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. Here the approximation used by Bogolyubov to study a weakly nonideal Bose gas, referred to as the weakly nonideal Bose gas approximation (WBGA), is applied in the analysis of three reaction-diffusion models: (i) $A+\overrightarrow{A}Ø,$ (ii) $A+\overrightarrow{B}Ø,$ and (iii) $A+A,B+B,A+\overrightarrow{B}Ø$ (the $\mathrm{ABBA}$ model). Two types of WBGA are considered, the simpler WBGA-I and the 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 the $A+B$ reaction well, it reproduces the correct $d/4\mathrm{}$ density decay exponent. However, it fails in the case of the $A+A$ reaction and the $\mathrm{ABBA}$ model. (To cure the deficiency of WBGA in dealing with the $A+A$ model, a hybrid of the WBGA and Kirkwood superposition approximations is suggested.) It is shown that the WBGA-I is identical to the dressed-tree calculation suggested by Lee [J. Phys. A 27, 2633 (1994)], and that the dressed-tree calculation does not lead to the $d/2\mathrm{}$ density decay exponent when applied to the $A+A$ reaction, as normally believed, but it predicts the $d/4\mathrm{}$ decay exponent. Last, the usage of the small $n_0$ approximation suggested by Mattis and Glasser [Rev. Mod. Phys. 70, 979 (1998)] is questioned if used beyond the $A+B$ reaction-diffusion model.

• Received 7 May 2003

DOI:https://doi.org/10.1103/PhysRevE.69.011106