On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics

Abstract

We study the zero-temperature limit for Gibbs measures associated to Frenkel-Kontoro- va models on(R d ) Z /Z d . We prove that equilibrium states concentrate on configurations of mini- mal energy, and, in addition, must satisfy a variational principle involving metric entropy and Lya- punov 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 equa- tion. 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. In statistical mechanics, Gibbs measures are probability measures on the configuration space, describing states of thermodynamical equilibrium. One of the major problems is to study the dependence of equilibrium states on the temperature (or other parameters): a lack of analyticity in this dependence is interpreted as the occurrence of a phase tran- sition, and the existence of several Gibbs measures at a given temperature, as the coexis- tence of several phases. In Part I of this paper, we are interested in the behaviour of Gibbs measures as temper- ature goes to zero, in the model where the particles of the system lie on the 1-dimensional lattice Z. This is not the favourite situation in statistical mechanics: in this case, and if the energy of interaction between particles satisfies reasonable assumptions, there is usually no phase transition. But even then, there is, to my knowledge, no general result describ- ing completely the behaviour of Gibbs measures at zero temperature: for instance, the existence or not of a limit of the equilibrium state. It is intuitive to think, and possible to prove, that such a limit must minimize the mean energy, but there are examples where it is not enough to conclude, as there may be several states of minimal mean energy ((Si82)). This paper deals with the case where the state of each particle is represented by an element of R d , so that a configuration of the system is described by a sequence = ( k)k2Z 2 (R d ) Z. We work in the Markovian case: the potential of interaction is of the form ¯