Thermostats are introduced, and illustrated via several mechanical examples, as models and tools to study stationary states in the presence of driving forces balanced by dissipation. This gives the opportunity to stress the role of the initial data randomness: Ruelle’s viewpoint on the privileged status of the initial data, as data generated by a protocol producing them as probability-one samples of any distribution with density over phase space. In presence of chaotic motions, the stationary states reached by the evolution are determined, independently of the protocol for the initial data, if chaoticity is intended in the precise sense of hyperbolic evolution on the surfaces attracting the motions. The latter property is formalized as “Chaotic hypothesis” (CH). And it is stressed that its adoption solves the problem of identifying, in equilibrium as well as out of equilibrium, which is the distribution, among the infinitely many stationary ones, physically relevant to provide the time averages of the observables: such remarkable property holds because, in systems verifying the CH, on each attracting surface (often unique) there is a unique stationary state, called SRB distribution, controlling the average of physical observables. Multiplicity of attracting surfaces is similar to phase coexistence of equilibria and in Sect.  5.6 is proposed among the causes of “intermittency”. The idea emerges that the CH is a paradigm for chaotic motions playing the role that the quasi-periodic motions play for ordered evolutions: the picture implied by the CH is that in phase space one or more surfaces which attract nearby points exist and are smooth and all their points (up to a set of zero area) have trajectories that cover them densely. Furthermore in conservative systems undergoing chaotic evolutions the CH implies the EH so that the two hypotheses cannot contradict each other. Still the CH has to be regarded as a law at the same level as the EH: it also covers systems well beyond the equilibrium, but tertium datur! as shown by easy counterexamples. Accent is also on examples of non-equilibrium problems and of thermostats, finite or infinite, that control dissipation, with wide discussion of the phase space contraction interpretation as heat ceded to thermostats. Difficulties about the interpretation due to the dependence of phase space volume on the metric used to measure it are avoided by showing that the amount of heat ceded does not depend on the metric. At the same time it is stressed that it is different to consider phase space contraction on the whole phase space or on the attracting surfaces (often much smaller and of difficult access).

Stationary Nonequilibrium

章节
发表日期：

2025年

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Giovanni Gallavotti

摘要

Thermostats are introduced, and illustrated via several mechanical examples, as models and tools to study stationary states in the presence of driving forces balanced by dissipation. This gives the opportunity to stress the role of the initial data randomness: Ruelle’s viewpoint on the privileged status of the initial data, as data generated by a protocol producing them as probability-one samples of any distribution with density over phase space. In presence of chaotic motions, the stationary states reached by the evolution are determined, independently of the protocol for the initial data, if chaoticity is intended in the precise sense of hyperbolic evolution on the surfaces attracting the motions. The latter property is formalized as “Chaotic hypothesis” (CH). And it is stressed that its adoption solves the problem of identifying, in equilibrium as well as out of equilibrium, which is the distribution, among the infinitely many stationary ones, physically relevant to provide the time averages of the observables: such remarkable property holds because, in systems verifying the CH, on each attracting surface (often unique) there is a unique stationary state, called SRB distribution, controlling the average of physical observables. Multiplicity of attracting surfaces is similar to phase coexistence of equilibria and in Sect. 5.6 is proposed among the causes of “intermittency”. The idea emerges that the CH is a paradigm for chaotic motions playing the role that the quasi-periodic motions play for ordered evolutions: the picture implied by the CH is that in phase space one or more surfaces which attract nearby points exist and are smooth and all their points (up to a set of zero area) have trajectories that cover them densely. Furthermore in conservative systems undergoing chaotic evolutions the CH implies the EH so that the two hypotheses cannot contradict each other. Still the CH has to be regarded as a law at the same level as the EH: it also covers systems well beyond the equilibrium, but tertium datur! as shown by easy counterexamples. Accent is also on examples of non-equilibrium problems and of thermostats, finite or infinite, that control dissipation, with wide discussion of the phase space contraction interpretation as heat ceded to thermostats. Difficulties about the interpretation due to the dependence of phase space volume on the metric used to measure it are avoided by showing that the amount of heat ceded does not depend on the metric. At the same time it is stressed that it is different to consider phase space contraction on the whole phase space or on the attracting surfaces (often much smaller and of difficult access).