Lifting the fog - a case for non-reversible "lifted" Markov chains

Abstract

Phase transitions appear all over science, and are familiar from everyday life, as water boiling, sugar melting into caramel or as nematic molecules turning smectic in liquid-crystal displays. The dynamics of phase transitions can be extremely slow, as for example when fog in winter does not lift, that is when the coarsening takes much time from many tiny water droplets to fewer but larger rain drops that feel the pull of gravity. The dynamics of phase transitions is relevant also for the performance of computer algorithms. In the ubiquitous Metropolis Monte Carlo algorithm, the mixing dynamics towards equilibrium leads towards the solution of a sampling problem. It is governed by the same reversibility and detailed-balance principles as the overdamped physical dynamics of fog. For the phase-separated Lennard-Jones system, we describe here how the coarsening dynamics of non-reversible "lifted" variants of the Metropolis algorithm proceeds on much faster time scales, with the microscopic non-reversibility translating into large-scale relative motion of droplets that is impossible under the Ostwald-ripening condition of reversibility. A density-displacement coupling moves droplets relative to each other through a lensing effect. Efficient implementations of the long-range Metropolis algorithm and its non-reversible lifting (event-chain Monte Carlo) allow us to show that, in consequence, the coarsening growth exponent is larger under lifting. For large system sizes, the computing problem is thus solved infinitely faster than before, with the outcome strictly unchanged with respect to the Metropolis algorithm. We also discuss the larger setting of our findings, namely that "lifted" non-reversible algorithms can be set up for generic reversible sampling methods, with applications going much beyond our example of lifting fog.