Minimum entropy production, detailed balance and Wasserstein distance for continuous-time Markov processes

TL;DR

This paper explores how to minimize entropy production in continuous-time Markov processes, linking it to Wasserstein distance, and introduces new bounds and optimal transport formulations for such stochastic dynamics.

Contribution

It establishes a connection between entropy production minimization, Wasserstein distance, and activity constraints in Markov jump processes, providing new theoretical insights and formulations.

Findings

Minimum entropy production can be achieved at diverging activity without constraints.

Optimal dynamics for fixed activity are governed by conservative forces.

A new speed limit relates dissipation, transitions, and Wasserstein distance.

Abstract

We investigate the problem of minimizing the entropy production for a physical process that can be described in terms of a Markov jump dynamics. We show that, without any further constraints, a given time-evolution may be realized at arbitrarily small entropy production, yet at the expense of diverging activity. For a fixed activity, we find that the dynamics that minimizes the entropy production is given in terms of conservative forces. The value of the minimum entropy production is expressed in terms of the graph-distance based Wasserstein distance between the initial and final configuration. This yields a new kind of speed limit relating dissipation, the average number of transitions and the Wasserstein distance. It also allows us to formulate the optimal transport problem on a graph in term of a continuous-time interpolating dynamics, in complete analogy to the continuous space setting. We demonstrate our findings for simple state networks, a time-dependent pump and for spin flips in the Ising model.