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.

Figures (5)

• Figure 1: Coarsening of the two-dimensional Lennard-Jones system ($N=10^4$ particles at density $\rho = 0.1$) from super saturation (a) into a single liquid drop surrounded by vapor (d). After nucleating several droplets (b), the system goes through a coarsening process, that gradually reduces the number of liquid clusters while increasing their average size (c). In the absence of gravity, the equilibrium state contains a single floating drop. The Metropolis algorithm, that mimics the physics of Ostwald ripening, needs about $10^{12}$ steps to complete this process, while its lifted counterpart, event-chain Monte Carlo, reaches equilibrium after around $10^9$ events.
• Figure 2: Density--velocity coupling of event-chain Monte Carlo in a periodic liquid ribbon. (a): The ribbon in its periodic simulation box ($\sim 10^5$ liquid particles at density $\rho=0.75$ surrounded by vapor, see Supplementary Material for details). The event chain's direction of motion is towards the right ($+x$) (b): Typical event chain, from its entry into the ribbon to its exit from it. (b1): Inside the ribbon, the active particle interacts---and the event chain grows---symmetrically around the direction of motion. (b2): At the ribbon boundary, the active particle interacts symmetrically around the density gradient and the event chain grows with a component perpendicular to the direction of motion (here, in the $-y$ direction). (c): Mean vertical growth component $\delta y$ of the event chain as a function of its horizontal position $\hat{x} \in [-\Delta, \Delta]$ in the ribbon. The opposite boundary effects are due to the opposite sign of the density gradient at the two interfaces. (d): Probability distribution of the active particle's $\hat{x}$ coordinate. The event chain grows predominantly near the forward boundary, as already evident from (b). The curves in panels (c) and (d) are smoothed, schematic, versions of the plots from the Supplementary Material.
• Figure 3: Lensing effect of event-chain Monte Carlo and relative motion of droplets. (a): Circular liquid droplet of radius $R$ and pair of droplets in their periodic simulation boxes. (b): Lensing effect of event chain Monte Carlo inside a liquid drop. The event chain, entering the drop uniformly undergoes multiple gradient effects and then leaves it predominantly near its tip. (c): Relative motion of two droplets created by the lensing effect. An event chain entering drop A, even at its extremity, likely exits it without touching drop B (see histogram). The expected time spent in A, and thus the displacement of A in $+x$ direction is larger than the time spent in and the displacement of B. (d): Probability distribution of the variation $\delta_f - \delta_\parallel$ in the horizontal separation between the droplets. Its negative mean value shows that the two droplied approach each other. The curves in panels (b), (c) and (d) are smoothed versions of histograms from the Supplementary Material.
• Figure 4: Equilibration dynamics of the factorized Metropolis algorithm and of event-chain Monte Carlo from an initial configuration with two spherical droplets (with 2565 particles each, at local density $\rho=0.75$) surrounded by 3872 vapor particles at local density $\rho=0.05$, and overall density $\rho=0.1$. Under the Metropolis dynamics, the system equilibrates through Ostwald ripening, that is, through the evaporation--condensation of vapor at the surface of immobile liquid droplets (here, after around $10^{11}$ moves). Under event-chain Monte Carlo dynamics, the two droplets move relative to each other, allowing them to coalesce much faster (here, after about $10^8$ events). We attribute this acceleration to the lensing effect.
• Figure 5: Coarsening dynamics of event-chain Monte Carlo and factorized Metropolis for a two-dimensional Lennard-Jones system in the phase-coexistence regime ($\rho = 0.1$). The average droplet radius $\left\langle R \right\rangle$ depends on time as a power law of growth exponent $\alpha$, with the time $t$ corresponding to the total distance traveled by all particles in the system. Event-chain Monte Carlo outperforms Metropolis both in terms of scaling and of absolute CPU time. Yet, for finite systems, the growth exponent $\alpha$ is found to depend on the event-chain length. Roughly, event-chain Monte Carlo mixes 100 times faster than the factorized Metropolis algorithm for $N=10^4$, and 1000 times faster for $N=10^5$.