2010/11/30 by Lorenzo Bertini, Alessandra Faggionato, Davide Gabrielli · 1 citation
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Large deviations theory #Lattice (music) #Lattice gas automaton #Nonlinear system #Rate function #Stochastic process #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Torus #cond-mat.stat-mech #math-ph #math.MP #math.PR
paper · pdf · doi:10.1214/11-aap805
published as Annals of Applied Probability 2013, Vol. 23, No. 1, 1-65 · Published in at http://dx.doi.org/10.1214/11-AAP805 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2013/01/25 · arxiv created 2013/02/12 · arxiv updated 2013/02/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider a lattice gas on the discrete d-dimensional torus (ℤ/Nℤ)d with a generic translation invariant, finite range interaction satisfying a uniform strong mixing condition. The lattice gas performs a Kawasaki dynamics in the presence of a weak external field E/N. We show that, under diffusive rescaling, the hydrodynamic behavior of the lattice gas is described by a nonlinear driven diffusion equation. We then prove the associated dynamical large deviation principle. Under suitable assumptions on the external field (e.g., E constant), we finally analyze the variational problem defining the quasi-potential and characterize the optimal exit trajectory. From these results we deduce the asymptotic behavior of the stationary measures of the stochastic lattice gas, which are not explicitly known. In particular, when the external field E is constant, we prove a stationary large deviation principle for the empirical density and show that the rate function does not depend on E.