Mean field model of coagulation and annihilation reactions in a medium of quenched traps: Subdiffusion

We present a mean field model for coagulation $\left(A+A\to A\right)$ and annihilation $\left(A+A\to 0\right)$ reactions on lattices of traps with a distribution of depths reflected in a distribution of mean escape times. The escape time from each trap is exponentially distributed about the mean for that trap, and the distribution of mean escape times is a power law. Even in the absence of reactions, the distribution of particles over sites changes with time as particles are caught in ever deeper traps, that is, the distribution exhibits aging. Our main goal is to explore whether the reactions lead to further (time dependent) changes in this distribution.

Figure 1

(Color online) The reactant concentrations $c_1\left(t\right)$ and $2c_2\left(t\right)$ in three dimensions as functions of time for $\gamma =0.5$ and $\gamma =0.8$ and two lattice sizes. The symbols denote numerical simulation results. The solid lines result from the mean field theory, and the dashed lines from the connection with the number of distinct sites visited.

Figure 2

(Color online) The reactant concentrations $c_1\left(t\right)$ and $2c_2\left(t\right)$ in one dimension as functions of time for $\gamma =0.5$ and $\gamma =0.8$. The symbols denote the results of numerical simulations, and the lines are linear fits to these results.

Figure 3

(Color online) For all results in this figure, $\gamma =0.5$. The solid lines are calculated from the analytic result Eq. (32) and the symbols are the simulation results for lattices with quenched traps scaled as described in the text. From high to low on the right side of each panel the curves are for $t={10}^2$, ${10}^3$, ${10}^4$, and ${10}^5$. Top panel: $3d$ lattice; bottom panel: lattice with equally likely jumps to any site, that is, a complete graph.