Abstract
In this paper, the first microscopic approach to Brownian motion is developed in the case where the mass density of the suspending bath is of the same order of magnitude as that of the Brownian (B) particle. Starting from an extended Boltzmann equation, which describes correctly the interaction with the fluid, we derive systematically via multiple-time-scale analysis a reduced equation controlling the thermalization of the B particle, i.e., the relaxation toward the Maxwell distribution in velocity space. In contradistinction to the Fokker-Planck equation, the derived new evolution equation is nonlocal both in time and in velocity space, owing to correlated recollision events between the fluid and particle B. In the long-time limit, it describes a non-Markovian generalized Ornstein-Uhlenbeck process. However, in spite of this complex dynamical behavior, the Stokes-Einstein law relating the friction and diffusion coefficients is shown to remain valid. A microscopic expression for the friction coefficient is derived, which acquires the form of the Stokes law in the limit where the meanfree path in the gas is small compared to the radius of particle B.
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References
N. Wax (ed.),Selected Papers on Noise and Stochastic Processes (Dover, New York, 1954).
A. Einstein,Investigations on the Theory of the Brownian Movement (Dover, New York, 1956).
N. G. van Kampen,Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1990).
J. L. Lebowitz and E. Rubin,Phys. Rev. 131:2381 (1963); P. Résibois and H. T. Davis,Physica 30:1077 (1964); J. L. Lebowitz and P. Résibois,Phys. Rev. 139:1101 (1965).
R. I. Cukier and J. M. Deutch,Phys. Rev. 177:240 (1969).
J. Mercer and T. Keyes,J. Stat. Phys. 32:35 (1983).
J. M. Deutch and I. Oppenheim,J. Chem. Phys. 54:3541 (1971); T. J. Murphy and J. L. Aguirre,J. Chem. Phys. 57:2098 (1972).
L. Bocquet, J. Piasecki, and J. P. Hansen,J. Stat. Phys. 76:505 (1994).
L. Bocquet, J. P. Hansen and J. Piasecki,Nuovo Cimento 16:981 (1994); J. Piasecki, L. Bocquet, and J. P. Hansen,Physica A 218:125 (1995).
E. H. Hauge and A. Martin-Löf,J. Stat. Phys. 7:259 (1973).
E. J. Hinch,J. Fluid. Mech. 72:499 (1975).
A. J. Masters,Mol. Phys. 57:303 (1986).
J. N. Roux,Physica A 188:526 (1992).
H. A. Lorentz,Lessen over Theoretische Natuurkunde, Vol. V. Kinetische Problemen, (E. J. Brill, Leiden, 1921).
D. Bedeaux and P. Mazur,Physica A 76:247 (1974).
M. Duerr, D. Goldstein, and J. Lebowitz,Commun. Math. Phys. 78:507 (1981).
H. van Beijeren and J. R. Dorfman,J. Stat. Phys. 23:335 (1980).
H. Spohn,Rev. Mod. Phys. 53:569 (1980).
P. Résibois and M. De Leener,Classical Kinetic Theory of Fluids (Wiley, New York, 1977).
B. J. Alder and T. E. Wainwright,Phys. Rev. A 1:18 (1970); R. Zwanzig and M. Bixon,Phys. Rev. A 2:2005 (1970).
J. L. Lebowitz, J. K. Percus, and J. Sykes,Phys. Rev. 188:487 (1969).
Y. Pomeau and P. Résibois,Phys. Rep. 19C:64 (1975).
D. Forster,Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions, (Benjamin, Reading, Massachusetts, (1975).
L. Bocquet, J. P. Hansen, and J. Plasecki,J. Stat. Phys. 76:527 (1994).
A. J. C. Ladd,Phys. Rev. Lett. 70:1339 (1993);J. Fluid Mech. 271:311 (1994).
P. D. Fedele and Y. W. Kim,Phys. Rev. Lett. 44:691 (1980); J. X. Zhu, D. J. Durian, J. Mueller, D. A. Weitz, and D. J. Pine,Phys. Rev. Lett. 68:2559 (1992).
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Knowing the interest of Matthieu Ernst in the subtle and fundamental problems of kinetic theory, we have the pleasure to dedicate this study to him.
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Bocquet, L., Piasecki, J. Microscopic derivation of non-Markovian thermalization of a Brownian particle. J Stat Phys 87, 1005–1035 (1997). https://doi.org/10.1007/BF02181268
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DOI: https://doi.org/10.1007/BF02181268