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**Additional info for Advances in Chemical Physics: Memory Function Approahes to Stochastic Problems in Condensed Matter, Volume 62**

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3) m2 ~ + +~i + l = -A~ Ki "i+l Ki+ 1 -Ai+1 mi+ 1 M YlO YB * YS Y7 Y6 YS Y4 Y3 y2 Y1 Figure 1. scheme of the one-dimensional system under study. x 36 P. GRIGOLINI AND F. MARCHESONI The Liouville equation corresponding to Eq. , A,, Al,. , and Readers can convince themselves that Eq. 4) is completely equivalent to Eq. 3) as follows. ). In the “Heisenberg picture” (see also Chapter I), corresponding to the “SchrMinger picture” of Eq. 4), the time evolution of a is driven by where When a = u from Eq.

Evans, G. J. Evans, W. T. CofTey, and P. Grigolini, Molecular wnamics, Wiley-Interscience, New York, 1982,Chapters 9 and 10. 20. R. F. Fox, Phys. , 48,179 (1978). 21. R. Zwanzig, in Stochastic Processes in Chemical Physics: The Master Equation, I. Oppenheim, K. E. Shuler, and G. H. , MIT Press, Cambridge, 1977,p. 255. 22. P. Rksibois, Physica. 27,541 (l%l); I. Prigogine and P. ReSibois, Physica, 27,629(l%l); P. Rksibois, Physica, 29, 721 (1963). 23. E. W. Montroll, Fundanrental Problems in Statistical Mechanics, compiled by E.

10). Let us focus on the third term on the right-hand side of Eq. 15). , much larger than the lifetime of K ( t ) ] ,Eq. 17) can be rewritten as where The time expansion of u(7) around t = 7 will be proven to be responsible for corrections of order higher than the result discussed in Section 11. Let us BASIC DESCRIPTION OF RULES LEADING TO mIAm-rIc ELIMINATION 47 expand into a power series the time-ordered exponential of Eq. 16). We get By making the following change of variables: + m p,'(b)e94'0/b0d7, k-3 0 .