Perturbative Calculation of Quasi-Potential in Non-equilibrium Diffusions: A Mean-Field Example

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 (Freidlin and Wentzell, Random perturbations of dynamical systems, 2012 ). In many interacting particle systems, the particle density is described by fluctuating hydrodynamics governed by Macroscopic Fluctuation Theory (Bertini et al., arXiv:1404.6466 , 2014 ), 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 (Stochastics 20:247–308, 1987 ; Stochastic differential systems, vol 96, pp 1–10, 1987 ). 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 (Prog Theoret Phys 75:1105–1110, [ 74 ]). This is one of few systems for which non-equilibrium free energies can be computed and analyzed in an effective way, at least perturbatively.