Solution of the Metropolis dynamics on a complete graph with application to the Markov chain Mpemba effect

We find analytically the complete set of eigenvalues and eigenvectors associated with Metropolis dynamics on a complete graph. As an application, we use this information to study a counter-intuitive relaxation phenomenon, called the Mpemba effect. This effect describes situations when upon performing a thermal quench, a system prepared in equilibrium at high temperatures relaxes faster to the bath temperature than a system prepared at a temperature closer to that of the bath. We show that Metropolis dynamics on a complete graph does not support weak nor strong Mpemba effect, however, when the graph is not complete, the effect is possible.