Optimal transport and anomalous thermal relaxations

We study connections between optimal transport and anomalous thermal relaxations. A prime example of anomalous thermal relaxations is the Mpemba effect, which occurs when a hot system overtakes an identical warm system and cools down faster. Conversely, optimal transport is a resource-efficient way to transport the source distribution to a target distribution in a finite time. By resource-efficient, i.e., optimal transport, what is often meant is transport with the least amount of entropy production, and this is the definition we will use. Our paradigm for a continuum system is a particle diffusing on a potential landscape, while for a discrete system, we use a three-state Markov jump process. The Mpemba effect is generically associated with high entropy production in the continuous case. At large yet finite times, the system evolution toward the target is not optimal with respect to entropy production. However, in the discrete case, we show that for specific dynamics, the optimal transport and the strong variant of the Mpemba effect can occur for the same relaxation protocol.