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We find analytically the complete set of eigenvalues and eigenvectorsassociated with Metropolis dynamics on a complete graph. As an application, weuse this information to study a counter-intuitive relaxation phenomenon, calledthe Mpemba effect. This effect describes situations when upon performing athermal quench, a system prepared in equilibrium at high temperatures relaxesfaster to the bath temperature than a system prepared at a temperature closerto that of the bath. We show that Metropolis dynamics on a complete graph doesnot support weak nor strong Mpemba effect, however, when the graph is notcomplete, the effect is possible.

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