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ON THE ZEROTEMPERATURE OR VANISHING VISCOSITY LIMIT FOR CERTAIN MARKOV PROCESSES ARISING FROM
 

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 (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 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

  

Source: Anantharaman, Nalini - Centre de Mathématiques Laurent Schwartz, École Polytechnique

 

Collections: Mathematics