The Kirkwood superposition approximation

The next approximation level requires the use of the three-point correlation functions 53 expressed through the Kirkwood superposition approximation (Section 2.3.1) 

Note that the introduction of the correlation functions pm in (5.2.4) instead of (m -t- l)-point densities pi rn in feet enabled us to reduce the number of variables. For instance, the molecular field approximation, p2 = 91 (ri )5i (ra), corresponds to that for superposition approximation (equation (2.3.55)) for pi,2 whereas, in its turn, equation (5.2.13) for 53 corresponds to the higher-order superposition approximation (equation (2.3.56)) for pi,3. When substituting (5.2.13) into (5.2.12) with m = 2, we obtain an exact equation for gi with 

The equations derived above, describing the A + B B reaction kinetics in terms of the correlation functions g and g2, have the form of the nonlinear generalised multi-dimensional diffusion equation. Ignoring the multidimensionality of the operator terms in (5.2.11), these equations could be formally considered as similar to the basic non-linear equations for the A - -B —0 reaction (Section 5.1). Equations studied in both Sections 5.1 and 5.2 are derived with the help of the Kirkwood superposition approximation, the use of which leads to several equations for the correlation functions of similar and dissimilar reactants. 

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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... 

Chapter 5 deals with derivation of the basic equations of the fluctuation-controlled kinetics, applied mainly to the particular bimolecular A + B 0 reaction. The transition to the simplified treatment of the density fluctuation spectrum is achieved by means of the Kirkwood superposition approximation. Its accuracy is estimated by means of a comparison of analytical results for some test problems of the chemical kinetics with the relevant computer simulations. Their good agreement permits us to establish in the next Chapters the range of the applicability of the traditional Waite-Leibfried approach. 

Fig. 2.21. The idea of the cut off of the infinite hierarchy of equations for the correlation functions by means of the Kirkwood superposition approximation.

The accuracy of the Kirkwood superposition approximation was questioned recently [15] in terms of the new reaction model called NAN (nearest available neighbour reaction) [16-20], Unlike previous reaction models, in the NAN scheme AB pairs recombine in a strict order of separation the closest pair in an initially random distribution is removed first, then the next one and so on. Thus for NAN, the recombination distance R, e.g., the separation of the closest pair of dissimilar particles at any stage of the recombination, replaces real time as the ordering variable time does not enter at all the NAN scheme. R is conveniently measured in units of the initial pair separation. At large R in J-dimensions, NAN scaling arguments [16] lead rapidly to the result that the pair population decreases asymptotically as cR d/2 (c... 

Therefore, the approximate treatment of the A+B — 0 reaction for charged particles inavoidably requires a combination of several approximations the Kirkwood superposition approximation for the reaction terms and the Debye-Hvickel approximation for modification of the drift terms with self-consistent potentials. Not discussing here the accuracy of the latter approximation, note... 

Here the first term arises from the diffusive approach of reactants A into trapping spheres around B s it is nothing but the standard expression (8.2.14). The second term arises due to the direct production of particles A inside the reaction spheres (the forbidden for A s fraction of the system s volume). Unlike the Lotka-Volterra model, the reaction rate is defined by an approximate expression (due to use of the Kirkwood superposition approximation), therefore first equations (8.3.9) and (8.3.10) of a set are also approximate. 

The written above equations for the temporal evolution contain three-point probabilities, i.e., we obtain a hierarchy of equations. We must truncate the infinite set of master equations in order to obtain a finite system of non-linear equations. To this end we use the Kirkwood superposition approximation (see Section 9.1.1). 

In our opinion, this book demonstrates clearly that the formalism of many-point particle densities based on the Kirkwood superposition approximation for decoupling the three-particle correlation functions is able to treat adequately all possible cases and reaction regimes studied in the book (including immobile/mobile reactants, correlated/random initial particle distributions, concentration decay/accumulation under permanent source, etc.). Results of most of analytical theories are checked by extensive computer simulations. (It should be reminded that many-particle effects under study were observed for the first time namely in computer simulations [22, 23].) Only few experimental evidences exist now for many-particle effects in bimolecular reactions, the two reliable examples are accumulation kinetics of immobile radiation defects at low temperatures in ionic solids (see [24] for experiments and [25] for their theoretical interpretation) and pseudo-first order reversible diffusion-controlled recombination of protons with excited dye molecules [26]. This is one of main reasons why we did not consider in detail some of very refined theories for the kinetics asymptotics as well as peculiarities of reactions on fractal structures ([27-29] and references therein). 

Note that the only approximation made in the derivation of Eq. (177) is the use of the Kirkwood superposition approximation for the triplet distribution function of the liquid [21]. In a dense liquid at low temperature (near its triple point), this is not a bad approximation [21],... 

It is worthwhile to discuss the relative contributions of the binary and the three-particle correlations to the initial decay. If the triplet correlation is neglected, then the values of the Gaussian time constants are equal to 89 fs and 93 fs for the friction and the viscosity, respectively. Thus, the triplet correlation slows down the decay of viscosity more than that of the friction. The greater effect of the triplet correlation is in accord with the more collective nature of the viscosity. This point also highlights the difference between the viscosity and the friction. As already discussed, the Kirkwood superposition approximation has been used for the triplet correlation function to keep the problem tractable. This introduces an error which, however, may not be very significant for an argon-like system at triple point. 

More exact than the quasi-chemical approximation (QCA) is the Kirkwood superposition approximation since if takes into account the indirect correlations. The form of the components in the right-hand side of... 

Singer, A., Maximum entropy formulation of the Kirkwood superposition approximation. J. Chem. Phys. 121, 3657-3666 (2004). 

Calculation of the second-order term in Eq. (3.5.4) and the first-order term in Eq. (3.5.5) requires knowledge of the triplet distribution function in the reference fluid which is usually replaced by the Kirkwood superposition approximation. Following Smith, we will refer to the approach as a whole as the reference averaged Mayer (RAM) function theory. Another choice of reference system based upon a division of the Mayer function is that of hard spheres with a diameter chosen so that the first-order term in the free energy vanishes. This gives rise to the so called blip function theory. ... 

Here we shall prove that the BGY g(r) yields the exact pressure for onedimensional hard rods if the virial theorem is used. This result is independent of any approximation on such as the Kirkwood superposition approximation (4). We shall also prove that the BGY2 radial distribution function for one-dimensional hard rods is exact. This proof is independent of the approximation (7) used for the quadruplet correlation function 4234. 

Computer simulation results for S2 are somewhat sparse and involve the usual uncertainties involved in extrapolating results for a truncated T(r) used in a periodic box to untruncated T(r) in an infinite system." Nevertheless for polarizable hard-sphere and Lennard-Jones particles, it is probably safe to say that the estimates currently available from the combined use of analytic and simulation input are enough to provide a reliable guide to the p and dependence of Sj over the full fluid range of those variables. The most comprehensive studies of have been made by Stell and Rushbrooke" and by Graben, Rushbrooke, and Stell," for the hard-sphere and Lennard-Jones cases, respectively. Both these works utilize the simulation results of Alder, Weis, and Strauss," as well as exact density-expansion results, and numerical results of the Kirkwood superposition approximation... 

Finally, we mention that very recently three other integral equation approaches to treating polymer systems have been proposed. Chiew [104] has used the particle-particle perspective to develop theories of the intermolecular structure and thermodynamics of short chain fluids and mixtures. Lipson [105] has employed the Born-Green-Yvon (BGY) integral equation approach with the Kirkwood superposition approximation to treat compressible fluids and blends. Initial work with the BGY-based theory has considered lattice models and only thermodynamics, but in principle this approach can be applied to compute structural properties and treat continuum fluid models. Most recently, Gan and Eu employed a Kirkwood hierarchy approximation to construct a self-consistent integral equation theory of intramolecular and intermolecular correlations [106]. There are many differences between these integral equation approaches and PRISM theory which will be discussed in a future review [107]. 

Eq.(1.147) shows that the SSM model maps the adsorption on a flat surface onto a two dimensional lattice problem of a very general kind, which is in general not amenable to analytic treatment. It can be simplified by introducing the Kirkwood superposition approximation... 

We can now turn to the question of matrix correlations, which arises because of the application of two conflicting approximations in the course of the calculation the Gaussian approximation for large and the continuinn approximation for small number densities of the matrix units. A reasonable way out of this dilemma is to retain the Gaussian approximation and to introduce matrix correlations. Here the only correlation effect considered is the principle that two matrix units cannot be located at the same position. This means that the factorization used in Eq. 2 cannot be applied. Matrix correlations can be accounted for, within the Ifamework of the stochastic model, by introducing a three-particle distribution function g2 R,R ) [16-18]. Applying the Kirkwood superposition approximation [19],... 

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