Memory Function Approaches to Stochastic Problems in by Myron W. Evans, Paolo Grigolini, Giuseppe Pastori

By Myron W. Evans, Paolo Grigolini, Giuseppe Pastori Parravicini

Copyright date is 1985.

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Extra info for Memory Function Approaches to Stochastic Problems in Condensed Matter (Advances in Chemical Physics)(Vol.62)

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This is made possible by the fact that the variables b, are influenced by friction and stochastic forces simulating virtually infinite degrees of freedom. The thermal baths of the variables b, are assumed to be characterized by an infinitely short time scale. In consequence, the variables b, when thought of as interacting with nothing but their own thermal bath, can be considered Markovian. If the variables b, are also Gaussian, that is, coupled to their thermal baths via linear couplings, no conceptual difficulty is found in building up their own Fokker-Planck equations and, ultimately, the Fokker-Planck equation of the system (a, b,v).

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. 58) b=-yu+f(t) that is, the standard Langevin equation. This makes it clear why the hierarchy equations of Eq.

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.

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