Tartero Shiratani Krauth 2026
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| - | ''G. Tartero, S. Shiratani, W. Krauth'' ''''' Lifting the fog - a case for non-reversible "lifted" Markov chains '''''' ''' arXiv2603.16855 (2026) ''''' | + | '''G. Tartero, S. Shiratani, W. Krauth''' ''''' Lifting the fog - a case for non-reversible "lifted" Markov chains '''''' ''' arXiv2603.16855 (2026) ''''' |
| '''Abstract''' | '''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. | 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. | ||
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| + | '''Context''' | ||
| + | In this 2026 preprint, my coauthors Gabriele Tartero (PhD candidate at ENS / Sorbonne), Sora Shiratani (PhD candidate at the University of Tokyo) and I considered the case of a two-dimensional Lennard-Jones system, that is a model of interacting classical particles that presents itself in many different phases, namely vapor, liquid, hexatic, and solid. This looks complicated, but we only looked at a certain number N of particles in a two-dimensional box of volume (that is, area) V, but tweaked the parameters such that there is no spatially homogeneous phase with local density N / V all over the box. If we start a simulation at such a uniform density, we're in fact creating a super-saturated vapor, which has a tendency to nucleate small bubbles. These bubbles become bigger and bigger, and fewer and fewer, until one ends up with the equilibrium state characterized by a single liquid bubble (of density much larger than N/V) and of vapor (of density somewhat smaller than N/V). The process of bubbles getting bigger but few is called ''coarsening''. How coarsening takes place is a fascinating story. In fact, under the conditions of reversibility, bubbles can't really move, and the growth of large bubble at the expense of small ones takes place through a process called "Ostwald ripening": particles preferentially evaporate from immobile small bubbles and preferentially adsorb onto immobile large bubbles. This is what is realized in the below left panel, which implements the Metropolis algorithm. Ostwald ripening is notoriously slow, and the mean bubble size grows with time t as t^{1/3}. On the other hand, event-chain Monte Carlo proceeds much faster, it overcomes the limitations of Ostwald ripening, as is shown in the below right panel, a movie which runs at 1000 times the speed of the Metropolis algorithm. | ||
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| + | [[Image:Metropolis.gif|frameless|100px]] | ||
| + | [[Image:ECMCCoarsening.gif|frameless|100px]] | ||
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| + | Event-chain Monte Carlo, as shown in the above right panel, has big bubbles move around a lot. As further discussed in our paper, this is a non-equilibrium effect. In equilibrium (that is, in the steady state, where there is only a single bubble), this buoyancy stops all of a sudden, as shown in the below panel. | ||
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| + | [[Image:ECMC_Equilibrium.gif|frameless|100px]] | ||
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| [http://arxiv.org/pdf/2603.16855 Electronic version (from arXiv)] | [http://arxiv.org/pdf/2603.16855 Electronic version (from arXiv)] | ||
Current revision
G. Tartero, S. Shiratani, W. Krauth Lifting the fog - a case for non-reversible "lifted" Markov chains ' arXiv2603.16855 (2026)
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.
Context
In this 2026 preprint, my coauthors Gabriele Tartero (PhD candidate at ENS / Sorbonne), Sora Shiratani (PhD candidate at the University of Tokyo) and I considered the case of a two-dimensional Lennard-Jones system, that is a model of interacting classical particles that presents itself in many different phases, namely vapor, liquid, hexatic, and solid. This looks complicated, but we only looked at a certain number N of particles in a two-dimensional box of volume (that is, area) V, but tweaked the parameters such that there is no spatially homogeneous phase with local density N / V all over the box. If we start a simulation at such a uniform density, we're in fact creating a super-saturated vapor, which has a tendency to nucleate small bubbles. These bubbles become bigger and bigger, and fewer and fewer, until one ends up with the equilibrium state characterized by a single liquid bubble (of density much larger than N/V) and of vapor (of density somewhat smaller than N/V). The process of bubbles getting bigger but few is called coarsening. How coarsening takes place is a fascinating story. In fact, under the conditions of reversibility, bubbles can't really move, and the growth of large bubble at the expense of small ones takes place through a process called "Ostwald ripening": particles preferentially evaporate from immobile small bubbles and preferentially adsorb onto immobile large bubbles. This is what is realized in the below left panel, which implements the Metropolis algorithm. Ostwald ripening is notoriously slow, and the mean bubble size grows with time t as t^{1/3}. On the other hand, event-chain Monte Carlo proceeds much faster, it overcomes the limitations of Ostwald ripening, as is shown in the below right panel, a movie which runs at 1000 times the speed of the Metropolis algorithm.
Event-chain Monte Carlo, as shown in the above right panel, has big bubbles move around a lot. As further discussed in our paper, this is a non-equilibrium effect. In equilibrium (that is, in the steady state, where there is only a single bubble), this buoyancy stops all of a sudden, as shown in the below panel.
