Stochastic Mode-Reduction in Models with Conservative Fast Sub-Systems

TL;DR

This paper develops a stochastic mode reduction method for multiscale models with energy-conserving fast subsystems, accounting for the slow evolution of the fast subsystem's energy as an additional variable.

Contribution

It introduces a novel approach to stochastic mode reduction that incorporates the slow dynamics of the fast subsystem's energy in energy-conserving systems.

Findings

Successfully models the slow evolution of fast subsystem energy.

Provides a framework for stochastic reduction in energy-conserving multiscale systems.

Enhances understanding of energy exchange in coupled deterministic-stochastic systems.

Abstract

A stochastic mode reduction strategy is applied to multiscale models with a deterministic energy-conserving fast sub-system. Specifically, we consider situations where the slow variables are driven stochastically and interact with the fast sub-system in an energy-conserving fashion. Since the stochastic terms only affect the slow variables, the fast-subsystem evolves deterministically on a sphere of constant energy. However, in the full model the radius of the sphere slowly changes due to the coupling between the slow and fast dynamics. Therefore, the energy of the fast sub-system becomes an additional hidden slow variable that must be accounted for in order to apply the stochastic mode reduction technique to systems of this type.