Memory-function expressions

With the introduction of G the memory function expression (50) can now be written in the form... 

Our next task is to deduce a tractable representation of the G KG term. Notice that when we put G KG into the memory function expression (79), at each end of the interaction K one has... 

The approximation wherein one retains the first term in the memory function expression (105) is of special interest because it leads to a kinetic equation that is closely related to the Boltzmann-Enskog equation in transport theory. In this section we will investigate this particular approximation in some detail not only from the standpoint of further analytical analysis but also from the standpoint of practical calculations. We will see that within certain limitations the approximation results in a reasonably realistic description of dense gases and liquids, and in this sense represents the first step in a systematic microscopic calculation. 

The central result of our analysis is a memory function expression in the form of (121). The memory function consists of two parts, a Boltzmann-Enskog term, which treats uncorrelated binary collisions, and a recollision term, which describes correlated binary collisions. We have shown that the Boltzmann-Enskog term alone already provides a first-order calculation of dense fluid correlations. The inclusion of recollision effects makes the problem... 

Let us now describe the application of the GEE formalism to the description of collective diffusion. As a result, we shall derive exact memory-function expressions for the intermediate scattering function F(k, f) and for its self-diffusion counterpart F k, t). We then explain the approximations that transform these exact results in an approximate self-consistent system of equations for these properties. 

The exact memory function expressions for F(k, t) and F Hk, t) in Equations 1.23 and 1.24 were extended to colloidal mixtures in reference [20]. Written in matrix form and in Laplace space, these exact expressions for the matrices F(]c, t) and F k, t) (when convenient, their A -dependence will be explicitly exhibited) in terms of the corresponding irreducible memory function matrices C(k, t) and 0 k, t) read... 

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

A short-term memory function determines how long a tabu restriction remains active. This can be expressed as the number of iterations a tabu condition is enforced once it is imposed. 

In generalized Rouse models, the effect of topological hindrance is described by a memory function. In the border line case of long chains the dynamic structure factor can be explicitly calculated in the time domain of the NSE experiment. A simple analytic expression for the case of local confinement evolves from a treatment of Ronca [63]. In the transition regime from unrestricted Rouse motion to confinement effects he finds ... 

The description of the chain dynamics in terms of the Rouse model is not only limited by local stiffness effects but also by local dissipative relaxation processes like jumps over the barrier in the rotational potential. Thus, in order to extend the range of description, a combination of the modified Rouse model with a simple description of the rotational jump processes is asked for. Allegra et al. [213,214] introduced an internal viscosity as a force which arises due to a transient departure from configurational equilibrium, that relaxes by reorientational jumps. Thereby, the rotational relaxation processes are described by one single relaxation rate Tj. From an expression for the difference in free energy due to small excursions from equilibrium an explicit expression for the internal viscosity force in terms of a memory function is derived. The internal viscosity force acting on the k-th backbone atom becomes ... 

Constitutive equations for the Rouse and Zimm models have been derived, and are found to be expressible in the form of Lodge s elastic liquid equation [Eq.(6.15)], with memory function given by (101) ... 

This latter expression has been used to simplify KD(t)- Note that the time dependences of the linear and angular momentum autocorrelation functions depend only on interactions between a molecule and its surroundings. In the absence of torques and forces these functions are unity for all time and their memories are zero. There is some justification then for viewing these particular memory functions as representing a molecule s temporal memory of its interactions. However, in the case of the dipolar correlation function, this interpretation is not so readily apparent. That is, both the dipolar autocorrelation function and its memory will decay in the absence of external torques. This decay is only due to the fact that there is a distribution of rotational frequencies, co, for each molecule in the gas phase. In... 

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... 

Inversion symmetry shows that 0 2k(/) = 2 (0 so that the memory function can be expressed as... 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

Note that in the above expression the memory function is proportional to the auto-correlation function of the random force. This is the well-known second fluctuation-dissipation theorem. 

In the above expression, summation over repeated indices is implied. C,v(q) — CMV(q, z = 0). The matrix elements of the frequency and memory function are given by... 

Note that the above expression is known as the generalized Einstein equation and that the memory function, ((z), is the frequency-dependent friction. 

To write down the expression for the dynamic structure factor, we need explicit expressions for the components of the frequency matrix, memory function matrix, and the normalization matrix C(q). 

The expression for the long-time part of the memory function is given by Eq. 

Equation (228) is the normalized density correlation function in the Fourier frequency plane and has the same structure as Eq. (210), which is the density correlation function in the Laplace frequency plane. i/ (z) in Eq. (228) is the memory function in the Fourier frequency plane and can be identified as the dynamical longitudinal frequency. Equation (228) provides the expression of the density correlation function in terms of the longitudinal viscosity. On the other hand, t]l itself is dependent on the density correlation function [Eq. (229)]. Thus the density correlation function should be calculated self-consistently. To make the analysis simpler, the frequency and the time are scaled by (cu2)1 2 and (cu2)-1 2, respectively. As the initial guess for rjh the coupling constant X is considered to be weak. Thus rjt in zeroth order is... 

One uses expression (6.18) for the stress tensor in which the memory function can be chosen in the simplest way... 

To calculate the dynamic modulus, we turn to the expression for the stress tensor (6.46) and refer to the definition of equilibrium moments in Section 4.1.2, while memory functions are specified by their transforms as... 

Defining the memory function as M(t) = —dG(t)/df, one can express a large linear deformation as the sum of all the small linear deformations that in turn can be written as the integral over all past times ... 

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