Abstract
We discuss the problem of eliminating the momentum variable in the phase space Langevin equations for a system of Brownian particles in two related situations: (i) position-dependent damping and (ii) existence of hydrodynamic interactions. We discuss the problems associated with the conventional elimination and we develop an alternative elimination procedure, in the Lagevin framework, which leads to the correct Smoluchowski equation. We give a heuristic argument on the basis of stochastic differential equations for the Smoluchowski limit and establish rigorously the limit for the general case of position-dependent friction and diffusion coefficents.
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References
H. Haken,Synergetics, An Introduction, Springer Series in Synergetics, vol. 1, 2a ed. (Springer, Berlin, 1978).
M. San Miguel and J. M. Sancho,J. Stat. Phys. 22:605 (1980).
M. San Miguel and J. M. Sancho,Z. Phys. B43:361 (1981).
P. Ryter,Z. Phys. B41:39 (1981).
H. Hasegawa, M. Mabuchi, and T. Baba,Phys. Lett. 79A:273 (1980).
P. HÄnggi, Connection between Deterministic and Stochastic Descriptions of Nonlinear Systems (unpublished 1981).
H. Mori, T. Morita, and K. T. Nashiyama,Progr. Theor. Phys. 63:1865 (1980).
R. F. Fox,Phys. Rep. 48C:181 (1979).
N. G. Van Kampen,J. Stat. Phys. 24:175 (1981).
T. J. Murphy and J. L. Aguirre,J. Chem. Phys. 57:2098 (1972).
G. Wilenski,J. Stat. Phys. 14:153 (1976).
U. M. Titulaer,Physica 100A:251 (1980).
H. Risken, H. D. Vollmer, and M. Mörsch,Z. Phys. B40:343 (1981).
K. Kaneko,Progr. Theor. Phys. 66:129 (1981).
G. Gavish,Phys. Rev. Lett. 44:1150 (1980).
W. Hess and R. Klein,Physica 94A:71 (1978).
T. Morita, H. Mori, and K. T. Nashiyama,Progr. Theor. Phys. 64:500 (1980).
R. Kubo,J. Math. Phys. 4:174 (1963).
E. A. Novikov,Sov. Phys. JETP 20:1290 (1965).
L. Arnold,Stochastic Differential Equations (Wiley, New York, 1974).
E. Nelson,Dynamical Theories of Brownian Motion (Princeton University Press, 1967).
G. C. Papanicolaou, D. Strook, and R. S. Varadhan: Martingale Approach to Some Limit Theorems, Duke University Mathematics Series III, (1977).
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Sancho, J.M., Miguel, M.S. & Dürr, D. Adiabatic elimination for systems of Brownian particles with nonconstant damping coefficients. J Stat Phys 28, 291–305 (1982). https://doi.org/10.1007/BF01012607
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DOI: https://doi.org/10.1007/BF01012607