Analytical study of tunneling times in flat histogram Monte Carlo

TL;DR

This paper develops an analytical model for tunneling times in flat histogram Monte Carlo simulations, revealing how energy space dynamics and density of states influence equilibration and tunneling times, especially in complex systems like spin glasses.

Contribution

It provides a complete analytic characterization of tunneling times in multicanonical methods based on the density of states and relates these to equilibration times in spin glass models.

Findings

Tunneling times can be significantly larger than equilibration times in certain models.

Transitions between low-density energy states dominate long-time dynamics.

The model applies to single spin flip dynamics and similar processes.

Abstract

We present a model for the dynamics in energy space of multicanonical simulation methods that lends itself to a rather complete analytic characterization. The dynamics is completely determined by the density of states. In the \pm J 2D spin glass the transitions between the ground state level and the first excited one control the long time dynamics. We are able to calculate the distribution of tunneling times and relate it to the equilibration time of a starting probability distribution. In this model, and possibly in any model in which entering and exiting regions with low density of states are the slowest processes in the simulations, tunneling time can be much larger (by a factor of O(N)) than the equilibration time of the probability distribution. We find that these features also hold for the energy projection of single spin flip dynamics.