Memory function moments

This statement suffices for the present purpose. In fact, a look on eqs. (2.14) and (2.15) which hold for the interesting moment of the memory function, makes the expectation acceptable that only a restricted number of the longest relaxation times will actually be of influence on the final results for slow steady shear flow, provided the g3 s are not too different. For a further discussion of the validity of the stress-optical law see Chapter 5. 

The major advantage of this memory function is that all of its moments are finite. The corresponding velocity correlation function cannot be determined analytically, but must be studied numerically. More will be said about this approximation later. 

Let the factor multiplying tc1/2/2 be called p. Thus we see that if we assume a functional form for the memory function, then it is possible to determine the parameters in the functional form by using the moment theorems of Eq. (162) and to determine, thereby, the transport coefficients, such as the friction coefficient. Moreover, the time correlation function, i /(t), can also be determined. 

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... 

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. 

In the case b 0, the moments a of the memory function are related to the moments of the correlation function via the simple PD recursion relation... 

In this section we briefly survey a number of physical problems whose Hamiltonians can be conveniently described in a local basis. Hamiltonians of this kind allow a reasonably simple calculation of moments and are thus natural candidates for the memory function methods we are going to describe. 

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. 

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. 

A number of moments (3.65) are reported for convenience in the first row of Table I. Using the memory function PD algorithm (see Chapter III and ref. 25), summarized by Eq. (3.54), we obtain the moments for the first few memory functions (also rq>orted in Table I), and hence the continued fraction parameters (3.64) are recovered. Notice also that the last row of Table I is constituted by powers of the same number, as foreseen by Eq. (3.54) in the case of exact truncation. 

Product-Difference Memory Function Procedure for Evaluating the Parameters of the Continued Fraction Expansioif Starting from the Moments - (1/3)6" [l-K-iTI... 

The correlation function corresponds to the memory function, which indicates to which degree values of one function at time t are comparable to values of another function at time t — a before. For statistical signals, the similarity usually decreases rapidly with increasing shift a. For white noise, all values are independent of the others, and the auto-correlation function is proportional to a delta function. The proportionality factor is the second moment (12 of the noise signal. 

To calculate stresses for a system of weakly coupled macromolecules, nonequilibrium correlation functions (71), (74) and (75), specified for the memory function (45), can be used to write down the stresses in linear approximation with respect to the velocity gradients. In this way, the stresses are determined by the velocity gradients in all the previous moments of time. Further... 

The processes such as collision-induced absorption or scattering, vibrational relaxation due to intermolecular forces, etc, are not considered. However, the remaining calculation is exact. The four lowest- order moments are determined in this way an approximate expression for the memory function is presented too. 

For t = o this function is identical to the ordinary second moment. At larger times M2(t), a cross orientation-angular velocity correlation function, resembles the angular velocity correlation function C (t) for systems close to the diffusion limit 46-48. Contrary to the memory functions discussed above, M2(t) has the... 

In the regime of the rigid rotation the waves pass through all points of the medium with the same time interval T equal to the rotation period of the spiral. When more complex regimes are considered this time interval may be different for different points of the medium and may also vary in time so that T = T x, y, t) where x and y are the Cartesian coordinates of a point. To find T(x, y, t) one must keep in memory the moments T x, y) of the last arrival of the curve at any point x, y) of the medium. Then the function T x, y, t) is calculated as... 

I quantities x and y are different, then the correlation function js sometimes referred to ross-correlation function. When x and y are the same then the function is usually called an orrelation function. An autocorrelation function indicates the extent to which the system IS a memory of its previous values (or, conversely, how long it takes the system to its memory). A simple example is the velocity autocorrelation coefficient whose indicates how closely the velocity at a time t is correlated with the velocity at time me correlation functions can be averaged over all the particles in the system (as can elocity autocorrelation function) whereas other functions are a property of the entire m (e.g. the dipole moment of the sample). The value of the velocity autocorrelation icient can be calculated by averaging over the N atoms in the simulation ... 

We can easily prove that these three equations yield the same time evolution for the second moment (x2(t)) if the first and the second refer to the same correlation function and if the memory kernel (t) of the third equation... 

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