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 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.
Highlights
In this paper we analyse the effects of breaking detailed balance for interacting particle systems, and their hydrodynamic scaling limits
Those with detailed balance are special—they correspond to Markov chains that are reversible with respect to an invariant measure π
We investigate how breaking detailed balance affects the hydrodynamic limit of the model—in this latter case, convergence to equilibrium is most analysed via large deviations of the empirical measure [35,36]
Summary
In this paper we analyse the effects of breaking detailed balance for interacting particle systems (as described by Markov processes [32]), and their hydrodynamic scaling limits (as described by Macroscopic Fluctuation Theory [9]). Those with detailed balance are special—they correspond to Markov chains that are reversible with respect to an invariant measure π These models are important because their steady states are time-reversal symmetric and lack any persistent currents, so they can be used to describe systems that relax to states of thermal equilibrium. They have applications outside physics, because given a (possibly non-normalised) measure ν, it is straightforward to design a reversible Markov chain whose invariant measure π is proportional to ν This construction is at the root of many Markov chain Monte Carlo (MCMC) methods [2,34], in which one typically aims to generate large numbers of uncorrelated samples from a prescribed distribution π. Such methods have widespread applications including Bayesian learning, protein folding and cryptography [14]
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