Abstract

One of the ways to, in principle, improve the performance of a Markov chain Monte Carlo algorithm is to optimize the choice of the transition rate matrix. In a related problem of developing an optimal thermal relaxation protocol, one wants to design a dynamic that achieves fast relaxation to a steady state or thermal equilibrium. In my talk, I will introduce the Mpemba effect, a prime example of anomalous thermal relaxations—where a system prepared at an initial hot temperature relaxes faster in the cold environment than a system prepared at a warm temperature. For a system with quenched disorder and Metropolis-Hasting dynamics, we study how the thermal relaxation changes in response to modifications of the transition rates. We show that Metropolis-Hastings dynamics on a complete graph does not support the Mpemba effect; however, when the graph is not complete, the effect is possible. We show that rank-one modifications of the transition rates can introduce anomalous thermal relaxation effects. I will comment on potential applications of anomalous thermal relaxation in designing MCMC algorithms.  This material is based upon work supported by the National Science Foundation under Grant No. DMR -1944539.