Thermodynamic unification of optimal transport*

Optimal transport is a mature field of mathematics and statistics, focused on the theory of optimal planning and cost associated with the transportation of probability distributions. Recently, a profound connection between optimal transport and stochastic thermodynamics has emerged, particularly in the context of continuous-state overdamped Langevin dynamics. This connection has revealed that the problem of minimizing entropy production can be mapped to the optimal transport problem. Moreover, this connection has led to critical applications, including the establishment of tight speed limits and the finite-time Landauer principle.

In this talk, I will introduce a thermodynamic framework for discrete optimal transport, which will illustrate the analogous relationship between stochastic thermodynamics and optimal transport within discrete-state systems. Specifically, I will present variational formulas that establish connections between stochastic thermodynamics of discrete Markov processes and discrete Wasserstein distances. These formulas not only unify the relationship between thermodynamics and optimal transport theory in both discrete and continuous cases but also extend it to the quantum realm. Notably, they yield remarkable applications of stochastic and quantum thermodynamics, such as stringent thermodynamic speed limits and the finite-time Landauer principle at arbitrary temperatures.