Entropy production under coarse-graining limits: Stationary preservation and nonstationary convergence

Abstract

In this paper, we examine the preservation and loss of entropy production under coarse-graining limits of singularly perturbed diffusion semigroups. Under the coefficient convergence assumptions, we establish the lower semicontinuity of the stationary rate ${\sigma }_{\mathrm{ss}}$ and provide a necessary and sufficient condition for its convergence in terms of the weighted $L^2$ approximation. In the nonstationary setting, by assuming the convergence of normalized semigroup orbits, the semigroup gradient estimate, uniform in the perturbation parameter, yields the convergence of ${\sigma }_{\mathrm{ex}}$ at every positive time. Meanwhile, the lower semicontinuity and convergence results for the housekeeping entropy production ${\sigma }_{\mathrm{hk}}$ follow by arguments analogous to those in the stationary case. Finally, the obtained results are applied to slow-fast averaging and a stiff-potential limit as two concrete examples.

arXiv comment1 figure, 46 pages