Abstract
We study the large deviations of current-type observables defined for Markov diffusion processes evolving in smooth bounded regions of with reflections at the boundaries. We derive for these the correct boundary conditions that must be imposed on the spectral problem associated with the scaled cumulant generating function, which gives, by Legendre transform, the rate function characterizing the likelihood of current fluctuations. Two methods for obtaining the boundary conditions are presented, based on the diffusive limit of random walks and on the Feynman–Kac equation underlying the evolution of generating functions. Our results generalize recent works on density-type observables, and are illustrated for an N-particle single-file diffusion on a ring, which can be mapped to a reflected N-dimensional diffusion.
Highlights
The main property of nonequilibrium systems that distinguishes them from equilibrium systems is the existence of energy or particle currents produced by non-conservative internal or external forces, or coupling to reservoirs at different temperatures or chemical potentials [1]
We study the large deviations of current-type observables defined for Markov diffusion processes evolving in smooth bounded regions of Rd with reflections at the boundaries
We derive for these the correct boundary conditions that must be imposed on the spectral problem associated with the scaled cumulant generating function, which gives, by Legendre transform, the rate function characterizing the likelihood of current fluctuations
Summary
The main property of nonequilibrium systems that distinguishes them from equilibrium systems is the existence of energy or particle currents produced by non-conservative internal or external forces, or coupling to reservoirs at different temperatures or chemical potentials [1]. The fluctuations of observables, such as currents, can be studied for nonequilibrium systems using the theory of large deviations, which provides a number of general techniques for obtaining the distribution of observables in a given scaling limit (e.g. low-noise, long-time or large-volume limit) relevant to the system studied [7,8,9] Many of these techniques were successfully applied in recent years for Markov models of physical interest, including random walks, interacting particle models, such as the exclusion and zero-range models, as well as Markov diffusions described by stochastic differential equations (SDEs) (see [10,11,12,13] for useful reviews). Applications to other models, including diffusions with partial or sticky reflections at boundaries, are discussed in the final section of the paper
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