Error Control and Analysis in Coarse - Graining of Stochastic Lattice Dynamics

The coarse-grained Monte Carlo (CGMC) algorithm was originally proposed in the series of works [15, 16]. In this paper we further investigate the approximation properties of the coarse-graining procedure and relation between the coarse-grained and microscopic processes. We provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for the error control of quantities that may also depend on the path. We also demonstrate that CGMC leads to a significant CPU speed-up of simulations of metastable phenomena, e.g., estimation of switching times or nucleation of new phases. Numerical evidence guided by analytical results suggests that CGMC probes the energy landscape in path-wise agreement to MC simulations at the microscopic level. 1. Introduction. Microscopic computational models for complex systems such as Molecular Dynamics (MD) and Monte Carlo (MC) algorithms are typically formulated in terms of simple rules describing interactions between individual particles or spin variables. The large number of variables, or degrees of freedom and even larger number of interactions between them present the principal limitation for efficient simulations. A related limiting factor is represented by the essentially sequential nature of resolving the time evolution in many particle systems that yields a substantial slowdown in the resolution of dynamics especially in metastable regimes. In [15, 16, 18] the authors started developing systematic mathematical strategies for the coarse-graining of microscopic models, focusing on the paradigm of stochastic lattice dynamics and the corresponding MC simulators. In principle, coarse-grained models are expected to have fewer observables than the original microscopic system making them computation-ally more efficient than direct numerical simulations. In these papers a hierarchy of coarse-grained stochastic models–referred to as coarse-grained MC (CGMC) – was derived from the microscopic rules through a stochastic closure argument. The CGMC hierarchy is reminiscent of Multi-Resolution Analysis approaches to the discretization of operators [1], spanning length/time scales from the microscopic to the mesoscopic. The resulting stochastic coarse-grained processes involve Markovian birth-death and generalized exclusion processes and their combinations, and as demonstrated in [15, 16, 18], they share the same ergodic properties with their microscopic counterparts. More specifically, the full hierarchy of the coarse-grained stochastic dynamics satisfies detailed balance relations and as a result not only yields self-consistent random fluctuation mechanism, but also consistent with the underlying microscopic fluctuations and the unresolved degrees of freedom (DOFs). From the computational complexity perspective, a comparison of CGMC with conventional …