Statistical mechanics pair correlation function

The interfacial pair correlation functions are difficult to compute using statistical mechanical theories, and what is usually done is to assume that they are equal to the bulk correlation function times the singlet densities (the Kirkwood superposition approximation). This can be then used to determine the singlet densities (the density and the orientational profile). Molecular dynamics computer simulations can in... 

From the many tools provided by statistical mechanics for determining the EOS [36, 173, 186-188] we consider first integral equation theories for the pair correlation function gxp(ra,rp) of spherical ions which relates the density of ion / at location rp to that of a at ra. In most theories gafi(ra,rp) enters in the form of the total correlation function hxp(rx,rp) = gxp(rx,rp) — 1. The Omstein-Zemike (OZ) equation splits up hap(rx,rp) into the direct correlation function cap(ra, rp) for pair interactions plus an indirect term that reflects these interactions mediated by all other particles y ... 

Several kinds of approximations are involved in the derivation of Eq. (5). First, it implies that the interactions between charges can be described by an average potential, which is a mean field type of approximation. A theory devoid of this assumption can be developed in the general framework of statistical mechanics, using pair correlation functions. One of the successful approaches, based on the... 

As mentioned above, a set of experimental data does not necessarily correspond to a unique molecular structure. Moreover, even unphysical structures may be consistent with a set of experimental data. It is therefore necessary to carefully choose a set of constraints to limit the number of possible structures. The uniqueness theorem of statistical mechanics [30, 31] provides a guide to the number and type of constraints that should be appfied in the RMC method in order to get a unique structure [32]. For systems in which only two- and three-body forces are important, the uniqueness theorem states that a given set of pair correlation function and three-body correlation function determines aU the higher correlation functions. In other words, assuming that only two- and three-body forces are important, the RMC method must be implemented along with constraints that describe the three-body correlations [27]. 

In this section, we review some of the important formal results in the statistical mechanics of interaction site fluids. These results provide the basis for many of the approximate theories that will be described in Section III, and the calculation of correlation functions to describe the microscopic structure of fluids. We begin with a short review of the theory of the pair correlation function based upon cluster expansions. Although this material is featured in a number of other review articles, we have chosen to include a short account here so that the present article can be reasonably self-contained. Cluster expansion techniques have played an important part in the development of theories of interaction site fluids, and in order to fully grasp the significance of these developments, it is necessary to make contact with the results derived earlier for simple fluids. We will first describe the general cluster expansion theory for fluids, which is directly applicable to rigid nonspherical molecules by a simple addition of orientational coordinates. Next we will focus on the site-site correlation functions and describe the interaction site cluster expansion. After this, we review the calculation of thermodynamic properties from the correlation functions, and then we consider the calculation of the dielectric constant and the Kirkwood orientational correlation parameters. 

Compared to the effort devoted to experimental work, theoretical studies of the partial molar volume have been very limited [61, 62]. The computer simulations for the partial molar volume were started a few years ago by several researchers, but attempts are still limited. As usual, our goal is to develop a statistical-mechanical theory for calculating the partial molar volume of peptides and proteins. The Kirkwood-Buff (K-B) theory [63] provides a general framework for evaluating thermodynamic quantities of a liquid mixture, including the partial molar volume, in term of the density pair correlation functions, or equivalently, the direct correlation functions. The RISM theory is the most reliable tool for calculating these correlation functions when the solute molecule comprises many atoms and has a complicated conformation. 

During World War II, little or no work was done on solution theory, but after the war, activity began again. Now, the emphasis of many theories began to fall on the properties and usefulness of molecular distribution functions, in particular the pair correlation function. This was due, in part, I believe, to the thesis of Jan de Boer (De Boer 1940,1949). As an aside, I once asked J. E. Mayer why he used the canonical ensemble in his early work on statistical mechanics and the grand ensemble in his later works. He replied, Oh, I switched after I read de Boer s thesis and saw how easy the grand ensemble made things. De Boer s work was for pure fluids, not solutions, and other authors, in particular John G. Kirkwood (Kirkwood 1935), also developed the correlation function method. 

This could be achieved with rigor if the statistical mechanical theory of the liquid state were quantitatively accurate. The difficulty is that complete description of the liquid structure involves specification of not only the two-body correlations, but three-body and higher correlations as well. Some properties of the three-body correlation function g R, R2,Ri) are available experimentally from the density derivative of S Q) or g R) (Egelstaff, 1992), but the experiments in general yield a direct measure of only the pair correlation function. Therefore essentially all the existing liquid state theories invoke an assumed form for the three-particle correlation function (see, e.g., Hansen and McDonald, 1976 Egelstaff, 1992) and theoretical models. 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

However, Waite s approach has several shortcomings (first discussed by Kotomin and Kuzovkov [14, 15]). First of all, it contradicts a universal principle of statistical description itself the particle distribution functions (in particular, many-particle densities) have to be defined independently of the kinetic process, but it is only the physical process which determines the actual form of kinetic equations which are aimed to describe the system s time development. This means that when considering the diffusion-controlled particle recombination (there is no source), the actual mechanism of how particles were created - whether or not correlated in geminate pairs - is not important these are concentrations and joint densities which uniquely determine the decay kinetics. Moreover, even the knowledge of the coordinates of all the particles involved in the reaction (which permits us to find an infinite hierarchy of correlation functions = 2,...,oo, and thus is... 

As earlier for the case U b = 0 (Fig. 6.35), the correlation functions XA(r,t) and Y(r, t) shown in Fig. 6.38 (a) and (b) demonstrate appearance of the domain structure in a reaction volume with interacting particles, having the distinctive size = y/Dt. Interaction within AB pairs holds at the relative distances r re (at long times rc < takes place) and only slightly modifies the AB pair distribution on the domain boundaries, where the reaction takes place, but do not influence essentially the entire mechanism of the domain formation (the effect of statistical aggregation). 

The fundamental problem in classical equilibrium statistical mechanics is to evaluate the partition function. Once this is done, we can calculate all the thermodynamic quantities, as these are typically first and second partial derivatives of the partition function. Except for very simple model systems, this is an unsolved problem. In the theory of gases and liquids, the partition function is rarely mentioned. The reason for this is that the evaluation of the partition function can be replaced by the evaluation of the grand canonical correlation functions. Using this approach, and the assumption that the potential energy of the system can be written as a sum of pair potentials, the evaluation of the partition function is equivalent to the calculation of... 

As mentioned above, in the absence of an external field, e can be expressed in terms of the response of each particle to the field set up by the others. In the model under consideration that field is a sum of pair terms, so the key statistical mechanical quantity involved in the expression of e is the two-body correlation function. Its systematic use unavoidably entails a heavy dose of terminology and notation, which we now introduce in the language of Refs. 4 and 5. 

Fortunately, in the classical statistical mechanics of fluids there is a general way out of this problem, which is based on the powerful concept of the direct correlation function [22, 148]. This key mathematical object was introduced by Omstein and Zemike [178] one hundred years ago to deal with the classical fluctuations of the fluid density near the critical point. The so-called Omstein-Zemike equation at the pair level (OZ2) relates the direct correlation function between a pair of atoms cfR ) to the total correlation function through the integral... 

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