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

Abstract

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.

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• Anomalous heating in a colloidal system
• Mpemba effect in driven granular Maxwell gases
• Paths towards equilibrium in molecular systems: a mesoscopic Mpemba-like effect in water
• Mpemba effect in an anisotropically driven granular gas
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• Mpemba effect in anisotropically driven inelastic Maxwell gases
• Mpemba effect in the relaxation of an active Brownian particle in a trap without metastable states