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Persistent energy flow for a stochastic wave equation model in nonequilibrium statistical mechanics

We consider a one-dimensional partial differential equation system modeling heat flow around a ring. The system includes a Klein-Gordon wave equation for a field satisfying spatial periodic boundary conditions, as well as Ornstein-Uhlenbeck stochastic differential equations with finite rank dissipation and stochastic driving terms modeling heat baths. There is an energy flow around the ring. In the case of a linear field with different (fixed) bath temperatures, the energy flow can persist even when the interaction with the baths is turned off. A simple example is given.