Action functional and quasi-potential for the Burgers equation in a bounded interval

Abstract

Consider the viscous Burgers equation $u_t+f(u{)}_x=\epsilon u_{xx}$ on the interval $[0,1]$ with the inhomogeneous Dirichlet boundary conditions $u(t,0)={\rho }_0$, $u(t,1)={\rho }_1$. The flux $f$ is the function $f(u)=u(1-u)$, $\epsilon >0$ is the viscosity, and the boundary data satisfy $0<{\rho }_0<{\rho }_1<1$. We examine the quasi-potential corresponding to an action functional, arising from non-equilibrium statistical mechanical models, associated to the above equation. We provide a static variational formula for the quasi-potential and characterize the optimal paths for the dynamical problem. In contrast with previous cases, for small enough viscosity, the variational problem defining the quasi potential admits more than one minimizer. This phenomenon is interpreted as a non-equilibrium phase transition and corresponds to points where the super-differential of the quasi-potential is not a singleton.

References in corpus (4)

• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Stochastic interacting particle systems out of equilibrium
• Dynamical large deviations for the boundary driven weakly asymmetric exclusion process
• Strong asymmetric limit of the quasi-potential of the boundary driven weakly asymmetric exclusion process

Cited by in corpus (3)

• Macroscopic fluctuation theory
• Large deviation principles for nongradient weakly asymmetric stochastic lattice gases
• Totally asymmetric limit for models of heat conduction