2015/09/30 by Freddy Bouchet, Krzysztof Gawedzki, Cesare Nardini · 2 citations
Physics and Astronomy · #cond-mat.stat-mech
paper · pdf · doi:10.1007/s10955-016-1503-2
arxiv created 2016/03/24 · arxiv updated 2016/05/04
In stochastic systems with weak noise, the logarithm of the stationary distribution becomes proportional to a large deviation rate function called the quasi-potential. The quasi-potential, and its characterization through a variational problem, lies at the core of the Freidlin-Wentzell large deviations theory%.~\citefreidlin1984.In many interacting particle systems, the particle density is described by fluctuating hydrodynamics governed by Macroscopic Fluctuation Theory%, ~\citebertini2014,which formally fits within Freidlin-Wentzell's framework with a weak noise proportional to 1/√(N), where N is the number of particles. The quasi-potential then appears as a natural generalization of the equilibrium free energy to non-equilibrium particle systems. A key physical and practical issue is to actually compute quasi-potentials from their variational characterization for non-equilibrium systems for which detailed balance does not hold. We discuss how to perform such a computation perturbatively in an external parameter λ, starting from a known quasi-potential for λ=0. In a general setup, explicit iterative formulae for all terms of the power-series expansion of the quasi-potential are given for the first time. The key point is a proof of solvability conditions that assure the existence of the perturbation expansion to all orders. We apply the perturbative approach to diffusive particles interacting through a mean-field potential. For such systems, the variational characterization of the quasi-potential was proven by Dawson and Gartner%. ~\citedawson1987,dawson1987b. Our perturbative analysis provides new explicit results about the quasi-potential and about fluctuations of one-particle observables in a simple example of mean field diffusions: the Shinomoto-Kuramoto model of coupled rotators%. ~\citeshinomoto1986. This is one of few systems for which non-equilibrium free energies can be computed and analyzed in an effective way, at least perturbatively.