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Extra info for Advances in Chemical Physics: Memory Function Approahes to Stochastic Problems in Condensed Matter, Volume 62
58) b=-yu+f(t) that is, the standard Langevin equation. This makes it clear why the hierarchy equations of Eq. 49) for k = 0,1,2,... are referred to as generalized Langevin equations. The linear version of the RMT2* is based on making the Markovian assumption of Eq. 56) at the k th order with k > 0. This means that it is possible to extract from the thermal bath of A k suitable linear combinations of the variables b. These linear combinations can be regarded as being variables as slow as A itself.
By substituting Eq. 64) into the time derivative of Eq. 66) can be rewritten as The vector left conjugate to that between the square brackets reads The reader can readily check that this equation formally coincides with Eq. 54). [We remind the reader of Eq. ] A still more rigorous demonstration of the complete equivalence of the Mori and Nakajima-Zwanzigapproach has been derived by G r a b e ~ ? ~~ simplicity we shall not illustrate his demonstration. We shall limit ourselves to remark that this clearly establishes that the two approaches are equivalent in that they are related in the same way as the Heisenberg and Schrodinger pictures are related.
61 A. One-Dimensional Stochastic Differe Equation of the Langevin Type B. Two-Dimensional Stochastic Differential Equation of the Langevin Type VI. Rotational Brownian .................... VII. On the Choice of the VIII. ConcludingRemarks . References . . . . . . . . . . . . . . . 70 I. INTRODUCTION The major aim of the present volume is to use the same rigorous and generally valid theoretical structure to explain a variety of physical phenomena in a self-consistent way.