Correlation functions matrix

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... 

The transformation to obtain the difference correlation function equation proceeds in the same way. The difference correlation function matrix is defined by... 

Taking the scalar product of Eq. (11.3.26) with A+ gives the equation for the correlation function matrix... 

The elements of the correlation function matrix C (t) have the following transformation properties under time reversal... 

It proves informative first to revisit the expression for the equilibrium average of a general operator but now in the phase-space centroid perspective. This simple analysis identifies the centroid-constrained correlation function matrix in Eqs. (2.56) and (2.57) as providing the effective centroid width factors in phase space. In the phase-space path integral perspective [11], the equilibrium average of an operator A is given by the expression... 

Flere the vector variable is a Gaussian vector with associated width matrix given by 0 (0, q ). The result above from Paper III is the simplest generalization of the expression obtained in Paper I for coordinate-dependent operators. The expression reveals the role played by the centroid-constrained correlation function matrix [Eq. (2.57)] in defining the effective width factor in phase space for the centroid quasiparticle. A more careful treatment of the operator ordering problems demonstrates that the derivation of the equations above involves additional approximation beyond second-order truncation of the cumulant expansion [59]. 

Here (f) is the phase-space centroid trajectory which obeys the CMD equation of motion in Eq. (3.16), and the time-dependent Gaussian width matrix C O, q (f)) for the vector is given by the centroid-constrained correlation function matrix in Eq. (2.56) with the position centroid located at qc(0- As shown in the appendix of Paper II, the general centroid correlation function in Eq. (3.56) is an approximation to the Kubo-transformed version of the exact correlation function C g(t). Therefore, to calculate C g(t) one makes use of the Fourier relationship... 

Fig. 8.4 Matrix summary of distance matrix, connection algorithm, and temporal ordering algorithm. The shade of each matrix element represents the distance between two species, calculated from the correlation function matrix by means of eqs. (7.3) and (7.4). The darker the shade, the smaller the distance (white, dij = 1.3, black, dij = 0.2 linear gray scale). A plus or minus sign within a matrix element denotes that the connection algorithm has assigned a significant connection between these two species further, a plus (minus) sign indicates that response of the row species follows (precedes) variation in the column species. (From [1].)...

Given these closures, the OZ equation can be solved in terms of the so-called Baxter factor correlation function matrix Q(r)(ll), whose elements can be easily found once the T parameter (in a sense the equivalent of the parameter k of the DH theory) is found, r can be obtained by solving (for example by means of the Newton-Raphson method) the equation... 

Generalizing Eq. 32, it can be easily proved that the complex cross-correlation function matrix of the zero-mean response process, which collects the cross-correlation of the pre-envelope response processes, can be evaluated as... 

Tanner D J and Weeks D E 1993 Wave packet correlation function formulation of scattering theory—the quantum analog of classical S-matrix theory J. Chem. Phys. 98 3884... 

Theoretical investigations of quenched-annealed systems have been initiated with success by Madden and Glandt [15,16] these authors have presented exact Mayer cluster expansions of correlation functions for the case when the matrix subsystem is generated by quenching from an equihbrium distribution, as well as for the case of arbitrary distribution of obstacles. However, their integral equations for the correlation functions... 

To define the correlation functions of partly quenched systems requires one to consider fluctuations. There are two types of fluctuations thermal fluctuations for a given configuration of matrix species, and fluctuations induced by disorder. We characterize the average over disorder of thermal fluctuations by the variance... 

In close relation to the fluctuations, one may introduce the correlation functions. The pair density distribution function for fluid particles (ri, r2) is defined as the average over all realizations of the matrix structure of the... 

The direct correlation function c is the sum of all graphs in h with no nodal points. The cluster expansions for the correlation functions were first obtained and analyzed in detail by Madden and Glandt [15,16]. However, the exact equations for the correlation functions, which have been called the replica Ornstein-Zernike (ROZ) equations, have been derived by Given and Stell [17-19]. These equations, for a one-component fluid in a one-component matrix, have the following form... 

The equlibrium between the bulk fluid and fluid adsorbed in disordered porous media must be discussed at fixed chemical potential. Evaluation of the chemical potential for adsorbed fluid is a key issue for the adsorption isotherms, in studying the phase diagram of adsorbed fluid, and for performing comparisons of the structure of a fluid in media of different microporosity. At present, one of the popular tools to obtain the chemical potentials is an approach proposed by Ford and Glandt [23]. From the detailed analysis of the cluster expansions, these authors have concluded that the derivative of the excess chemical potential with respect to the fluid density equals the connected part of the fluid-fluid direct correlation function (dcf). Then, it follows that the chemical potential of a fluid adsorbed in a disordered matrix, p ), is... 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

First we are looking for the adsorption of a fluid consisting of particles of species m, in a slit-like pore of width H. The pore walls are chosen normal to the z axis and the pore is centered at z = 0. Adsorption of the fluid m, i.e., the matrix, occurs at equihbrium with its bulk counterpart at the chemical potential The matrix fluid is then characterized by the density profile, p (z) and by the inhomogeneous pair correlation function A (l,2). The structure of that fluid is considered... 

Now, we would like to investigate adsorption of another fluid of species / in the pore filled by the matrix. The fluid/ outside the pore has the chemical potential at equilibrium the adsorbed fluid / reaches the density distribution pf z). The pair distribution of / particles is characterized by the inhomogeneous correlation function /z (l,2). The matrix and fluid species are denoted by 0 and 1. We assume the simplest form of the interactions between particles and between particles and pore walls, choosing both species as hard spheres of unit diameter... 

The simulations are repeated several times, starting from different matrix configurations. We have found that about 10 rephcas of the matrix usually assure good statistics for the determination of the local fluid density. However, the evaluation of the nonuniform pair distribution functions requires much longer runs at least 100 matrix replicas are needed to calculate the pair correlation functions for particles parallel to the pore walls. However, even as many as 500 replicas do not ensure the convergence of the simulation results for perpendicular configurations. 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

This matrix must be diagonalized to obtain the largest eigenvalue and its eigenvector, which allows the partial coverages and the correlation functions to be obtained. This is trivial for no interactions (yi = yi = 3 12 = 1) and gives... 

Here the relative intensities of the components of each branch are determined by the Boltzmann factor Correlation function K (t, J), corresponding to Gq(a>, J), is obviously the correlation function of a transition matrix element in Heisenberg representation... 

The pair-correlation function for the segmental dynamics of a chain is observed if some protonated chains are dissolved in a deuterated matrix. The scattering experiment then observes the result of the interfering partial waves originating from the different monomers of the same chain. The lower part of Fig. 4 displays results of the pair-correlation function on a PDMS melt (Mw = 1.5 x 105, Mw/Mw = 1.1) containing 12% protonated polymers of the same molecular weight. Again, the data are plotted versus the Rouse variable. 

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