Derivation of Langevin dynamics in a nonzero background flow field
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 6, pp. 1583-1626.

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.

DOI : 10.1051/m2an/2013077
Classification : 82C05, 82C31
Mots clés : nonequilibrium, Langevin dynamics, multiscale, molecular simulation
@article{M2AN_2013__47_6_1583_0,
     author = {Dobson, Matthew and Legoll, Fr\'ed\'eric and Leli\`evre, Tony and Stoltz, Gabriel},
     title = {Derivation of {Langevin} dynamics in a nonzero background flow field},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1583--1626},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {6},
     year = {2013},
     doi = {10.1051/m2an/2013077},
     mrnumber = {3110489},
     zbl = {1287.82017},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2013077/}
}
TY  - JOUR
AU  - Dobson, Matthew
AU  - Legoll, Frédéric
AU  - Lelièvre, Tony
AU  - Stoltz, Gabriel
TI  - Derivation of Langevin dynamics in a nonzero background flow field
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2013
SP  - 1583
EP  - 1626
VL  - 47
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2013077/
DO  - 10.1051/m2an/2013077
LA  - en
ID  - M2AN_2013__47_6_1583_0
ER  - 
%0 Journal Article
%A Dobson, Matthew
%A Legoll, Frédéric
%A Lelièvre, Tony
%A Stoltz, Gabriel
%T Derivation of Langevin dynamics in a nonzero background flow field
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 1583-1626
%V 47
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2013077/
%R 10.1051/m2an/2013077
%G en
%F M2AN_2013__47_6_1583_0
Dobson, Matthew; Legoll, Frédéric; Lelièvre, Tony; Stoltz, Gabriel. Derivation of Langevin dynamics in a nonzero background flow field. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 6, pp. 1583-1626. doi : 10.1051/m2an/2013077. http://www.numdam.org/articles/10.1051/m2an/2013077/

[1] M.P. Allen and D.J. Tildesley, Computer simulation of liquids. Clarendon Press, New York, NY, USA (1989). | Zbl

[2] 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). | MR | Zbl

[3] P. Calderoni, D. Dürr and S. Kusuoka, A mechanical model of Brownian motion in half-space. J. Stat. Phys. 55 (1989) 649-693. | MR | Zbl

[4] G. Ciccotti, R. Kapral and A. Sergi, Non-equilibrium molecular dynamics. In Handbook of Materials Modeling, edited by S. Yip (2005) 745-761.

[5] R. Cont and P. Tankov, Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL (2004). | MR | Zbl

[6] D. Dürr, S. Goldstein and J. Lebowitz, A mechanical model for the Brownian motion of a convex body. Z. Wahrscheinlichkeit 62 (1983) 427-448. | MR | Zbl

[7] D. Dürr, S. Goldstein and J.L. Lebowitz. A mechanical model of Brownian motion. Comm. Math. Phys. 78 (1981) 507-530. | MR | Zbl

[8] B. Edwards, C. Baig and D. Keffer, A validation of the p-SLLOD equations of motion for homogeneous steady-state flows. J. Chem. Phys. 124 (2006).

[9] D.J. Evans and G.P. Morriss, Statistical mechanics of nonequilibrium liquids. ANU E Press, Canberra (2007). | Zbl

[10] N.G. Hadjiconstantinou, Discussion of recent developments in hybrid atomistic-continuum methods for multiscale hydrodynamics. Bull. Pol. Acad. Sci-Te. 53 (2005) 335-342. | Zbl

[11] J.H. Irving and J.G. Kirkwood, The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18 (1950) 817-829. | MR

[12] R. Joubaud and G. Stoltz, Nonequilibrium shear viscosity computations with Langevin dynamics. Multiscale Model. Simul. 10 (2012) 191-216. | MR | Zbl

[13] P. Kotelenez, Stochastic ordinary and stochastic partial differential equations. In vol. 58 of Stoch. Modell. Appl. Probab. (2008). | MR | Zbl

[14] T.G. Kurtz, Semigroups of conditioned shifts and approximation of Markov processes. Ann. Probab. 3 (1975) 618-642. | MR | Zbl

[15] S. Kusuoka and S. Liang, A Classical Mechanical Model of Brownian Motion with Plural Particles. Rev. Math. Phys. 22 (2010) 733-838. | MR

[16] C. Le Bris and T. Lelièvre, Micro-macro models for viscoelastic fluids: modelling, mathematics and numerics. Sci. China Math. 55 (2012) 353-384. | MR

[17] F. Legoll, M. Luskin and R. Moeckel, Non-ergodicity of the Nosé-Hoover thermostatted harmonic oscillator. Arch. Ration. Mech. Anal. 184 (2007) 449-463. | MR | Zbl

[18] F. Legoll, M. Luskin and R. Moeckel, Non-ergodicity of Nosé-Hoover dynamics. Nonlinearity 22 (2009) 1673-1694. | MR | Zbl

[19] M. Mcphie, P. Daivis, I. Snook, J. Ennis and D. Evans, Generalized Langevin equation for nonequilibrium systems. Phys. A 299 (2001) 412-426. | Zbl

[20] S.T. O'Connell and P.A. Thompson, Molecular dynamics-continuum hybrid computations: A tool for studying complex fluid flows. Phys. Rev. E 52 (1995) R5792-R5795.

[21] W. Ren and W. E, Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. J. Comput. Phys. 204 (2005) 1-26. | MR | Zbl

[22] R. Rowley and M. Painter, Diffusion and viscosity equations of state for a Lennard-Jones fluid obtained from molecular dynamics simulations. Int. J. Thermophys. 18 (1997) 1109-1121.

[23] A.V. Skorokhod, Limit theorems for Markov processes. Theor. Probab. Appl. 3 (1958) 202-246. | Zbl

[24] T. Soddemann, B. Dünweg and K. Kremer, Dissipative particle dynamics: A useful thermostat for equilibrium and nonequilibrium molecular dynamics simulations. Phys. Rev. E 68 (2003) 046702.

[25] B. Todd and P.J. Daivis, A new algorithm for unrestricted duration nonequilibrium molecular dynamics simulations of planar elongational flow. Comput. Phys. Commun. 117 (1999) 191-199.

[26] B.D. Todd and P.J. Daivis, Homogeneous non-equilibrium molecular dynamics simulations of viscous flow: techniques and applications. Mol. Simulat. 33 (2007) 189-229. | Zbl

[27] M.E. Tuckerman, C.J. Mundy, S. Balasubramanian and M.L. Klein, Modified nonequilibrium molecular dynamics for fluid flows with energy conservation. J. Chem. Phys. 106 (1997) 5615-5621.

[28] T. Werder, J.H. Walther and P. Koumoutsakos, Hybrid atomistic-continuum method for the simulation of dense fluid flows. J. Comp. Phys. 205 (2005) 373-390. | MR | Zbl

Cité par Sources :