The solution of the Metropolis dynamics on a complete graph with applications to anomalous thermal relaxations

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 an unusual 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.