Class of Monte Carlo algorithms for dynamic problems leads to an adaptive method

Abstract

We introduce a class of Monte Carlo algorithms that solve a dynamic problem defined by the transition rates and the initial state of a discrete system. This class contains the method of Bortz, Kalos, and Lebowitz ͑BKL͒ ͓J. Comp. Phys. 17, 10 ͑1975͔͒ as a limit. We show that introducing a constant time step in a Metropolis algorithm leads to an approximation of the solution in which the system relaxation times are underestimated. This can be corrected if the time step is an adequate stochastic variable. Thus, we are able to define kinetic Metropolis algorithms and generalize them in a case of nonconstant numbers of attempt configurations. The algorithm class allows us to introduce a useful method in which the calculation of transition rates are exploited for the next step in an adaptive way. This method corresponds to a kinetic Metropolis algorithm when the rejection probability is reasonable and becomes similar to the BKL method otherwise. We describe and compare four different algorithms applied to a physical example about the diffusion in lattice gases.

Dipolar interaction and incoherent quantum tunneling: a Monte Carlo study of magnetic relaxation

“…where α is a constant. The compound [(tacn) 8 Fe 8 O 8 (OH) 8 ] 8+ (where tacn is 1-4-7-triazacyclononane) pertains, at low temperature T < 1K, to a new class which has not been studied until very recently [1,2]. This material [3], hereafter called 'Fe 8 ', is a paramagnet and the quantity m(t) of interest is the magnetization per spin, more precisely its component along a well defined axis z which is an easy magnetization axis.…”

Monolayers of hard rods on planar substrates. II. Growth

“…In each KMC iteration step, a single, currently allowed event having rate k i is chosen randomly among all such events with a relative probability k i k tot ({C}) , k tot ({C}) = j k j , where {k j } j {C} k j is the full list of allowed events at this configuration {C}. (Note that this list could include forbidden events, 45 but at the computational cost of rejecting them.) KMC is therefore effectively "rejection-free," at least in the variant of the algorithm used here, first proposed in Ref.…”

“…As events occur independently, the total random process of waiting for any among all events is also Poissonian with a mean waiting time 1/ j k j . [42][43][44][45][46] Specifically, this probability distribution of waiting times has the form P wait (t) = e − j kj /t . In each KMC iteration step, a single, currently allowed event having rate k i is chosen randomly among all such events with a relative probability k i k tot ({C}) , k tot ({C}) = j k j , where {k j } j {C} k j is the full list of allowed events at this configuration {C}.…”

Specific heat of the Coulomb glass

“…It becomes increasingly advantageous in comparison with the standard Metropolis procedure as T is lowered. Moreover, problems which can arise from time discretization (Adam et al 1999) are avoided in this way.…”