 
Summary: ON THE ZEROTEMPERATURE OR VANISHING VISCOSITY
LIMIT FOR CERTAIN MARKOV PROCESSES ARISING FROM
LAGRANGIAN DYNAMICS
NALINI ANANTHARAMAN
Abstract. We study the zerotemperature limit for Gibbs measures associ
ated to FrenkelKontorova models on (Rd)Z/Zd. We prove that equilibrium
states concentrate on configurations of minimal energy, and, in addition, must
satisfy a variational principle involving metric entropy and Lyapunov expo
nents, a bit like in the RuellePesin inequality. Then we transpose the result
to certain continuoustime stationary stochastic processes associated to the
viscous HamiltonJacobi equation. As the viscosity vanishes, the invariant
measure of the process concentrates on the socalled "Mather set" of classi
cal mechanics, and must, in addition, minimize the gap in the RuellePesin
inequality.
In statistical mechanics, Gibbs measures are probability measures on the config
uration 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 transition, and the existence of several Gibbs measures at a
given temperature, as the coexistence of several phases.
