GENERIC formalism of a Vlasov–Fokker–Planck equation and connection to large-deviation principles

Abstract

In this paper we discuss the connections between a Vlasov-Fokker-Planck equation and an underlying microscopic particle system, and we interpret those connections in the context of the GENERIC framework (Öttinger 2005). This interpretation provides (a) a variational formulation for GENERIC systems, (b) insight into the origin of this variational formulation, and (c) an explanation of the origins of the conditions that GENERIC places on its constitutive elements, notably the so-called degeneracy or non-interaction conditions. This work shows how the general connection between large-deviation principles on one hand and gradient-flow structures on the other hand extends to non-reversible particle systems.

Variational approach to coarse-graining of generalized gradient flows

“…Gradient flows and large-deviation principles As mentioned in the introduction, this approach using the duality formulation of the rate functionals is motivated by our recent results on the connection between generalised gradient flows and large-deviation principles [2,3,24,26,27,52]. We want to discuss here how the two overlap but are not the same.…”

“…dissipative systems. The approach of this paper also applies to certain variational-evolutionary systems that include non-dissipative effects, such as GENERIC systems [26,62]; our examples illustrate this. Since our approach only uses the duality structure of the rate functionals, which holds true for more general systems, this method also works for other limits in non-gradient-flow systems such as the Langevin limit of the Nosé-Hoover-Langevin thermostat [31,61,68].…”

“…In particular it has been shown that the sequence (ρ n ) has a large-deviation property [9,22,26] which characterizes the probability of finding the empirical measure far from the limit ρ, written informally as…”

“…If we assume that the initial data X n i are chosen to be deterministic, and such that the initial empirical measure ρ n (0) converges narrowly to some ρ 0 , then I has the form [26] I (ρ) := sup…”

Symmetries and Geometrical Properties of Dynamical Fluctuations in Molecular Dynamics

“…It may be checked directly from the path probabilities (13) that this gives the correct heat transfer in our case. Our inference from [24] is that one should not regard (16) as a fundamental formula for heat flow: one should instead compute the heat transfer to the bath directly using (36) and then derive the corresponding formula in terms of path probabilities. Based on that assumption, it is easily verified that for the conditioned ensembles as defined here, one should take…”