The Moment Map: Nonlinear dynamics of density evolution via a few moments

TL;DR

This paper introduces moment maps as a tool to analyze the nonlinear dynamics of density evolution in high-dimensional systems, enabling understanding of metastability and multiscale algorithms.

Contribution

It defines and studies moment maps for low-order moments, revealing how nonlinearity stabilizes metastable states across various models.

Findings

Nonlinearity in moment maps stabilizes metastable states.

Moment maps effectively reduce high-dimensional dynamics.

Applications demonstrated from stochastic equations to molecular dynamics.

Abstract

We explore situations in which certain stochastic and high-dimensional deterministic systems behave effectively as low-dimensional dynamical systems. We define and study moment maps, maps on spaces of low-order moments of evolving distributions, as a means of understanding equations-free multiscale algorithms for these systems. We demonstrate how nonlinearity arises in these maps and how this results in the stabilization of metastable states. Examples are shown for a hierarchy of models, ranging from simple stochastic differential equations to molecular dynamics simulations of a particle in contact with a heat bath.