Steady state large deviations for one-dimensional, symmetric exclusion processes in weak contact with reservoirs

Abstract

Consider the symmetric exclusion process evolving on an interval and weakly interacting at the end-points with reservoirs. Denote by $I_{[0,T]}(\cdot )$ its dynamical large deviations functional and by $V(\cdot )$ the associated quasi-potential, defined as $V(\gamma )={\inf }_{T>0}{\inf }_u{I}_{[0,T]}(u)$, where the infimum is carried over all trajectories $u$ such that $u(0)=\bar{\rho }$, $u(T)=\gamma$, and $\bar{\rho }$ is the stationary density profile. We derive the partial differential equation which describes the evolution of the optimal trajectory, and deduce from this result the formula obtained by Derrida, Hirschberg and Sadhu \cite{DHS2021} for the quasi-potential through the representation of the steady state as a product of matrices.

References in corpus (8)

• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Macroscopic fluctuation theory
• Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process
• Microscopic versus macroscopic approaches to non-equilibrium systems
• Exclusion process with slow boundary
• Large deviations in the symmetric simple exclusion process with slow boundaries
• A quasi-potential for conservation laws with boundary conditions
• Static large deviations of boundary driven exclusion processes

Cited by in corpus (1)

• Large Deviations in the Symmetric Simple Exclusion Process with Slow Boundaries: A Hydrodynamic Perspective