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Memory function equation

There have been a number of attempts to calculate time-correlation functions on the basis of simple models. Notable among these is the non-Markovian kinetic equation, the memory function equation for time-correlation functions first derived by Zwanzig33 and studied in great detail by Berne et al.34 This approach is reviewed in this article. Its relation to other methods is pointed out and its applicability is extended to other areas. The results of this theory are compared with the results of molecular dynamics. [Pg.9]

Linear response theory is reviewed in Section II in order to establish contact between experiment and time-correlation functions. In Section III the memory function equation is derived and applied in Section IV to the calculation of time-correlation functions. Section V shows how time-correlation functions can be used to guess time-dependent distribution functions and similar methods are then applied in Section VI to the determination of time-correlation functions. In Section VII a succinct review is given of other exact and experimental calculations of time-correlation functions. [Pg.9]

Ku(t) is called the memory function/ and the equation for the time-correlation function that we derived is called the memory function equation.33,34,42 Note that the propagator in this equation contains the projection operator Pt. Further note that the memory function is an even function of the time,... [Pg.39]

The memory function equation for the time-correlation function of a dynamical operator Ut can be cast into the form of a continued fraction as was first pointed out by Mori.43 We prove this in a different way than Mori. In order to proceed it is necessary to define the set of memory functions K0 t),. .., Kn t). .., such that... [Pg.46]

Time-correlation functions Cu(t) obey the memory function equation... [Pg.48]

From the memory function equation it follows that... [Pg.49]

The mean square torque is taken from computer experiments. Nevertheless, it could have been found from the infrared bandshapes. Likewise the integral in this expression can be found from the experimental spin rotation relaxation time, or it can be found directly from the computer experiment as it is here. The memory function equation can be solved for this memory. The corresponding angular momentum correlation function has the same form as v /(0 in Eq. (302) with... [Pg.113]

The long-time behavior of Fs(k, t) can be extracted from the memory function equation by using a technique originally due to Zwanzig.75 If s(k, S) and < 2k(S) denote the Laplace transforms of Fs(k, t) and 2k(r), then Laplace transformation of the memory function equation yields... [Pg.134]

Thus from the memory function equation we have succeeded in showing that Fs(k, t) satisfies the diffusion equation... [Pg.136]

F(k, t) and Fs(k, t) consequently satisfy memory function equations with corresponding memories... [Pg.137]

Consider first the memory function equation for f k, t). From Eqs. (167) and (168) it is seen that the short-time behavior of the memory function 02 (/) is... [Pg.137]

Fs(k, t) can be determined from Cs(k, t), which in turn satisfies the memory function equation... [Pg.138]

The advantages of this kind of formulation stand out not only in terms of elegance and beauty (the moment method, the Lanczos method, and the recursion method are relevant but particular cases of the memory function equations), but also in the possibility of providing insight into a number of problems, such as the asymptotic behavior of continued fraction parameters and their relationship with moments, the possible inclusion of nonlinear effects, the introduction of the concept of random forces, and so on. [Pg.150]

This is called the memory-function equation. In this equation the random force appears implicitly9 in K r). [Pg.283]

The memory function M(k, pp" t) represents effects from the dynamics of collisional processes. Before embarking on the survey of the results of the generalized kinetic theory, let us see briefly how the basic equations of classical kinetic theories can be recovered by means of the memory-function equation (5.38). For instance, the Vlasov equation can be obtained by completely ignoring the memory term in Eq. (5.38) ... [Pg.286]

In the discrete space the Laplace transform of the memory-function equation (5.38) reads... [Pg.288]

We next consider the static interaction W, which is similar to V, but is considerably more complicated. " We see that W plays essentially the same role in the equation for G that plays in the memory function equation for C... [Pg.197]

The generalized Langevin equation and the memory function equation simplify considerably when the set Ai, Am relaxes much more slowly than all other properties. If all such slowly relaxing variables are included in the set Ai,..., Am, the set is called a good set of variables. At the outset it is important to note that there are no rules by which a good set of variables can be chosen. Generally this is a matter of one s intuition. It is, however, the crucial step in the application of the Zwanzig-Mori formalism to specific problems. [Pg.243]

By taking the scalar product of (1.44) with A and using (1.31) we obtain the memory function equation... [Pg.118]

An alternative use of the G.L.E. which appears to be quite different from the multi-variable formalism is based on the realization that one can write an entire hierarchy of memory function equations. This arises from the fact that the memory function itself is a phase variable and thus obeys its own G.L.E. If the n th memory function is denoted by (t), we can write... [Pg.126]

It is the presence of the non-zero elements mi2 which brings about a coupling of the memory function equations for the correlation functions with different m subscripts (but a fixed j superscript). It can be seen that the matrix of "memory constant" is diagonal whenf (y these symmetric diffusers, the equations for each m decouple and one recovers the well-known result ... [Pg.129]

N variables (for fixed t) to be in A. In addition, these authors close the memory function equations by also including the and then assuming that the memory function matrix which characterize this 2N j5rariable set decay rapidly enought to be replaced by a matrix of constants. (Note that we have discussed similar... [Pg.130]


See other pages where Memory function equation is mentioned: [Pg.40]    [Pg.108]    [Pg.109]    [Pg.111]    [Pg.141]    [Pg.93]    [Pg.285]    [Pg.280]    [Pg.289]    [Pg.317]   
See also in sourсe #XX -- [ Pg.240 , Pg.243 ]




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