Preservation of thermodynamic structure in model reduction.

Based on the availability of an invariant manifold, we develop a model-reduction procedure that preserves thermodynamic structure. More concretely, we construct the Poisson and irreversible brackets of the general equation for the nonequilibrium reversible-irreversible coupling of nonequilibrium thermodynamics by means of the ideas originally introduced for handling constraints. The general ideas are then applied to the Kramers problem, that is, the description of transitions between two potential wells separated by a high barrier. This example reveals how a fortuitous cancellation mechanism that allows a logarithmic entropy to generate a linear diffusion equation is inherited by a master equation resulting from model reduction.