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Additional resources for Advances in Chemical Physics: Memory Function Approahes to Stochastic Problems in Condensed Matter, Volume 62
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. Note, however, that when the scalar product implies equilibrium as Eq. 5) does, the Mori theory appears 21 THEORETICAL FOUNDATIONS to be basically suitable to evaluate the equilibrium correlation function (A(O)A(r)),/(A*),, thereby resulting in a vanishing value for the mean stochastic force.
6), does not act on the variables a. 12) which in turn can be proved by means of integration by parts supplemented by the property that the heat bath equilibrium distribution properly vanishes in the boundary region. 7). 13) Note that the time-independent nature of the projection operator P [already exploited to derive Eq. 14) By substituting Eq. 5) (without the inhomogeneous term) into Eq. 13), P. GRIGOLINI AND F. 15) where To obtain this result we have used both Eq. 9) and Eq. 10). Let us focus on the third term on the right-hand side of Eq.
Note that The extension of Eq. 48) leads to By Laplace transforming Eq. 48) we get Let us substitute Eq. 42) into Eq. 50). We then obtain which is the Laplace transform of where Let us write Eq. 49) explicitly for A and come back from the quantumlike P. GRIGOLINI 18 to the classical formalism. 55c) The symbol ( )eq denotes averaging over the equilibrium distribution p,(A,b) [seeEq. 5)]. 57) Let us substitute Eq. 56) into Eq. 54) and assume A, = 0 while identifying A with the variable velocity u.
Advances in Chemical Physics: Memory Function Approahes to Stochastic Problems in Condensed Matter, Volume 62