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The Moment Map: Nonlinear Dynamics of Density Evolution via a Few Moments

Barkley, D. and Kevrekidis, I. G. and Stuart, A. M. (2006) The Moment Map: Nonlinear Dynamics of Density Evolution via a Few Moments. SIAM Journal on Applied Dynamical Systems, 5 (3). pp. 403-434. ISSN 1536-0040. https://resolver.caltech.edu/CaltechAUTHORS:20170612-131025773

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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 equation-free multiscale algorithms for these systems. The moment map itself is deterministic and attempts to capture the implied probability distribution of the dynamics. By choosing situations where the low-dimensional dynamics can be understood a priori, we evaluate the moment map. Despite requiring the evolution of an ensemble to define the map, this can be an efficient numerical tool, as the map opens up the possibility of bifurcation analyses and other high level tasks being performed on the system. 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1137/050638667DOIArticle
http://epubs.siam.org/doi/10.1137/050638667PublisherArticle
https://arxiv.org/abs/cond-mat/0508551arXivDiscussion Paper
Additional Information:© 2006 Society for Industrial and Applied Mathematics. Received by the editors August 23, 2005; accepted for publication (in revised form) by T. Kaper February 27, 2006; published electronically July 17, 2006.
Subject Keywords:equation-free, multiscale, moment map, metastable states
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ66
Issue or Number:3
Classification Code:AMS subject classifications. 65C30, 34F05, 60H10, 60H35
Record Number:CaltechAUTHORS:20170612-131025773
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170612-131025773
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78110
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:12 Jun 2017 20:55
Last Modified:03 Oct 2019 18:05

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