arXiv subjects

A. M. Stuart

Publications and source records attributed to A. M. Stuart.

39 records · Page 3Linked to original sources

arXiv2005

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

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.

cond-mat.other↗

arXiv2005

Periodic Homogenization for Inertial Particles

We study the problem of homogenization for inertial particles moving in a periodic velocity field, and subject to molecular diffusion. We show that, under appropriate assumptions on the velocity field, the large scale, long time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone. To achieve this we use a formal multiple scale expansion in the scale parameter. This expansion relies on the hypo-ellipticity of the underlying diffusion. An expression for the diffusivity tensor is found and various of its properties studied. In particular, an expansion in terms of the non-dimensional particle relaxation time $τ$ (the Stokes number) is shown to co-incide with the known result for passive (non-inertial) tracers in the singular limit $τ\to 0$. This requires the solution of a singular perturbation problem, achieved by means of a formal multiple scales expansion in $τ.$ Incompressible and potential fields are studied, as well as fields which are neither, and theoretical findings are supported by numerical simulations.

cond-mat.stat-mech↗

arXiv2005

Analysis of White Noise Limits for Stochastic Systems with Two Fast Relaxation Times

In this paper we present a rigorous asymptotic analysis for stochastic systems with two fast relaxation times. The mathematical model analyzed in this paper consists of a Langevin equation for the particle motion with time-dependent force constructed through an infinite dimensional Gaussian noise process. We study the limit as the particle relaxation time as well as the correlation time of the noise tend to zero and we obtain the limiting equations under appropriate assumptions on the Gaussian noise. We show that the limiting equation depends on the relative magnitude of the two fast time scales of the system. In particular, we prove that in the case where the two relaxation times converge to zero at the same rate there is a drift correction, in addition to the limiting Itô integral, which is not of Stratonovich type. If, on the other hand, the colored noise is smooth on the scale of particle relaxation then the drift correction is the standard Stratonovich correction. If the noise is rough on this scale then there is no drift correction. Strong (i.e. pathwise) techniques are used for the proof of the convergence theorems.

math-ph↗