Effect of compressibility on the annihilation process

Using the renormalization group in the perturbation theory, we study the influence of a random velocity field on the kinetics of the single-species annihilation reaction at and below its critical dimension $d_\mathrm{c}=2$. The advecting velocity field is modeled by a Gaussian variable self-similar in space with a finite-radius time correlation (the Antonov–Kraichnan model). We take the effect of the compressibility of the velocity field into account and analyze the model near its critical dimension using a three-parameter expansion in $\epsilon$, $\Delta$, and $\eta$, where $\epsilon$ is the deviation from the Kolmogorov scaling, $\Delta$ is the deviation from the (critical) space dimension two, and $\eta$ is the deviation from the parabolic dispersion law. Depending on the values of these exponents and the compressiblity parameter $\alpha$, the studied model can exhibit various asymptotic (long-time) regimes corresponding to infrared fixed points of the renormalization group. We summarize the possible regimes and calculate the decay rates for the mean particle number in the leading order of the perturbation theory.