Three-particle correlations

In [11] the alpha-particles were included into the EOS, and detailed comparisons of the outcome with respect to the alpha-particle contribution has been made. We will elaborate this item further, first by using a systematic quantum statistical treatment instead of the simplifying concept of excluded volume, second by including also other (two- and three-particle) correlations. 

Actually three-particle correlations such as that in Fig. 4(a) are introduced by the CSEs and ICSEs, even within a second-order ladder approximation. To understand why, consider the diagram in Fig. 4(d), which represents one of the terms in Within a second-order ladder approximation to A3, diagram 4(b) is included within diagram 4(d). Thus three- and higher-body effects are incorporated into the cumulants A3 and A4 by the CSEs or ICSEs, even when... 

The only problem necessary for developing the condensation theory is to add to the above-mentioned equation of the state the equation defining the function x(r)- Unfortunately, it turns out that the exact equation for the joint correlation function, derived by means of basic equations of statistical physics, contains f/iree-particle correlation function x 3), which relates the correlations of the density fluctuations in three points of the reaction volume. The equation for this three-particle correlations contains four-particle correlation functions and so on, and so on [9], This situation is quite understandable, since the use of the joint correlation functions only for description of the fluctuation spectrum of a system is obviously not complete. At the same time, it is quite natural to take into account the density fluctuations in some approximate way, e.g., treating correlation functions in a spirit of the mean-field theory (i.e., assuming, in particular, that three-particle correlations could be expanded in two-particle ones). 

Equation (2.3.62) corresponds to the three-particle correlation function (2.3.25)... 

Quantitative deviations are seen also from the correlation shown in Fig. 5.9. The correlation functions of dissimilar particles Y (r, t) are in good agreement with simulations, which results also in a reliable reproduction of the decay kinetics for nA(t) - unlike behaviour of the correlation functions of the similar particles Xv r,t) which is very well pronounced for XA(r,t). Positive correlations, Xu(r,t) > 1 as r < , argue for the similar particle aggregation, and the superposition approximation tends to overestimate their density. The obtained results permit to conclude that the approximation (2.3.63) of the three-particle correlation function could be in a serious error for the excess of one kind of reactants. 

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). 

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. 

Kirkwood derived an analogous equation that also relates two- and three-particle correlation functions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of three or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for = 1, however, is a convenient starting point for perturbation theories of inhomogeneous fluids in an external field. 

If we now make a time-dependent superposition approximation for this three-particle correlation function we obtain... 

In this expression K(0) is the so-called Einstein frequency , which can be calculated exactly from the information of the site-site intermolecular potential as well as of the density pair-correlation functions. K 0) is also an equilibrium quantity but with the three-particle correlation functions, whose approximate expression has been derived in [91]. 

Relaxation and electrophoretic effects are calculable with the help of Kirkwood s superposition approximation for the three-particle correlation function gikm-... 

In principle the memory kernel in the mode-coupling equation contains contributions from three particle correlations, however, for all systems studied so far in computer simulations, these only slightly modified the predictions of the theory and helped improve agreement between simulation and theory [31]. Also, there has been an extension of the theory taking chain connectivity into account [32] which improved agreement with the simulations of the bead-spring model, but it remains to be seen whether an application of this theory to the two models presented here can account for their strongly different dynamic behavior. 

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. 

---