A nonequilibrium-to-equilibrium mapping and its application to the perturbed sine-Gordon equation

Abstract

Given a partial differential equation (pde) in one time and D spatial dimensions which is driven by noise ζ( x ; t) with a known distribution P[ζ], one may find the distribution P[ψ] of the solution ψ( x ; t). Furthermore, the distribution may be written as P[ψ] ∼ e -β H[ψ] with H[ψ] an effective Hamiltonian in D + 1 dimensions (time having become an extra spatial dimension). Then, the most probable solution of the pde is that function which minimizes H[ψ]. Here, we describe this method and illustrate it for the damped, driven sine-Gordon equation in one spatial dimension ( D = 1).