Recurrence is discussed in detail under the perspective to present it in the light of a discrete representation of motion, in principle necessary in simulations and in their interpretation. A discrete viewpoint is important also in qualitative understanding of the statistical properties of motions on the attracting surfaces. It is stressed that studying recurrence leads to a formulation which unifies entirely different dynamic systems: under the Chaotic hypothesis (CH) the most diverse dynamical systems can be digitally represented as shifts of one dimensional chain of symbols on which the invariant distributions appear as stationary stochastic processes, natural generalizations of Markov processes. The “universal” representation of stationary states of Ch-verifying systems as, essentially, Markov processes is remarkable because stochasticity is not introduced a priori. Representation of phase space as an array of points, regularly filling it and moving according to a specific rule (“code”), is deeply different from the phase space cells used, at times, in equilibrium Statistical Mechanics: here the points are called “microcells”, to be conceived as regularly disposed on a lattice in phase space just as they are ideally conceived in simulations codes. CH and the hyperbolic nature of the time evolution allow to collect the microcells in much larger groups, forming a partition of phase space into “coarse cells”, labeled by a finite number of labels, small enough so that the (few) observables of interest can be considered constant on each cell. The motion will be dissipative, at least in non-equilibrium cases, and most microcells will not be recurrent: so the statistical properties will be carried by recurrent ones, i.e. by the periodic points on each attracting set. In discretized representations all motions will be eventually periodic and it is tempting to go back to the original proposal by Boltzmann and think that there is only one attracting periodic orbit: and the whole discussion (hinted in Chap.  1 ) about this point and the EH can be imagined and repeated. Hyperbolicity makes it possible to design the coarse cells so that each microcell is identified with the string formed by the labels of the coarse cells successively visited in its motion: and the regularity of the array of microcells can be seen as the source of the physical relevance of SRB distribution, as the heuristic interpretation of the SRB distribution shows in Sect.  3.8 . Discrete interpretation of the SRB distribution leads to an estimate of the number of microcells composing the attracting periodic orbit: it is presented because it gives the opportunity of showing the existence of a mathematical cancellation, interesting in itself, and to argue that the logarithm of the recurrent microcells number has the interpretation of entropy only in the case of equilibrium states while in non-equilibrium cannot be seen as a function of the state.

Discrete Phase Space

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发表日期：

2025年

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Giovanni Gallavotti

摘要

Recurrence is discussed in detail under the perspective to present it in the light of a discrete representation of motion, in principle necessary in simulations and in their interpretation. A discrete viewpoint is important also in qualitative understanding of the statistical properties of motions on the attracting surfaces. It is stressed that studying recurrence leads to a formulation which unifies entirely different dynamic systems: under the Chaotic hypothesis (CH) the most diverse dynamical systems can be digitally represented as shifts of one dimensional chain of symbols on which the invariant distributions appear as stationary stochastic processes, natural generalizations of Markov processes. The “universal” representation of stationary states of Ch-verifying systems as, essentially, Markov processes is remarkable because stochasticity is not introduced a priori. Representation of phase space as an array of points, regularly filling it and moving according to a specific rule (“code”), is deeply different from the phase space cells used, at times, in equilibrium Statistical Mechanics: here the points are called “microcells”, to be conceived as regularly disposed on a lattice in phase space just as they are ideally conceived in simulations codes. CH and the hyperbolic nature of the time evolution allow to collect the microcells in much larger groups, forming a partition of phase space into “coarse cells”, labeled by a finite number of labels, small enough so that the (few) observables of interest can be considered constant on each cell. The motion will be dissipative, at least in non-equilibrium cases, and most microcells will not be recurrent: so the statistical properties will be carried by recurrent ones, i.e. by the periodic points on each attracting set. In discretized representations all motions will be eventually periodic and it is tempting to go back to the original proposal by Boltzmann and think that there is only one attracting periodic orbit: and the whole discussion (hinted in Chap. 1 ) about this point and the EH can be imagined and repeated. Hyperbolicity makes it possible to design the coarse cells so that each microcell is identified with the string formed by the labels of the coarse cells successively visited in its motion: and the regularity of the array of microcells can be seen as the source of the physical relevance of SRB distribution, as the heuristic interpretation of the SRB distribution shows in Sect. 3.8 . Discrete interpretation of the SRB distribution leads to an estimate of the number of microcells composing the attracting periodic orbit: it is presented because it gives the opportunity of showing the existence of a mathematical cancellation, interesting in itself, and to argue that the logarithm of the recurrent microcells number has the interpretation of entropy only in the case of equilibrium states while in non-equilibrium cannot be seen as a function of the state.