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arXiv · 2402.11267

On the optimal relaxation rate for the Metropolis algorithm in one dimension

A. Patrón·A. D. Chepelianskii·A. Prados·E. Trizac
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Abstract

We study the relaxation of the Metropolis Monte Carlo algorithm corresponding to a single particle trapped in a one-dimensional confining potential, with even jump distributions that ensure that the dynamics verifies detailed balance. Previous work suggested that, for smooth jump distributions, the fastest relaxation rate is obtained as a result of the competition between diffusive and rejection-dominated dynamics. In this work, we show that a new regime comes into play for two-peaked jump distributions, where the relaxation dynamics is neither dominated by diffusion nor rejection: the eigenmodes adopt an oscillatory form, reminiscent of charge density waves (CDW) -- thus we term this new regime the CDW regime. Using a combination of numerical and analytical techniques, the parameter regions corresponding to diffusion, rejection, and CDW are characterised, as well as the transition lines between them -- i.e. a phase diagram is built. The optimal relaxation rate is located at the triple point of phase coexistence, where the transition lines (diffusive-rejection, diffusive-CDW, and CDW-rejection) intersect. Our theoretical framework is checked versus the numerical diagonalisation of the master equation. We also briefly discuss more sophisticated attempts at optimising the relaxation rate to equilibrium.

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A. Patrón, A. D. Chepelianskii, A. Prados, E. Trizac. 2024-02-17. On the optimal relaxation rate for the Metropolis algorithm in one dimension. https://doi.org/10.1088/1742-5468/ada7e6

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