Minimal Wasserstein Surfaces

TL;DR

This paper introduces the concept of minimal surfaces in Wasserstein space, providing mathematical formulations and explicit solutions, to facilitate understanding of geometric properties relevant in stochastic thermodynamics.

Contribution

It develops the first formal definition of minimal surfaces in Wasserstein space and derives associated PDEs, with explicit Gaussian solutions, bridging geometry and thermodynamics.

Findings

Derived a two-parameter minimal surface equation in Wasserstein space.

Presented explicit solutions using Gaussian covariance matrices.

Connected minimal surfaces to thermodynamic dissipation and work extraction.

Abstract

In finite-dimensions, minimal surfaces that fill in the space delineated by closed curves and have minimal area arose naturally in classical physics in several contexts. No such concept seems readily available in infinite dimensions. The present work is motivated by the need for precisely such a concept that would allow natural coordinates for a surface with a boundary of a closed curve in the Wasserstein space of probability distributions (space of distributions with finite second moments). The need for such a concept arose recently in stochastic thermodynamics, where the Wasserstein length in the space of thermodynamic states quantifies dissipation while area integrals (albeit, presented in a special finite-dimensional parameter setting) relate to useful work being extracted. Our goal in this work is to introduce the concept of a minimal surface and explore options for a suitable construction. To this end, we introduce the basic mathematical problem and develop two alternative formulations. Specifically, we cast the problem as a two-parameter Benamou-Breiner type minimization problem and derive a minimal surface equation in the form of a set of two-parameter partial differential equations. Several explicit solutions for minimal surface equations in Wasserstein spaces are presented. These are cast in terms of covariance matrices in Gaussian distributions.