Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems

Abstract

In this paper we present a new class of coarse-grained stochastic processes and Monte Carlo simulations, derived directly from microscopic lattice systems and describing mesoscopic length scales. As our primary example, we mainly focus on a microscopic spin-flip model for the adsorption and desorption of molecules between a surface adjacent to a gas phase, although a similar analysis carries over to other processes. The new model can capture large scale structures, while retaining microscopic information on intermolecular forces and particle fluctuations. The requirement of detailed balance is utilized as a systematic design principle to guarantee correct noise fluctuations for the coarse-grained model. We carry out a rigorous asymptotic analysis of the new system using techniques from large deviations and present detailed numerical comparisons of coarse-grained and microscopic Monte Carlo simulations. The coarse-grained stochastic algorithms provide large computational savings without increasing programming complexity or the CPU time per executed event compared to microscopic Monte Carlo simulations.

An applied mathematics perspective on stochastic modelling for climate

“…As shown in Katsoulakis et al (2003b), the coarse-grained birth/death process above satisfies detailed balance with respect to the Gibbs measure in (3.8) as well as a number of other attractive theoretical features. The simplest coarse-grained approximation given above assumes that the effect of the microscopic interactions on the mesoscopic scales occurs within the mesoscopic coarse-mesh scale Dx, otherwise systematic non-local couplings are needed (Katsoulakis et al 2003b). The accuracy of these approximations is tested for diverse examples from material science elsewhere (Katsoulakis & Vlachos 2003;Katsoulakis et al 2003a,b) and for the instructive idealized coupled models in (1.1) (KMS 2004(KMS , 2005(KMS , 2006(KMS , 2007.…”

Spatially distributed stochastic systems: Equation-free and equation-assisted preconditioned computations

“…Refs. 35,36 ). It can also be wrapped around different (non-kMC) types of fine scale or hybrid models such as Lattice-Boltzmann inner simulators 10 (with density PDE preconditioning), or around molecular, Brownian or dissipative particle dynamics simulators of condensed matter problems, with the preconditioning coming from traditional continuum closures (elasticity theory, non-Newtonian rheology).…”

Coarse-graining Calcium Dynamics on Stochastic Reaction-diffusion Lattice Model

“…In fact, fluctuations are always exist in ion channels and play a crucial role in Ca 2+ release mechanism [11,12]. Recently, Vlachos and coworkers proposed a multiscale approach for coarse graining stochastic processes and associated Monte Carlo (MC) simulations in surface reaction systems [13][14][15]. The method is efficient in describing much larger length scales than conventional MC simulations while still incorporating microscopic details, and resulting in significant computational savings.…”