High friction limit of the Kramers equation: The multiple time-scale approach

American Journal of Physics

Volumn 65, Issue 2, 1997, Pages 140-144

  Bocquet, Lydéric ^a  

^a UNIVERSITÉ DE LYON   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031538488     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18507     Document Type: Article

References (14)

1. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, Amsterdam, 1990), 6th ed.
2. S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover, New York, 1984), pp. 111-142.
3. O. Klein, "The statistical theory of suspensions and solutions," Ark. Mat. Astron. Fys. 16(5), 1-52 (1921).
4. H. A. Kramers, "Brownian motion in a field of force and the diffusion model of chemical reactions," Physica 7, 284-304 (1940).
5. M. von Smoluchowski, "Brownian molecular movement under the action of external forces and its connection with the generalized diffusion equation," Ann. Phys. 48, 1103-1112 (1916).
6. G. Wilemski, "On the derivation of Smoluchowski equations with corrections in the classical theory of Brownian motion," J. Stat. Phys. 14, 153-169 (1976).
7. U. M. Titulaer, "A systematic solution procedure for the Fokker-Planck equation of a Brownian particle in the high-friction case," Physica A 91, 321-344 (1978); "The Chapman-Enskog procedure as a form of degenerate perturbation theory," Physica A 100, 234-250; and "Corrections to the Smoluchowski equation in the presence of hydrodynamic interactions," Physica A 100, 251-265 (1980).
8. U. M. Titulaer, "A systematic solution procedure for the Fokker-Planck equation of a Brownian particle in the high-friction case," Physica A 91, 321-344 (1978); "The Chapman-Enskog procedure as a form of degenerate perturbation theory," Physica A 100, 234-250; and "Corrections to the Smoluchowski equation in the presence of hydrodynamic interactions," Physica A 100, 251-265 (1980).
9. U. M. Titulaer, "A systematic solution procedure for the Fokker-Planck equation of a Brownian particle in the high-friction case," Physica A 91, 321-344 (1978); "The Chapman-Enskog procedure as a form of degenerate perturbation theory," Physica A 100, 234-250; and "Corrections to the Smoluchowski equation in the presence of hydrodynamic interactions," Physica A 100, 251-265 (1980).
10. For a complete discussion of the Hilbert and Chapman-Enskog methods, see Ref. 7 and C. Cercignani, The Boltzmann Equation and its Applications (Springer-Verlag, Berlin, 1988), pp. 232-261.
11. For further details and applications of the multiple time-scale method, see A. Nayfeh, Perturbation Methods (Wiley, New York, 1973), pp. 228-307.
12. J. Piasecki, "Time scales in the dynamics of the Lorentz electron gas," Am. J. Phys. 61, 718-722 (1993).
13. J. L. Anderson, "Multiple time scale methods for adiabatic systems," Am. J. Phys. 60, 923-927 (1992).
14. L. Bocquet, J. Piasecki, J.-P. Hansen, "On the Brownian motion of a massive sphere in a hard-sphere fluid. I. Multiple-time-scale analysis and microscopic expression for the friction coefficient," J. Stat. Phys. 76, 505-526 (1994).