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

TL;DR

This paper analyzes the quasi-potential for the viscous Burgers equation with boundary conditions, revealing multiple minimizers for small viscosity, indicating a non-equilibrium phase transition.

Contribution

It provides a static variational formula for the quasi-potential and characterizes optimal paths, highlighting a novel phase transition phenomenon.

Findings

Multiple minimizers of the variational problem for small viscosity

Identification of a non-equilibrium phase transition

Characterization of optimal paths in the dynamical problem

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.