Multiscale Couplings in Prototype Hybrid Deterministic/stochastic Systems: Part Ii, Stochastic Closures∗

Couplings of microscopic stochastic models to deterministic macroscopic ordinary and partial differential equations are commonplace in numerous applications such as catalysis, deposition processes, polymeric flows, biological networks and parametrizations of tropical and open ocean convection. In this paper we continue our study of the class of prototype hybrid systems presented in [8]. These model systems are comprised of a microscopic Arrhenius dynamics stochastic process modeling adsorption/desorption of interacting particles which is coupled to an ordinary differential equation exhibiting a variety of bifurcation profiles. Here we focus on the case where phase transitions do not occur in the microscopic stochastic system and examine the influence of noise in the overall system dynamics. Deterministic mean field and stochastic averaging closures derived in [8] are valid under stringent conditions on the range of microscopic interactions and time-scale separation respectively. Furthermore, their derivation is valid only for finite time intervals where rare events will not trigger a large deviation from the average behavior in the zero noise limit. In this paper we study such questions in the context of simple hybrid systems, demonstrating that deterministic closures based on various separation of scales arguments cannot in general capture transient and long-time dynamics. For this purpose we develop coarse grained stochastic closures for this class of hybrid systems and compare them to deterministic, mean-field and stochastic averaging closures. We show that the proposed coarse grained closures describe correctly the microscopic hybrid system solutions in all test cases examined, including rare events and random transitions between multiple stable states.