Derivation of Langevin dynamics in a nonzero background flow field

¹ Department of Mathematics and Statistics, 710 N. Pleasant Street, University of Massachusetts, Amherst, MA 01003-9305, USA.
dobson@math.umass.edu
² Laboratoire Navier – Ecole des Ponts ParisTech, 6 et 8 avenue Blaise Pascal, Cité Descartes – Champs sur Marne, 77455 Marne la Vallée Cedex 2, France.
legoll@lami.enpc.fr
³ INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France
lelievre@cermics.enpc.fr
⁴ CERMICS – Ecole des Ponts ParisTech, 6 et 8 avenue Blaise Pascal, Cité Descartes – Champs sur Marne, 77455 Marne la Vallée Cedex 2, France.
stoltz@cermics.enpc.fr

Received: 16 March 2012
Revised: 15 October 2012

Abstract

We propose a derivation of a nonequilibrium Langevin dynamics for a large particle immersed in a background flow field. A single large particle is placed in an ideal gas heat bath composed of point particles that are distributed consistently with the background flow field and that interact with the large particle through elastic collisions. In the limit of small bath atom mass, the large particle dynamics converges in law to a stochastic dynamics. This derivation follows the ideas of [P. Calderoni, D. Dürr and S. Kusuoka, J. Stat. Phys. 55 (1989) 649–693. D. Dürr, S. Goldstein and J. Lebowitz, Z. Wahrscheinlichkeit 62 (1983) 427–448. D. Dürr, S. Goldstein and J.L. Lebowitz. Comm. Math. Phys. 78 (1981) 507–530.] and provides extensions to handle the nonzero background flow. The derived nonequilibrium Langevin dynamics is similar to the dynamics in [M. McPhie, P. Daivis, I. Snook, J. Ennis and D. Evans, Phys. A 299 (2001) 412–426]. Some numerical experiments illustrate the use of the obtained dynamic to simulate homogeneous liquid materials under shear flow.