Kirkwood approximation

The Quasi-Chemical Approximation. The mean-field approximation ignores all correlation in the occupation of neighboring sites. This is incorrect when there is a strong interaction between adsorbates at such sites. The simplest way to include some correlation is to work with probabilities of occupations of two sites (XY) instead of one site (X). Approximations that do this are generally called pair approximations (not to be confused with pair interactions). There are more possibilities to reduce multi-site probabilities as in eqn. (8) to 2-site probabilities than to 1-site probabilities. This leads to different types of pair approximations. The best-known approximation that is used for Ising models is the Kirkwood approximation, which uses for example ... 

Reducing a multi-site probability to a probability of fewer sites is called a decoupling scheme. They have been used extensively in equilibrium statistical physics for models like the Ising model.The difference with models of surface reactions is that we have sum rules like eqn. (25), which conflicts with some of the decoupling schemes from equilibrium statistical physics e.g., the Kirkwood approximation mentioned before is not consistent with sum rules. 

Three-particle densities with (m+m ) = 3 could be expressed through the Kirkwood approximation as products of single-particle (2.3.58) and two-particle (2.3.59)-(2.3.61) densities ... 

The kinetic equations [1-3, 12] were rewritten in [15] for a special choice of the recombination law cr(r), adequate to the NAN model, and solved numerically for d— 1. The general conclusion was drawn that the Kirkwood approximation is quite correct but leads to the error of the order of 10% for the critical exponent a in the asymptotic decay law n(R) oc R a. This quantity (10%) was suggested to be used as a measure of the accuracy of the Kirkwood approximation in the kinetics of the bimolecular reaction A + B -> 0. 

Before discussing mathematical formalism we should stress here that the Kirkwood approximation cannot be used for the modification of the drift terms in the kinetics equations, like it was done in Section 6.3 for elastic interaction of particles, since it is too rough for the Coulomb systems to allow us the correct treatment of the charge screening [75], Therefore, the cut-off of the hierarchy of equations in these terms requires the use of some principally new approach, keeping also in mind that it should be consistent with the level at which the fluctuation spectrum is treated. In the case of joint correlation functions we use here it means that the only acceptable for us is the Debye-Htickel approximation [75], equations (5.1.54), (5.1.55), (5.1.57). 

A set of kinetic equations is cut off by means of the Kirkwood approximation (2.3.62). Let us introduce now dimensionless variables (primes are omitted below) ... 

Using now this weak condition, we have to calculate the new resulting ansatz. To this end we have to make closer look at the Kirkwood approximation. The latter has a high symmetry with respect to the lattice vectors, but in the kinetic equations the three-point probabilities appear only in the form of equation (9.1.27) where the lattice sites play different roles l and n are then nearest neighbours and in nearly all terms i — n = 1, Z — m and n — m > 1 holds. Therefore we can suggest that the sites l and n are in all cases more important than the site m. (The case l — m = 1 is an exception which we do not study in detail.) Therefore we can introduce another ansatz instead of (9.1.20) which is non-symmetric ... 

If we would use the Kirkwood approximation, equation (9.1.20), directly in equation (9.1.46), we would neglect all three-particle terms. In particular, we would obtain... 

Comparing now this exact equation with equation (9.1.50) we see that the Kirkwood approximation takes only a few terms into account. Next we must express equation (9.1.59) in terms of the single particle densities C (A = 0, A, B) and pair correlation functions F (r). We see that if a rate R is infinite, the corresponding distribution function p(2) is zero and... 

FAfl(l) = 0. Therefore we should not write down the temporal evolution of F ( 1) and should not solve it with the Kirkwood approximation. Instead we must calculate the product Rx p = Fx, 0. To this end we use an extension of the idea of Kirkwood which takes into account all the probabilities up to p(1+m) (acentral site plus m = 1,..., z neighbour sites). Here we use the so-called Mamada-Takano approximation [21] ... 

Standard approximate methods, e.g., the Percus-Yevick or hyper-chain approximations, are applicable for systems with the Gibbs distribution and are based on the distinctive Boltzmann factor like exp —U r)/ ksT)), where U(r) is the potential energy of interacting particles. The basic kinetic equation (2.3.53) has nothing to do with the Gibbs distribution. The only approximate method neutral with respect to the ensemble averaging is the Kirkwood approximation [76, 77, 87]. 

Therefore we should not write down the temporal evolution of Fxn ) and should not solve it with the Kirkwood approximation. Instead we must calculate the product = Fxp, 0. To this end we use an... 

For spherical atoms, the polarizability in the Kirkwood approximation can be written as... 

Barrett, A. J., Intrinsic viscosity and friction coefficients for an excluded volume polymer in the Kirkwood approximations, Macromolecules, 17,1566-1572 (1984). 

The same expression can be derived without the use of the preaveraged Oseen tensor and the Kirkwood approximation [22]. Furthermore, if applied to a rigid sphere and an ellipsoid of rotation (either prolate or oblate), eq 2.26 happens to give the known exact results, the Stokes formula for the former and the Perrin formular for the latter [23]. Thus we may consider that eq 2.33 is exact as long as d/L is much smaller than unity. 

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