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The moment map: nonlinear dynamics of density evolution via a few moments. (English) Zbl 1210.65010

Summary: 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.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)

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