Finite-time and finite-size scalings in the evaluation of large-deviation functions: Numerical approach in continuous time

Abstract

Rare trajectories of stochastic systems are important to understand -because of their potential impact. However, their properties are by definition difficult to sample directly. Population dynamics provides a numerical tool allowing their study, by means of simulating a large number of copies of the system, which are subjected to selection rules that favor the rare trajectories of interest. Such algorithms are plagued by finite simulation time-and finite population size-effects that can render their use delicate. In this paper, we present a numerical approach which uses the finite-time and finite-size scalings of estimators of the large deviation functions associated to the distribution of rare trajectories. The method we propose allows one to extract the infinite-time and infinite-size limit of these estimators which -as shown on the contact process -provides a significant improvement of the large deviation functions estimators compared to the the standard one.

Sampling rare events across dynamical phase transitions

“…λ ≈ 0 or q ≈ q st = σ 0Â E, small fluctuation regime), as otherwise expected, these corrections mount up as z → 0 (equivalently λ → −E or q → 0) tending to overestimate the value of |µ(z)|. In any case, these corrections scale as 1/N c , in agreement with [91,92], so as to observe an excellent convergence toward the macroscopic fluctuation theory prediction (solid line) for large enough N c and the largest N explored, as displayed in the inset to Fig. 12.…”

“…One possibility recently explored consists in introducing an interpolation technique for the large deviation function based on the analysis of the systematic errors of a birth-death process [91]. An extension of such technique has led to a more efficient version of the cloning algorithm in continuous time which significantly improves the large deviation function estimators in the long time and large number of clones limit [92]. In addition, some rigorous bounds on convergence properties of the algorithm have been established [93,94].…”

Dynamical phase transitions for the activity biased Ising model in a magnetic field

“…The accuracy of the cloning method requires that the clone population N c is large, otherwise the model suffers from both systematic and random errors [67][68][69][70]. In principle, the method can yield accurate results whatever the values of the control parameters (J c 1 , J c 2 , K c 3 , h c ), but in practice one requires a good choice of these parameters, otherwise the number of clones required for accurate results may be prohibitively large.…”

Large deviations and optimal control forces for hard particles in one dimension

“…(2019) 123208 the detailed-balance property of TPS [51] means that it samples directly from P s or P V s as defined in (14) and (22). By contrast, cloning methods do not sample directly from a target distribution; they do allow estimation of averages with respect to these distributions but the associated statistical estimators have systematic errors (bias) which only disappear as the population size tends to infinity [33,74]. Estimation of statistical uncertainties is also simpler for TPS, see section 5.3.…”

“…The second method for deriving a suitable control potential is the standard mathematical approach: consider the adjoint (Hermitian conjugate) of the eigenproblem (74) which is…”