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Extra resources for Advances in Chemical Physics: Memory Function Approahes to Stochastic Problems in Condensed Matter, Volume 62
Haken. Synergetics. A n Introduction. Ye& Organization in Physics, Chemistry and Biologv, Springer-Verlag, Berlin, 1977. A. Schenzle and H. Brand, Phys. , Am,1628 (1979). R. Graham and A. Schenzle, Phys. , A25,1731 (1981). 106-141. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. THEORETICAL FOUNDATIONS 27 14. M. Suzuki, K. Kaneko, and F. Sasagawa, Prog. Theor. , 65, 828 (1981). 15. H. A. Kramers, Physica, 7,284 (1940). 16. M. Bhttiker and R. Landauer, in Nonlinear Phenomena at Phase Transition and Instabilities, T.
The projection technique has also been used by Chaturvedi and Shibata,lsbwho used a memoryless equation as the starting point of their treatment. 16-1*They emphasized that the reduction process produces unavoidably non-Markovian statistics. Nevertheless they noted that the existence of Fokker-Planck equations does not conflict with the non-Markovian character of the stochastic process, since the corresponding solution, in harmony with ref. 19, is valid only to evaluate one-time averages and is of no use in multitime averages.
3’) This means that this force is assumed to have a vanishingly short correlation time. In a sense, Einstein’s result provides information on the slow space variable r through a contraction of the fast variable v. This is the end result of any AEP. The excellent review of Chandrasekharlb provides a detailed account of the history of the subject, to which both Smoluchowski and Einstein made fundamental contributions. It is worth mentioning the well-known paper of Kramers,’ who provided a rigorous derivation of the Smoluchowski equation from the complete Fokker-Planck equation of a Brownian particle in an external potential.