Mpemba effect on near-complete graphs with independence Metropolis–Hastings dynamics

Abstract We use independence Metropolis–Hastings dynamics on a complete graph as a testbed to investigate the Mpemba effect—a striking anomalous relaxation phenomenon characterized by a nonmonotonic dependence of the relaxation dynamics on the initial condition. Such nonmonotonicity is exemplified whenever a system prepared at a high temperature cools down to an environment’s temperature more efficiently than an identical system initialized at a lower temperature, which is closer to the target. We show that Metropolis dynamics on a complete graph does not exhibit the Mpemba effect, whereas the effect can arise on non-complete graphs with the same dynamics. Alternatively, even on a complete graph with Metropolis dynamics, anomalous relaxation can occur if the initial states are at equilibrium in the presence of a potential that is absent when the system is coupled to the target bath. Finally, we consider the more general case of a system relaxing with independence Metropolis–Hastings dynamics. We show that, with nonuniform priors, the dynamics can exhibit nonanalyticity in the relaxation direction, akin to a dynamical phase transition, and thus the Mpemba effect–even on a complete graph.