Kirkwood superposition approximation

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. 

Kuzovkov and Kotomin [89-91] (see also [92, 93]) were the first to use the complete Kirkwood superposition approximation (2.3.62) in the kinetic calculations for bimolecular reaction in condensed media. This approximation allows us to cut off the infinite hierarchy of equations for the correlation functions describing spatial distribution of particles of the two kinds and to restrict ourselves to the treatment of minimal set of the kinetic equations which realistically could be handled (Fig. 2.21). In earlier studies [82, 84, 91, 94-97] a shortened superposition approximation was widely used... 

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 applications of the many-particle densities will be demonstrated on a full scale in further Chapters. It should be only said here that the many-particle density formalism being combined with the shortened Kirkwood superposition approximation, equation (2.3.64), results in the well-known equations of the standard kinetics for both neutral [83] and charged particles [100] giving just another way of their derivation. On the other hand, the use of the full-scale (complete) Kirkwood s approximation, equation (2.3.62), permits us to take into account the many-particle (cooperative) effects [81, 91, 99-102] we are studying in this book. 

In fact, the latter is a functional of the correlation function of dissimilar particles, i.e., to calculate K(t) we need to know either Y(r, t) or p. In its turn, equation (4.1.16) demonstrates that these latter are coupled with three-point densities etc. Therefore, to solve the problem, we have to cut off the infinite equation hierarchy, thus only approximately describing the fluctuation spectrum. Usually it is done by means of the complete Kirkwood superposition approximation, equations (2.3.62) and (2.3.63), or the shortened approximation, equations (2.3.64) and (2.3.65). 

In this Section we consider again the kinetics of bimolecular A + B -A 0 recombination but instead of the linearized approximation discussed above, the complete Kirkwood superposition approximation, equation (2.3.62) is used which results in emergence of two new joint correlation functions for similar particles, Xu(r,t), v = A,B. The extended set of the correlation functions, nA(f),nB(f),Xfi,(r,t),Xa(r,t) and Y(r,t) is believed to be able now to describe the intermediate order in the particle spatial distribution. 

The shortened Kirkwood superposition approximation (2.3.64) differs from the complete one, equation (2.3.63), by the additional condition imposed on the correlation function of similar particles Xv(r,t) = 1 at any time t. Its substitution into equation (5.1.4) and taking into account equation (5.1.5) leads, as one could expect, to the linearized equation (4.1.23) for the correlation dynamics. Therefore, the applicability range of the linearized kinetic... 

Fig. 5.1. The interrelation between the two kinetics of concentrations and correlations treated in terms of complete Kirkwood superposition approximation.

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

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. 

Lastly, in conclusion of the Chapter 5 a study of the effect of the shortened and complete Kirkwood superposition approximations on asymptotic kinetics of a number of single-species reaction should be noted [112, 113]. The general conclusion was drawn that for irreversible reactions the complete superposition approximation, (2.3.62), gives better results than its shortened... 

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... 

The single-species A + A —> A reaction allows us also to test the applicability of the shortened and the complete Kirkwood superposition approximations, (2.3.62) and (2.3.64) [98], The calculated quantities of the saturation concentrations... 

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). 

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