Memory functions matrix

K(%) is the memory function matrix, with elements given by Eq. (113). 

The frequency matrix Qy and the memory function matrix Ty, in the relaxation equation are equivalent to the Liouville operator matrix Ly and the Uy matrix, respectively. The later two matrices were introduced by Kadanoff and Swift [37] (see Section V). Thus the frequency matrix can be identified with the static variables (the wavenumber-dependent thermodynamic quantities) associated with the nondissipative part, and the memory kernel matrix can be identified with the transport coefficients associated with the dissipative part. 

The velocity autocorrelation function can be obtained from the relaxation equation [Eq. (76)], where Cv(z) = Cjt(q = 0z). Here the suffix s stands for single-particle property. For zero wavenumber, there is no contribution from the frequency matrix [that is, D v(q = 0) = 0] and the memory function matrix becomes diagonal. If we write (z) = Tfj (q = 0z), then the VACF in the frequency plane can be written as... 

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). 

For arbitrary finite k all the elements of the memory function matrix are nonzero, so that this matrix ipL(k, t) in the longitudinal case has the structure,... 

Let us consider now more in detail the simplest nontrivial case of a multi-component mixture that is a binary fluid. We first simplify some expressions presented above, and let us start from the elements of the memory functions matrix. Taking into account the relations (35) and (36), we can introduce the normalized generalized mutual diffusion coefficient D(k,z), the normalized generalized thermal diffusion coefficient L)- (k. z), and the cross-correlation coefficient ((k, z) as follows,... 

Now we have a formally exact equation of motion for F(A , t), Eq. (5.106). To obtain F k,t), it is required to develop an approximation scheme for the memory kernel K k,t). Here we propose a simple model for the memory-function matrix which is a direct generalization of the one for monatomic liquids developed by Lovesey [26, 27] (see Sec. 5.1.3). 

This means that the memory function matrix is [cf. Eq. (51)]... 

Then to lowest order in q the memory function matrix has only one nonvanishing element, ... 

We now construct the G.L.E. for these orientational variables. Since all the coupled variables are functions only of angle, the frequency matrix is identically zero. The memory function matrix for this problem will now be shown to have elements... 

Since the elements of the memory function matrix are here assumed to decay rapidly compared to the rate of change of the orientation angles, the time dependence of eq. (2.43) must come from changes in the angular velocities and we can thus write... 

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... 

S-matrix formalism 129 memory function formalism 30-8 Mori chain 5 methane... 

Initialize move value matrix Initialize short term memory function... 

The long-term memory function uses the matrix called move value matrix in Figure 10.5, whose (/, j) element is the number of times that job i has been scheduled in position j. This matrix is updated after every move by adding 1 to the (/, j) element if p(i) = j in the current sequence. Then, the fraction of time each job has spent in each position can be calculated by dividing these matrix elements by the... 

Note that the correlations in a multivariate process Uu..UN are described by the N x N matrix of time-correlation functions C(t) whose elements are the time-correlation functions Cji(t). The correlation function matrix evolves in time according to the memory function equation42... 

In the above discussion, only a single variable was considered. This can be extended to evaluate the time evolution of many coupled variables—for example, the five hydrodynamical variables. In that case, A is not a single variable but a column matrix and C(q, f) = (A(q, r) A+(q)) is now the correlation matrix. O(q) and T(q, x) are the frequency and the memory function matrices, respectively [20]. 

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... 

The equations of motion in the extended hydrodynamic theory (Section IV) are obtained from the relaxation equation, where the correlation function is normalized. As mentioned before, in the extended hydrodynamic theory, the memory kernel matrix is considered to be independent of frequency thus the transport coefficients are replaced by their corresponding Enskog values. 

The most important part of the integral equations is their kernel or the matrix memory function M(f), which is an operator of rank 4 defined by its Laplace transformation,... 

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. 

Once a tridiagonal representation of M is obtained, one can use standard methods to diagonalize the tridiagonal matrix and obtain the eigenvalues (in this particular aspect, the Lanczos algorithm, as commonly used in the literature, differs from the other memory function methods where the Green s... 

MEMORY FUNCTION METHODS IN SOUD STATE PHYSICS where the matrix elements p in the localized basis are defined as... 

Summing up, we see that the traditional approach to impurity problems within the Green s-function formalism exploits the basic idea of splitting the problem into a perfect crystal described by the operator and a perturbation described by the operator U. The matrix elements of < are then calculated, usually by direct diagonalization of or by means of the recursion method. Following this traditional line of attack, one does not fully exploit the power of the memory function methods. They appear at most as an auxiliary (but not really essential) tool used to calculate the matrix elements of... 

Hazards of parameter mismatch. In the present model all results are solvable. We know the exact memory kernels in (152) and we can solve for the time evolution from (156). However, in realistic applications, one must ordinarily resort to approximations. In this situation, we get equations of the type (150) but with incorrect values of the parameters in the memory function. In order to display the possible disasters that may ensue, we modify the functions /i and /2 arbitrarily and look at the time evolution of the density matrix. 

Now let us see how an approximate form for the memory function for F k,t), i.e., Ki, k,t) in Eq. (5.23), comes about on the basis of the results of the generalized kinetic theory. This can be done by relating the phase-space correlation function to more familiar ones, such as F k,t), Ch k,t) and Ct(A , t). For this purpose it is convenient to switch from a continuous to a discrete matrix representation of the phase-space correlation function by introducing a complete set of orthonormal momentum functions these are generally chosen to be the Hermite... 

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