Nonequilibrium Thermodynamics of Semilinear Stochastic Partial Differential Equations

Abstract

We develop a rigorous mathematical framework for the nonequilibrium thermodynamics of semilinear stochastic partial differential equations. The invariant Gaussian measure of the associated Ornstein-Uhlenbeck process, which need not be time-reversible, serves as the reference measure. We define thermodynamic quantities in terms of probability densities and derive the entropy balance equation. We obtain explicit expressions for the entropy production rate and prove that its vanishing in the stationary state is equivalent to time reversibility. Sample-path entropy production is identified with the logarithm of the Radon-Nikodym derivative of the forward path measure with respect to the time-reversed path measure and is shown to satisfy fluctuation theorems. For nonstationary systems, sample-path entropy production decomposes into housekeeping heat and free-energy dissipation, each satisfying an integral fluctuation theorem. Both have nonnegative expectations, and the mean free-energy dissipation equals the decrease in relative entropy with respect to the stationary distribution. The theory is applied to semilinear stochastic heat equations with spatially correlated noise, a random polymer model, a nonlinear model with a divergence-free drift, and two coupled stochastic heat equations in contact with heat baths at different temperatures.