Application of Bogolyubov’s theory of weakly nonideal Bose gases to theA+A,A+B,B+Breaction-diffusion system

Abstract

Theoretical methods for dealing with diffusion-controlled reactions inevitably rely on some kind of approximation and to find the one that works on a particular problem is not always easy. In here the approximation used by Bogolyubov to study weakly non-ideal Bose gas, to be refereed to as weakly non-ideal Bose gas approximation (WBGA), is applied in the analysis of of the three reaction-diffusion models (i) A + A −→ ∅, (ii) A + B −→ ∅ and (iii) A + A, B + B, A + B −→ ∅ (the ABBA model). The two types of WBGA are considered, the simpler WBGA-I and more complicated WBGA-II. All models are defined on the lattice to facilitate comparison with computer experiment (simulation). It is found that the WBGA describes A+B reaction well, it reproduces correct d/4 density decay exponent. However, it fails in the case of the A+A reaction and the ABBA model. (To cure deficiency of WBGA in dealing with A+A model the hybrid of WBGA and Kirkwood superposition approximation is suggested.) It is shown that the WBGA-I is identical to the dressed tree calculation suggested by Lee in J. Phys. A 27, 2633Phys. A 27, (1994, and that the dressed tree calculation does not lead to the d/2 density decay exponent when applied to the A+A reaction, as normally believed, but it predicts d/4 decay exponent. Last, the usage of the small n0 approximation suggested by Mattis and Glasser in Rev. Mod. Phys. 70(3), 979 (1998) is questioned if used beyond A+B reaction-diffusion model.

Multiparticle reaction noise characteristics

“…It is straight forward to generalize the methods discussed for spatially extended systems in [20] and [21] to account for higher order effects in a perfect mixing setup.…”

“…To close the equations the approaches discussed in [20] and [21] will be used and adapted to describe a well mixed case.…”

“…The paper [21] showed how to implement these ideas for spatially extended system, where multiple occupancy of lattice sites is allowed. It is straight forward to modify the techniques presented in [21] to describe a perfectly mixed reaction volume. A single lattice site (with a multiple occupancy allowed) is equivalent to a perfectly mixed reaction volume studied in here.…”

“…Please note that the exponents on the correlation functions count number of AA, PP, and AP pairs in the system with x A and y P particles. Equation (34) represents an instance of Kirkwood's superposition approximation [20,21], more specifically of the full Kirkwood's pair approximation. This approximation is central for workings of the PARNES method.…”

Diffusion Controlled Reactions, Fluctuation Dominated Kinetics, and Living Cell Biochemistry

“…Equations of motion have the same form regardless whether multiple occupancy or single occupancy of lattice sites are allowed (though form of boundary conditions might differ). Work in [16] focuses on multiple occupancy situation while work in [4] deals with single occupancy situation. For large volumes both approaches give essentially the same results.…”