Abstract
In this paper, we propose a new and shorter proof of the following fact: in a spin-flip process on {−1, +1}S, whereS is a countable set, the free energy is non-increasing.
Free energy is a well defined functional only for invariant measures under a convenient group of bijections ofS. We formalize this with the notion ofB-ameanability ofS. This frame contains the usual example ofZ d under translations but also many nice lattices that are not groups under groups of isometries.
For invariant measures, except Gibbs ones, the free energy is strictly decreasing. Among invariant measures, the only stationary measures for the spin-flip process are therefore Gibbs measures. From this result we also deduce an ergodic theorem.
The first result on this subject was obtained by Holley [1] for a finite range potential on\(\{ - 1, + 1\} ^{Z^d } \) and some extension by Higuchi, Shiga [2].
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References
Holley, R.: Free energy in a markovian model of lattice spin systems. Commun. math. Phys.23, 87–99 (1971)
Higushi, Y., Shiga, T.: Some results on Markov processes of infinite lattice spin systems. J. math. Kyoto Univ.15, 211–229 (1975)
Kieffer, J. C.: A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space. Ann. Prob.3, 1031–1037 (1975)
Moulin Ollagnier, J., Pinchon, D.: Fonctions thermodynamiques locales en mécanique statistique. Séminaire sur les processus markoviens a une infinité de particules. Ecole Polytechnique (1976)
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Communicated by J. L. Lebowitz.
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Moulin Ollagnier, J., Pinchon, D. Free energy in spin-flip processes is non-increasing. Commun.Math. Phys. 55, 29–35 (1977). https://doi.org/10.1007/BF01613146
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DOI: https://doi.org/10.1007/BF01613146