Long-range order in nonequilibrium systems of interacting Brownian linear oscillators

Long-range order (lro) is established with the help of a generalized Peierls argument for non-equilibrium lattice systems of one-dimensional ( linear) interacting oscillators whose equation of motion ( for a finite number of them) is the Smolouchowski equation for the density of a probability distribution. Interaction is mediated through the pair nearest-neighbor quadratic translation invariant potential. The initial density is Gibbsian with a potential energy satisfying the Ruelle superstability and regularity conditions.