Correlation functions, integral equations

As we discussed in Section II.B, site-site correlation functions provide a very useful formalism for describing the structure of fluids modeled with interaction site potentials. In this formalism, information equivalent to g l,2) is obtained from the set of site-site correlation functions and intramolecular correlation functions. For this reason, a great deal of effort has been put into the development of integral equation theories for these correlation functions. The seminal contribution in this area was made by Chandler and Andersen, who sought to write an integral equation of the Ornstein-Zernike form in which the set of site-site total correlation functions were related to a set of site-site direct correlation functions. Their equation has the form... 

The simplest fundamental connection between equations of state, expressed as Helmholtz functions of T, v, and x, and correlation function integrals is via the second composition derivative of the residual Helmholtz energy and DCFI,... 

The generalized Krichevskii parameter or its equivalent, the direct correlation function integral, Cu, is well behaved in the critical region, as shown in Figure 2.18, and on this is based Equation (2.85) (O Connell et al., 1996), which was used to fit the standard partial molar volume of nonelectrolytes all over the density range. 

The integration of Equations 9.49 and 9.51 is carried out using the second-order Heun s algorithm, with a very small time step of 0.001. These equations differ from the corresponding classical equations in two ways First, the noise correlation of c-number spin-bath variables r t) are quantum mechanical in nature, as evident from the correlation function in Equation 9.42, which is numerically fitted by the superposition of exponential functions with D, and X . Second, the knowledge of Q requires the quantum correction equations that yield quantum dispersion around the quantum mechanical mean values q and p for the system. Statistical averaging over noise is... 

An alternative approach to the integral expression cf the correlation function in equation 16 is to use a discrete Fourier transformation method. Then the integral expression for the correlation function is replaced by a summation (16) ... 

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

Most integral equations are based on the Ornstein-Zernike (OZ) equation [3-5]. The idea behind the OZ equation is to divide the total correlation function h ri2) iiito a direct correlation function (DCF) c r 12) that describes the fact that molecules 1 and 2 can be directly correlated, and an indirect correlation function 7( 12), that describes the correlation of molecule 1 with the other molecules that are also correlated with molecule 2. At low densities, when only direct correlations are possible, 7(r) = 0. At higher densities, where only triplet correlations are possible, we can write... 

In order to develop integral equations for the correlation functions, we consider the system composed of N polydisperse spheres. The average density of particles with diameter <7, is given by... 

In any relation given above, the knowledge of the total or direct pair correlation functions yields an equation for the density profile. The domain of integration in Eqs. (14)-(16) must include all the points where pQ,(r) 0. In the case of a completely impermeable surface, pQ,(r) = 0 inside the wall... 

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... 

We proceed with cluster series which yield the integral equations. Evidently the correlation functions presented above can be defined by their diagrammatic expansions. In particular, the blocking correlation function is the subset of graphs of h rx2), such that all paths between... 

The thermodynamic quantities and correlation functions can be obtained from Eq. (1) by functional integration. However, the functional integration cannot usually be performed exactly. One has to use approximate methods to evaluate the functional integral. The one most often used is the mean-field approximation, in which the integral is replaced with the maximum of the integrand, i.e., one has to find the minimum of. F[(/)(r)], which satisfies the mean-field equation... 

Integral equations can also be used to treat nonuniform fluids, such as fluids at surfaces. One starts with a binary mixture of spheres and polymers and takes the limit as the spheres become infinitely dilute and infinitely large [92-94]. The sphere polymer pair correlation function is then simply related to the density profile of the fluid. 

We could of course write down the explicit form of the general nth. order ring diagrams we prefer however to establish directly an algebraic equation for the whole series and deduce the pair correlation function from its exact solution. Indeed, it is easily verified that the nth order term is derived from the (n — l)th one by adding a loop either on the upper or on the lower line. This leads immediately to the integral equation of Fig. 9b which we now write in an analytic form. 

Following FerrelK, the second term in Equation 2 can be expressed as a Green-Kubo integral over a flux-flux correlation function. The transport is due to a velocity perturbation caused by two driving forces, the Brownian force and frictional force. The transport coefficient due to the segment-segment interaction can be calculated from the Kubo formula(9 ... 

Eq. (14) for the exchange-correlation energy in terms of the coupling constant integrated pair-correlation function. If we take the functional derivative of this equation we find that we can write as [66]... 

Z is the nuclear charge, R-r is the distance between the nucleus and the electron, P is the density matrix (equation 16) and (qv Zo) are two-electron integrals (equation 17). f is an exchange/correlation functional, which depends on the electron density and perhaps as well the gradient of the density. Minimizing E with respect to the unknown orbital coefficients yields a set of matrix equations, the Kohn-Sham equations , analogous to the Roothaan-Hall equations (equation 11). 

How could we take into account the fluctuations of the order parameter Let us return to the well-studied example of the gas-liquid system. A general equation of the state of gases and liquids proved in statistical physics [9] has a form p = nk T - n2G(x) where G(x) is some integral containing the interaction potential of particles and the joint correlation function x(r). Therefore, the equation for the long-range order parameter n contains in itself the functional of the intermediate-order parameter x r)-... 

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