The memory function formalism

a single-valued connection is established between the kernel of this equation R(t) (a memory function ) and KM(t). Their Laplace transforms R and Km are related by 

The memory function R(t) has an advantage over KM(t), since, unlike what occurred in Eq. (1.66), the integral of R(t) differs from zero  

Differentiation of Eq. (1.71) shows that the initial value of R(t) is positive. Thus one may consider R(t) to have typical correlation function properties for t = 0 it is equal to the correspondingly determined dispersion Ro and it decreases to zero in a time of the order tc. 

This conclusion can be confirmed by an alternative derivation of Eq. (1.71). According to Mori [52], Eq. (1.71) may be obtained from a generalized Langevin equation  

In this case Eq. (1.71) reduces to a differential equation. Its rigorous solution with initial conditions Kj(0) = rd, Kj(0) = 0 gives 

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Let us compare in detail the differential theory results with those obtained for rotational relaxation kinetics from the memory function formalism (integral theory). Using R(t) from Eq. (1.107) as a kernel of Eq. (1.71) we can see that in the low-density limit... 

In order to apply the memory function formalism to the collective coordinates of Eq. (367), it is necessary to define the dimensionless normalized collective coordinates,... 

In this article the memory function formalism has been used to compute time-correlation functions. It has been shown that a number of seemingly disparate attempts to account for the dynamical behavior of time correlation functions, such as those of Zwanzig,33,34 Mori,42,43 and Martin,16 are... 

They may be obtained by means of the matrix IET but only together with the kernel E(f) = F(t) specified by its Laplace transformation (3.244), which is concentration-independent. However, from the more general point of view, Eqs. (3.707) are an implementation of the memory function formalism in chemical kinetics. The form of these equations shows the essentially non-Markovian character of the reversible reactions in solution the kernel holds the memory effect, and the convolution integrals entail the prehistoric evolution of the process. In the framework of ordinary chemical kinetics S(/j = d(t), so that the system (3.707) acquires the purely differential form. In fact, this is possible only in the limit when the reaction is entirely under kinetic control. 

A Hankel determinant D is a function of 2n +1 independent parameters (the moments) yet when constructed explicitly it requires a matrix with (ra -fl) elements. The problem of finding efficient algorithms, which take into account the peculiar persymmetric structure of the Hankel matrices [left diagonals of (S.13) are formed with the same element], has been considered in the literature by several authors. We discuss here in detail a recent satisfactory solution of this problem, obtained within the memory function formalism, and then compare it with other algorithms. 

Out of the detailed mathematical aspects, some of them summarized in this section, there is a more general physical concept at the heart of the theory of error bounds. It is a fact that the memory function formalism provides in a natural manner a framework by which Ae short-time behavior, via the kernel of the integral equations, makes its effects felt in the long-time tail. The mathematical apparatus of continued fractions can adequately describe memory effects, and this explains the central role of this tool in the theory of relaxation. 

Finally, we wish to note that the chain of variables defined by Eqs. (3.21) form a complete set for describing the time evolution of the initial state /o). In fact, the time evolution in a successive infinitesimal interval is determined from knowledge of /q) and - /// /o> in the next step we also need ( — i/f) /o), and so on. This is the origin of the drep relationship between the recursion method and the memory function formalism that we are going to illustrate. 

To establish the relation between the memory function formalism and the moments, consider the Volterra integro-differential equations (3.44) for %it) ... 

The memory function formalism leads to several advantages, both from a formal point of view and from a practical point of view. It makes transparent the relationship between the recursion method, the moment method, and the Lanczos metfiod on the one hand and the projective methods of nonequiUbrium statistical mechanics on the other. Also the ad hoc use of Padd iqiproximants of type [n/n +1], often adopted in the literature without true justification, now appears natural, since the approximants of the J-frac-tion (3.48) encountered in continued fraction expansions of autocorrelation functions are just of the type [n/n +1]. The mathematical apparatus of continued fractions can be profitably used to investigate properties of Green s functions and to embody in the formalism the physical information pertinent to specific models. Last but not least, the memory function formaUsm provides a new and simple PD algorithm to relate moments to continued fraction parameters. 

The linearized transport equations (7), the equations for the equilibrium time correlation functions (13), and the equation for collective mode spectrum (14) form a general basis for the study of the dynamic behavior of a multicomponent fluid in the memory function formalism. 

Since its introduction in the f960s by Zwanzig and Mori [21, 22, 23], the memory-function formalism based on projection operators has pervaded many theoretical approaches dealing with the dynamics of strongly interacting systems. Indeed, the idea of describing a many-body system by a limited number of relevant variables characterized by a relatively simple dynamics appears to be extremely appealing. 

In the following we shall consider the application of the memory-function formalism to the calculation of the intermediate scattering function F k,t) for simple liquids. For this purpose a natural choice for the vector A is a set of conserved variables consisting of the microscopic density and the longitudinal current density ... 

As mentioned in Sec. 1.1, the memory-function formalism which leads to Eq. (5.23) is, in a sense, merely a formal rephrasing of the original Eq. (5.1), shifting the difficulty in determining F k,t) to that in evaluating the memory function K k,t). A priori, the only exact result that we know about the memory function is its initial value [18, 19, 20, 25]... 

As we have seen so far in this section, in the theoretical development for the time-correlation functions of liquids at wavelengths and frequencies of a molecular scale, the memory-function formalism based on projection operators has played a key role. By combining this convenient framework and the interaction-site representation of molecules. 

The rest of this chapter is organized as follows. Section 5.2 establishes the basic theoretical framework based on the memory-function formalism... 

In this section the generalized Langevin equation (GLE) for density correlation functions for molecular liquids is derived based on the memory-function formalism and on the interaction-site representation. In contrast to the monatomic liquid case, all functions appearing in the GLE for polyatomic fluids take matrix forms. Approximation schemes are developed for the memory kernel by extending the successful frameworks for simple liquids described in Sec. 5.1. 

In this chapter we have described a theory for dynamics of polyatomic fluids based on the memory-function formalism and on the interaction-site representation of molecular liquids. Approximation schemes for memory functions appearing in the generalized Langevin equation have been developed by assuming an exponential form for memory functions and by employing the mode-coupling approach. Numerical results were presented for longitudinal current spectra of a model diatomic liquid and water, and it has been discussed how the results can be interpreted in... 

One further approach to the understanding of orientational correlation functions is the memory function formalism 1, 39. A memory function K(t)is defined by the equation... 

More quantitatively, from a detailed analysis using the memory function formalism, the characteristic parameters D and u>c can be extracted from the lineshape (25). 

In this section we will first refer to the formalism originally introduced by Zwanzig and Bixon [96, 97], which was then applied to polymer dynamics by Schweizer [98, 99] and others [100-108]. This sort of theory is based on the Zwanzig/Mori projection operator technique in connection with treatments of the Generalized Langevin Equation [109-115]. It should be noted that this equation can be considered as the microscopic basis of phenomenological approaches based on the memory function formalism [116-122]. 

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