Large deviations and non-equilibrium statistical mechanics

Many interesting particle systems are described by stochastic dynamics without the basis of a known underlying Hamiltonian. Their stationary states will be in general non-equilibrium stationary states which cannot be easily computed as there is currently no counterpart to the Gibbs theory for equilibrium systems. Thus a challenging issue would be to provide a probabilistic description of the non-equilibrium states and describe their limiting structure when the number of particles diverges.

In this talk, we will review some results on the steady states of diffusive systems maintained off equilibrium by two heat baths at unequal temperatures. Using the framework of the hydrodynamic limits, we will discuss the large deviations of the heat current through these systems. In particular, we will explain the occurrence of a dynamical phase transition which may occur for some models.