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Additional resources for Advances in Chemical Physics: Memory Function Approahes to Stochastic Problems in Condensed Matter, Volume 62
10) where K is a normalization constant. 12) 38 P. GRIGOLINI AND F. 13) holds. From Chapter I we are led to define the projection operator P as follows: We are now in a position to apply the simple recipe described in Chapter I. From Eqs. 20) of that chapter we get It will become clear later that at the lowest significant perturbation order, we can replace Eq. 15) with With increasing time, Eq. 18’) the actual form of which can be derived straightforwardly by noticing that the second term on the right-hand side of Eq, (2,7’), - u a/aA,, gives a vanishing contribution when substituted in the first pl appearing in the first term on the right-hand side of Eq.
19). 76~) since the thermal bath of the auxiliary variable w is faster to an extreme extent than the variables v and w , thereby resulting immediately in (F(2)) = 0 even if strongly excited. ] If the variable w is not found in its equilibrium state ( w ) , = 0, then the stochastic force F ( t ) is also far from its equilibrium state. As the continued fraction structure of Eq. 43) and, at least in principle, the corresponding expansion parameters are provided by the Mori theory, we are in a position to state that this theory can be used to provide a fairly reliable description of the thermal bath excitations.
30. (a) S. Nordholm, R. Zwanzig, J . Stat. , W, 347 (1975); (b) H. Grabert, 2. , B26,79 (1977). 31. R. , 132, 198 (1980). 32. M. Bixon and R. Zwanzig, J. Stat. , 3, 245 (1971). 33. R. Zwanzig, J . Chem. , 60,2717(1971);S. W. Haan and R. Z w k g , J . Chem. , 68, 1879 (1978); M. Bixon and R. , 1896. 34. Balucani,R. Vallauri, V. Tognetti, P. Grigolini, and P. Marin, Z . , Condensed Matter, B49,181 (1982), and references therein. 35. N. van Kampen, Siochasric Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981.