A stochastic coupling method for atomic-to-continuum Monte-Carlo simulations

In this paper, we propose a multiscale coupling approach to perform Monte-Carlo simulations on systems described at the atomic scale and subjected to random phenomena. The method is based on the Arlequin framework, developed to date for deterministic models involving coupling a region of interest described at a particle scale with a coarser model (continuum model). The new method can result in a dramatic reduction in the number of degrees of freedom necessary to perform Monte-Carlo simulations on the fully atomistic structure. The focus here is on the construction of an equivalent stochastic continuum model and its coupling with a discrete particle model through a stochastic version of the Arlequin method. Concepts from the Stochastic Finite Element Method, such as the Karhünen-Loeve expansion and Polynomial Chaos, are extended to multiscale problems so that Monte-Carlo simulations are only performed locally in subregions of the domain occupied by particles. Preliminary results are given for a 1D structure with harmonic interatomic potentials.