Thermodynamic interpretation of Wasserstein distance

TL;DR

This paper establishes a fundamental link between dissipation in stochastic dynamics and the Wasserstein distance, providing bounds and optimal strategies for minimal dissipation during diffusion processes.

Contribution

It introduces a thermodynamic interpretation of Wasserstein distance, deriving bounds on dissipation and optimal forces for stochastic diffusion processes.

Findings

Dissipation is proportional to Wasserstein distance divided by process duration.

Lower bounds on dissipation depend only on initial and final means and covariances.

Optimal forces minimizing dissipation are derived for given state statistics.

Abstract

We derive a relation between the dissipation in a stochastic dynamics and the Wasserstein distance. We show that the minimal amount of dissipation required to transform an initial state to a final state during a diffusion process is given by the Wasserstein distance between the two states, divided by the total time of the process. This relation implies a lower bound on the dissipation for any diffusion process in terms of its initial and final state. Using a lower bound on the Wasserstein distance, we further show that we can give a lower bound on the dissipation in terms of only the mean and convariance matrix of the initial and final state. We apply this result to derive the optimal forces that minimize the dissipation for given initial and final mean and covariance.