Finite-Size Scaling of a First-Order Dynamical Phase Transition: Adaptive Population Dynamics and an Effective Model

Abstract

We analyze large deviations of the time-averaged activity in the one-dimensional Fredrickson-Andersen model, both numerically and analytically. The model exhibits a dynamical phase transition, which appears as a singularity in the large deviation function. We analyze the finite-size scaling of this phase transition numerically, by generalizing an existing cloning algorithm to include a multicanonical feedback control: this significantly improves the computational efficiency. Motivated by these numerical results, we formulate an effective theory for the model in the vicinity of the phase transition, which accounts quantitatively for the observed behavior. We discuss potential applications of the numerical method and the effective theory in a range of more general contexts.

Ergodicity and large deviations in physical systems with stochastic dynamics

“…Given a biased ensemble of interest, one may choose the controlled process (and hence g) in order to transform the problem into a form that is more tractable. This is very useful for numerical work [61][62][63][64][65]. It also enables analytic progress.…”

“…However, the manifestation of this phenomenon may differ between thermodynamic and dynamical transitions. This can be illustrated by the finite-size scaling behaviour at these transitions [16,51,62,75]. We summarise the associated behaviour, a more detailed analysis can be found in [62,78].…”

“…(d) Sketch of the same probability distribution in a finite system where τ is very large, compared to L. In this case the distribution is unimodal and phase coexistence involves multiple domains arranged along the time-like axis. See [79,80] and [62,78]…”

Exact Fluctuations of Nonequilibrium Steady States from Approximate Auxiliary Dynamics

“…A formally exact importance sampling can be derived through Doob's h transform, although this requires the exact eigenvector of the tilted operator that generates the biased path ensemble [24][25][26]. As this is not practical, approximate importance sampling schemes have been introduced [21,27,28], including a sophisticated iterative algorithm to improve sampling based on feedback and control [21,28].In this Letter, we will show that guiding distribution functions (GDF), used to implement importance sampling in diffusion Monte Carlo (DMC) calculations of quantum ground states, can be extended to provide an approximate, but improvable, importance sampling for the simulation of nonequilibrium steady states. We show the potential of the GDF method by computing the large deviation functions of time integrated currents at large bias values that capture very rare fluctuations within two widely studied models: a driven diffusion model and an interacting lattice model.…”

“…A formally exact importance sampling can be derived through Doob's h transform, although this requires the exact eigenvector of the tilted operator that generates the biased path ensemble [24][25][26]. As this is not practical, approximate importance sampling schemes have been introduced [21,27,28], including a sophisticated iterative algorithm to improve sampling based on feedback and control [21,28].…”