * Why does the molecular dynamics give the proper results? The Khinchin's approach: The ergodic hypothesis is basically true if * N 1 * we consider "suitable" observables * we allow for a failure of (1) in a "small region" (i.e. with small probability) Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 11 / 41 Khinchin has been able to show that for the class of sum functions with the shape N X f (X) = fn(qn; pn) n=1 one has jδf (X)j c c Prob ≥ 1 ≤ 2 N N1=4 N1=4 where δf (X) is the difference between the time average starting from X and the ensemble average. Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 12 / 41 Technical notes The original work of Khinchin was only for non interacting system, i.e. X H = h1(qn; pn) n Mazur and van der Linden generalised the result to the more physical interesting case of (weakly) interacting particles. X X H = h1(qn; pn) + V (jqn − qn0 j) n n;n0 Many, but not all, macroscopic observables are sum functions. Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 13 / 41 The essence of the Khinchin, and Mazur and van der Linden result Basically Khinchin, Mazur and van der Linden showed that, although the ergodic hypothesis mathematically does not hold, it is "physically" valid if we are "tolerant" i.e. we accept that in system with N 1, ergodicity can fails in regions whose probability is small O(N−1=4), i.e. vanishing in the limit N ! 1. * The dynamics has a marginal role * The very relevant ingredient is the large number of particles * The technical aspect is the Law of large numbers (which needs N 1) On the other hand at physical level, one can say that the Khinchin result is not enough, e.g. given generic initial conditions, and observable A, it is not easy to find the minimum value of measurement time T such that time average gives the correct result, i.e. A¯T '< A >, see difficulties in the FPUT problem. Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 14 / 41 The old and debated problem of the irreversibility The origin of the difficulties: * The microscopic world is ruled by laws (Hamilton eq.s) which are invariant under time reversal: t ! −t ; q ! q ; p ! −p : * The macroscopic world is described by irreversible equations, e.g. the Fick equation for the diffusion @t C = D∆C : How is it possible to conciliate the two above facts? Angelo VULPIANI (2020) Probability, Typicality and Emergence in StatisticalSeminario Mechanics delle Meccaniche, Roma, 11 Maggio 2020 (online talk) 15 / 41 An aged debate The two celebrated criticisms to the Boltzmann's H theorem by • Loschmidt (about reversibility) and • Zermelo (about recurrency).
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