Large deviations and gradient flows

Abstract

In recent work we uncovered intriguing connections between Otto’s characterization of diffusion as an entropic gradient flow on the one hand and large-deviation principles describing the microscopic picture (Brownian motion) on the other. In this paper, we sketch this connection, show how it generalizes to a wider class of systems and comment on consequences and implications. Specifically, we connect macroscopic gradient flows with large-deviation principles, and point out the potential of a bigger picture emerging: we indicate that, in some non-equilibrium situations, entropies and thermodynamic free energies can be derived via large-deviation principles. The approach advocated here is different from the established hydrodynamic limit passage but extends a link that is well known in the equilibrium situation.

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.…”

“…It is well known how such duality structures give rise to good convergence properties such as (3), but the focus in this paper is on how this duality structure combines well with coarse-graining. In this paper we define coarse-graining to be a shift to a reduced, lower dimensional description via a coarse-graining map ξ : X → Y which identifies relevant information and is typically highly non-injective.…”

“…In recent years we have discovered that many gradient flows, and also many 'generalized' gradient systems, can be matched one-to-one to the largedeviation characterization of some stochastic process [2,3,24,26,27,52]. The large-deviation rate functional, in this connection, can be seen to define the generalized gradient system.…”

Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

“…(1) is a Wasserstein gradient flow (we describe this below), and that the Wasserstein gradient flows have many interesting and useful properties, we asked the question "how is the Wasserstein gradient-flow structure of (1) related to the modeling of (1)?" We answered this question [1][2][3] by connecting the Wasserstein gradient-flow structure to the large-deviation behavior of stochastic particle systems. Before turning to the content of the present paper, we briefly describe these previous results.…”

“…We contrast these gradient-flow structures with those for processes describing the diffusion of mass, most importantly the class of Wasserstein gradient-flow systems. The linear and nonlinear heat-equation gradient-flow structures are each driven by entropy terms of the form − log ρ; they involve dissipation or mobility terms of order ρ 2 …”