We study the zero-temperature limit for Gibbs measures associated to Frenkel-Kontorova models on (R d) Z/Z d . We prove that equilibrium states concentrate on configurations of minimal energy, and, in addition, must satisfy a variational principle involving metric entropy and Lyapunov exponents, a bit like in the Ruelle-Pesin inequality. Then we transpose the result to certain continuous-time stationary stochastic processes associated to the viscous Hamilton-Jacobi equation. As the viscosity vanishes, the invariant measure of the process concentrates on the so-called Mather set of classical mechanics, and must, in addition, minimize the gap in the Ruelle-Pesin inequality.
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Nalini Anantharaman, On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics. J. Eur. Math. Soc. 6 (2004), no. 2, pp. 207–276DOI 10.4171/JEMS/9