Boundary driven Kawasaki process with long range interaction: dynamical large deviations and steady states

Mustapha Mourragui (1) , Enza Orlandi (2)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 Dipartimento di Matematica [Roma TRE]

A particle system with a single locally-conserved field (density) in a bounded interval with different densities maintained at the two endpoints of the interval is under study here. The particles interact in the bulk through a long range potential parametrized by $\beta\ge 0$ and evolve according to an exclusion rule. It is shown that the empirical particle density under the diffusive scaling solves a quasi-linear integro-differential evolution equation with Dirichlet boundary conditions. The associated dynamical large deviation principle is proved. Furthermore, for $\beta$ small enough, it is also demonstrated that the empirical particle density obeys a law of large numbers with respect to the stationary measures (hydrostatic). The macroscopic particle density solves a non local, stationary, transport equation.

Mots clés: Kawasaki process; Kac potential; Large deviations; Stationary nonequilibrium states

Domaines: Probabilités [math.PR]

Mustapha Mourragui, Enza Orlandi. Boundary driven Kawasaki process with long range interaction: dynamical large deviations and steady states. 2011. ⟨hal-00634205v2⟩