  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.
