Acceleration of convergence to equilibrium in Markov chains by breaking detailed balance

TL;DR

This paper investigates how breaking detailed balance in Markov chains accelerates convergence to equilibrium, providing theoretical insights, quantitative measures, and geometric interpretations of the phenomenon.

Contribution

It offers a comprehensive analysis of the effects of breaking detailed balance on convergence rates, including a quantitative expression and geometric interpretation within hydrodynamic limits.

Findings

Irreversible processes converge faster than reversible ones.

Quantitative expression for acceleration in hydrodynamic limit.

Geometric interpretation involving antisymmetric currents.

Abstract

We analyse and interpret the effects of breaking detailed balance on the convergence to equilibrium of conservative interacting particle systems and their hydrodynamic scaling limits. For finite systems of interacting particles, we review existing results showing that irreversible processes converge faster to their steady state than reversible ones. We show how this behaviour appears in the hydrodynamic limit of such processes, as described by macroscopic fluctuation theory, and we provide a quantitative expression for the acceleration of convergence in this setting. We give a geometrical interpretation of this acceleration, in terms of currents that are \emph{antisymmetric} under time-reversal and orthogonal to the free energy gradient, which act to drive the system away from states where (reversible) gradient-descent dynamics result in slow convergence to equilibrium.