  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
"a resource-efficient way," what is often meant is with the least amount of
entropy production. 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. In the continuous case, the Mpemba effect is generically
associated with high entropy production. As such, at large yet finite times,
the system evolution toward the target is not optimal in this respect. 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.
