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Additional resources for Advances in Chemical Physics: Memory Function Approahes to Stochastic Problems in Condensed Matter, Volume 62
An interesting aspect of their work is the development of the contracted distribution function a(a; f) (see also Section 11) inside the time-convolution integral. They pointed out that this provides perturbation terms neglected erroneously by Brinkman. This interesting feature of their approach is included in the AEP illustrated in this chapter. They explicitly evaluated correction terms up to order Y - ~ . The projection technique has also been used by Chaturvedi and Shibata,lsbwho used a memoryless equation as the starting point of their treatment.
The systematic approach illustrated in the next section will make it clear that the higher-order corrections have no effect except that of renormalizing 40 P. GRIGOLINI AND F. 26) which is indeed a special case of a general result in the following sections [see Eqs. 65)l. In Chapter VI, Ferrario et al. discuss the precise conditions under which the heat bath driven by the operator Pb of Eq. 25)) In this case, the system of Eq. 27) where F J t ) is a Gaussian white noise defined by This is the simple system described in Section 111 of Chapter I.
23) k c k) ( zbeing ) the Laplace transform of K ( k ) ( t ) Equation . 24) where 9 is a zero-order perturbation order, whereas H j k ) is ( k + r)th perturbation order. The lowest order term contributing to Dr is ( r + 2)th order. #0 Note that little attention has been given in the literature to the case 9= except in recent Let us now consider this case explicitly. First of all, let us rewrite K ( & given ) by Eqs. 21”) as follows: This was made possible by using Eqs. 11). The main problem we shall have to face is to provide explicit expressions for the commutator In the explicit case considered, this is possible if pa= 0.
Advances in Chemical Physics: Memory Function Approahes to Stochastic Problems in Condensed Matter, Volume 62