Effect of Compressibility on the Annihilation Process

Abstract

Annihilation processes, where the reacting particles are influenced by some external advective field, are one of the simplest examples of nonlinear statistical systems. This type of processes can be observed in miscellaneous chemical, biological or physical systems. In low space dimensions usual description by means of kinetic rate equation is not sufficient and the effect of density fluctuations must be taken into ac- count. Using perturbative renormalization group we study the influ- ence of random velocity field on the kinetics of single-species annihila- tion reaction at and below its critical dimension $d_c = 2$. The advecting velocity field is modelled by the self-similar in space Gaussian variable finite correlated in time (Antonov-Kraichnan model). Effect of the compressibility of velocity field is taken into account and the model is analyzed near its critical dimension by means of three-parameter expansion in $\epsilon$, $\Delta$ and $\eta$. Here $\epsilon$ is the deviation from the Kolmogorov scaling, $\Delta$ is the deviation from the (critical) space dimension 2 and {\eta} is the deviation from the parabolic dispersion law. Depending on the value of these exponents and the value of compressiblity parameter {\alpha}, the studied model can exhibit various asymptotic (long-time) regimes corresponding to the infrared (IR) fixed points of the renormalization group. The possible regimes are summarized and the decay rates for the mean particle number are calculated in the leading order of the perturbation theory.