Application of the Bogolyubov's theory of weakly non-ideal Bose gas on the A+A, A+B, B+B reaction-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->0, (ii) A+B->0 and (iii) A+A,B+B,A+B->0 (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, 2633 (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 n_0 approximation suggested by Mattis and Glasser in Rev. Mod. Phys. 70, 979 (1998) is questioned if used beyond A+B reaction-diffusion model.