Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-15T05:55:01.080Z Has data issue: false hasContentIssue false
  • English
  • Français

Derivation of Langevin dynamics in a nonzero background flowfield

Published online by Cambridge University Press:  20 August 2013

Matthew Dobson,
Frédéric Legoll,
Tony Lelièvre and
Gabriel Stoltz
Matthew Dobson
Affiliation:
Department of Mathematics and Statistics, 710 N. Pleasant Street, University of Massachusetts, Amherst, MA 01003-9305, USA.. dobson@math.umass.eduINRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France; lelievre@cermics.enpc.frCERMICS – 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
Frédéric Legoll
Affiliation:
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.frINRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France; lelievre@cermics.enpc.fr
Tony Lelièvre
Affiliation:
INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France; lelievre@cermics.enpc.frCERMICS – 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
Gabriel Stoltz
Affiliation:
INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France; lelievre@cermics.enpc.frCERMICS – 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

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We propose a derivation of a nonequilibrium Langevin dynamics for a large particleimmersed in a background flow field. A single large particle is placed in an ideal gasheat bath composed of point particles that are distributed consistently with thebackground flow field and that interact with the large particle through elasticcollisions. In the limit of small bath atom mass, the large particle dynamics converges inlaw to a stochastic dynamics. This derivation follows the ideas of [P. Calderoni, D. Dürrand 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 nonzerobackground flow. The derived nonequilibrium Langevin dynamics is similar to the dynamicsin [M. McPhie, P. Daivis, I. Snook, J. Ennis and D. Evans, Phys. A299 (2001) 412–426]. Some numerical experiments illustrate the useof the obtained dynamic to simulate homogeneous liquid materials under shear flow.

Keywords

NonequilibriumLangevin dynamicsmultiscalemolecular simulation

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013, pp. 1583 - 1626
Copyright
© EDP Sciences, SMAI, 2013

References

M.P. Allen and D.J. Tildesley, Computer simulation of liquids. Clarendon Press, New York, NY, USA (1989).
P. Billingsley, Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley and Sons Inc., New York, second edition (1999).
Calderoni, P., Dürr, D. and Kusuoka, S., A mechanical model of Brownian motion in half-space. J. Stat. Phys. 55 (1989) 649–693. Google Scholar
G. Ciccotti, R. Kapral and A. Sergi, Non-equilibrium molecular dynamics. In Handbook of Materials Modeling, edited by S. Yip (2005) 745–761.
R. Cont and P. Tankov, Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL (2004).
Dürr, D., Goldstein, S. and Lebowitz, J., A mechanical model for the Brownian motion of a convex body. Z. Wahrscheinlichkeit 62 (1983) 427–448. Google Scholar
Dürr, D., S. Goldstein and J.L. Lebowitz. A mechanical model of Brownian motion. Comm. Math. Phys. 78 (1981) 507–530. Google Scholar
Edwards, B., Baig, C. and Keffer, D., A validation of the p-SLLOD equations of motion for homogeneous steady-state flows. J. Chem. Phys. 124 (2006). Google ScholarPubMed
D.J. Evans and G.P. Morriss, Statistical mechanics of nonequilibrium liquids. ANU E Press, Canberra (2007).
Hadjiconstantinou, N.G., Discussion of recent developments in hybrid atomistic-continuum methods for multiscale hydrodynamics. Bull. Pol. Acad. Sci-Te. 53 (2005) 335–342. Google Scholar
Irving, J.H. and Kirkwood, J.G., The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18 (1950) 817–829. Google Scholar
Joubaud, R. and Stoltz, G., Nonequilibrium shear viscosity computations with Langevin dynamics. Multiscale Model. Simul. 10 (2012) 191–216. Google Scholar
P. Kotelenez, Stochastic ordinary and stochastic partial differential equations. In vol. 58 of Stoch. Modell. Appl. Probab. (2008).
Kurtz, T.G., Semigroups of conditioned shifts and approximation of Markov processes. Ann. Probab. 3 (1975) 618–642. Google Scholar
Kusuoka, S. and Liang, S., A Classical Mechanical Model of Brownian Motion with Plural Particles. Rev. Math. Phys. 22 (2010) 733–838. Google Scholar
Le Bris, C. and Lelièvre, T., Micro-macro models for viscoelastic fluids: modelling, mathematics and numerics. Sci. China Math. 55 (2012) 353–384. Google Scholar
Legoll, F., Luskin, M. and Moeckel, R., Non-ergodicity of the Nosé-Hoover thermostatted harmonic oscillator. Arch. Ration. Mech. Anal. 184 (2007) 449–463. Google Scholar
Legoll, F., Luskin, M. and Moeckel, R., Non-ergodicity of Nosé-Hoover dynamics. Nonlinearity 22 (2009) 1673–1694. Google Scholar
McPhie, M., Daivis, P., Snook, I., Ennis, J. and Evans, D., Generalized Langevin equation for nonequilibrium systems. Phys. A 299 (2001) 412–426. Google Scholar
O’Connell, S.T. and Thompson, P.A., Molecular dynamics-continuum hybrid computations: A tool for studying complex fluid flows. Phys. Rev. E 52 (1995) R5792–R5795. Google Scholar
Ren, W. and E, W., Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. J. Comput. Phys. 204 (2005) 1–26. Google Scholar
Rowley, R. and Painter, M., Diffusion and viscosity equations of state for a Lennard-Jones fluid obtained from molecular dynamics simulations. Int. J. Thermophys. 18 (1997) 1109–1121. Google Scholar
Skorokhod, A.V., Limit theorems for Markov processes. Theor. Probab. Appl. 3 (1958) 202–246. Google Scholar
Soddemann, T., Dünweg, B. and Kremer, K., Dissipative particle dynamics: A useful thermostat for equilibrium and nonequilibrium molecular dynamics simulations. Phys. Rev. E 68 (2003) 046702. Google ScholarPubMed
Todd, B. and Daivis, P.J., A new algorithm for unrestricted duration nonequilibrium molecular dynamics simulations of planar elongational flow. Comput. Phys. Commun. 117 (1999) 191–199. Google Scholar
Todd, B.D. and Daivis, P.J., Homogeneous non-equilibrium molecular dynamics simulations of viscous flow: techniques and applications. Mol. Simulat. 33 (2007) 189–229. Google Scholar
Tuckerman, M.E., Mundy, C.J., Balasubramanian, S. and Klein, M.L., Modified nonequilibrium molecular dynamics for fluid flows with energy conservation. J. Chem. Phys. 106 (1997) 5615–5621. Google Scholar
Werder, T., Walther, J.H. and Koumoutsakos, P., Hybrid atomistic-continuum method for the simulation of dense fluid flows. J. Comp. Phys. 205 (2005) 373–390. Google Scholar