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Entropy functional (free energy) for dynamical systems and their random perturbations. (English) Zbl 0553.60097
Stochastic analysis, Proc. Taniguchi Int. Symp., Katata & Kyoto/Jap. 1982, North-Holland Math. Libr. 32, 437-467 (1984).
[For the entire collection see Zbl 0538.00017.]
In the present paper the author presents a formulation of the Gibbs variational principle in an abstract manner for a triplet (X,F,m) where (X,F) is a compact dynamical system and m is a probability measure on X. This approach unifies the Donsker-Varadhan theory for Markov chains [see. e.g., M. D. Donsker and S. R. S. Varadhan, Commun. Pure Appl. Math. 36, 183-212 (1983; Zbl 0512.60068)], the equilibrium classical statistical mechanics of lattice systems [see, e.g. G. Gallavotti and S. Miracle-Sole [Commun. Math. Phys. 5, 317-323 (1967; Zbl 0154.465)] as well as the theories for symbolic dynamics, expanding maps and Anosov diffeomorphisms.
Reviewer: K.Schürger
MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
54H20 Topological dynamics (MSC2010)
28Dxx Measure-theoretic ergodic theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)