Learning nonequilibrium statistical mechanics and dynamical phase transitions

Abstract

Nonequilibrium statistical mechanics exhibit a variety of complex phenomena far from equilibrium. It inherits challenges of equilibrium, including accurately describing the joint distribution of a large number of configurations, and also poses new challenges as the distribution evolves over time. Characterizing dynamical phase transitions as an emergent behavior further requires tracking nonequilibrium systems under a control parameter. While a number of methods have been proposed, such as tensor networks for one-dimensional lattices, we lack a method for arbitrary time beyond the steady state and for higher dimensions. Here, we develop a general computational framework to study the time evolution of nonequilibrium systems in statistical mechanics by leveraging variational autoregressive networks, which offer an efficient computation on the dynamical partition function, a central quantity for discovering the phase transition. We apply the approach to prototype models of nonequilibrium statistical mechanics, including the kinetically constrained models of structural glasses up to three dimensions. The approach uncovers the active-inactive phase transition of spin flips, the dynamical phase diagram, as well as new scaling relations. The result highlights the potential of machine learning dynamical phase transitions in nonequilibrium systems.

K-core attack, equilibrium K-core, and kinetically constrained spin system

“…given system size L (and the number of vertices N = L 3 ), when the initial fraction p of occupied vertices is smaller than certain critical value p c (L), there is a high chance that the random initial configuration 𝑐 0 contains no 4-core. When p > p c (L), with high probability there is an extensive 4-core, and its relative size f is close to the value predicted by the mean-field theory (12). The value of p c (L) for each L is determined as the point of p at which fifty percent of the independent samples 𝑐 0 contain an extensive 4-core.…”

“…[74] We also see from Fig. 5(a) that, when the initial occupation density p exceeds 0.75, the 3-core relative size can again be well described by the mean field formula (12).…”

“…The fraction r of samples containing an extensive 4-core is also shown in panel (b), and here the value of f is averaged only over these non-trivial samples. The solid curves are the predictions of the mean-field (MF) formula (12). The total number of vertices are N = 32768, 262144 and 2097152, corresponding to side lengths L = 32, 64 and 128, respectively.…”

“…The FA model and other kinetically constrained spin models have served as basic theoretical platforms to study the dynamical phase transitions of supercooled liquids. [8,9,[12][13][14] For simplicity, we will consider only the FA model here (Section 2).…”

“…The relative size f of the frozen occupied K-core with K = 3 versus the initial fraction p of occupied vertices, for the regular random graph ensemble of degree D = 6. The solid line is theoretical prediction(12). Circles and error bars (denoting standard deviation) are simulation results obtained on a single graph instance of size N = 32768 [68].…”