Variational approach to coarse-graining of generalized gradient flows

Abstract

In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with nondissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the smallnoise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom. Mathematics Subject Classification

Formulation of the relativistic heat equation and the relativistic kinetic Fokker–Planck equations using GENERIC

“…Note that the right-hand side of (33) is well-defined due to (30)- (32). We emphasize that the coupling is only one-direction: Eq.…”

“…Research on GENERIC from rigorously mathematical perspectives is actively carried out, see e.g., Refs. [24,[30][31][32].…”

Analysis of the mean squared derivative cost function

“…Based on the representation obtained in Theorem 1.2 of the present paper, in a companion paper [24], we provide an elementary proof for [11, Proposition 3.1] and extend [19,20] to the forward Kolmogorov equation associated with (12) of which the Kramers equation is a special case.…”

“…Equations of this type have been studied from various points of view such as trends to equilibrium [10], Gaussian estimates for the fundamental solution [11], connections to particle systems and coarse-graining [12][13][14].…”

“…The diffusion/heat equation and the aforementioned ultra‐parabolic equation are special cases of the following hypo‐elliptic equation: Equations of this type have been studied from various points of view such as trends to equilibrium , Gaussian estimates for the fundamental solution , connections to particle systems and coarse‐graining .…”