Role of diffusion in branching and annihilation random walk models

Different branching and annihilating random walk models are investigated by the cluster mean-field method and simulations in one and two dimensions. In the case of the $A\to 2A$, $2A\to \stackrel{̸}{0}$ model the cluster mean-field approximations show diffusion dependence in the phase diagram as was found recently by the nonperturbative renormalization group method [L. Canet et al., Phys. Rev. Lett. 92, 255703 (2004)]. The same type of survey for the $A\to 2A$, $4A\to \stackrel{̸}{0}$ model results in a reentrant phase diagram, similar to that of the $2A\to 3A$, $4A\to \stackrel{̸}{0}$ model [G. Ódor, Phys. Rev. E 69, 036112 (2004)]. Simulations of the $A\to 2A$, $4A\to \stackrel{̸}{0}$ model in one and two dimensions confirm the presence of both the directed percolation transitions at finite branching rates and the mean-field transition at zero branching rate. In two dimensions the directed percolation transition disappears for strong diffusion rates. These results disagree with the predictions of the perturbative renormalization group method.

Figure 1

(Color online) (a) Phase diagram of the $A\to 2A$, $2A\to \stackrel{̸}{0}$ model determined by $N=3,4,\dots ,8$ (right to left curves) cluster approximations. (b) The ${({\lambda }_c∕D)}_{tr}$ critical end point values of the ${\sigma }_c>0$ transitions as a function of $1∕N$. Boxes correspond to the $A\to 2A$, $2A\to \stackrel{̸}{0}$ and circles to the $A\to 2A$, $4A\to \stackrel{̸}{0}$ model. The line shows a fitting of the form (10).

Figure 2

(Color online) Steady state density results in the $N=7$ approximations of the $A\to 2A$, $4A\to \stackrel{̸}{0}$ model. Different curves correspond to $D=1,0.7,0.6,0.4,0.371,0.36,0.2$ (top to bottom). The ${\sigma }_c>0$ critical point disappears for $D^{*}>\mathrm{0.370\hspace{0.17em}83}$.

Figure 3

(Color online) Simulation results for the steady state density of the $A\to 2A$, $4A\to \stackrel{̸}{0}$ model. Crosses correspond to $D=1.0$ and $+$ signs to $D=0.2$ diffusion in 1D. Circles denote $D=0.01$ and squares $D=1$ data in 2D. Error bars are smaller than the symbol sizes. The lines serve to guide the eye. The inset shows the data magnified in the neighborhood of the $\sigma =0$ transition point. The solid line shows a power-law fitting with the exponent $\beta =0.33\left(1\right)$.

Figure 4

(Color online) (a) Local slopes of the density decay exponent $\alpha$ as a function of $t^{-0.5}$ of the one-dimensional $A\to 2A$, $4A\to \stackrel{̸}{0}$ model at $D=0.2$. Different curves correspond to $\sigma =0.8399$, $\mathrm{0.839\hspace{0.17em}85}$, $0.8398$, $\mathrm{0.839\hspace{0.17em}75}$, $0.8397$ (top to bottom). The critical point is located at ${\sigma }_c=\mathrm{0.839\hspace{0.17em}80}\left(2\right)$. (b) The same as in (a) in two dimensions, for $\sigma =0.1426$, $\mathrm{0.142\hspace{0.17em}62}$, $\mathrm{0.142\hspace{0.17em}63}$, $\mathrm{0.142\hspace{0.17em}64}$, $\mathrm{0.142\hspace{0.17em}65}$, $0.1427$ (top to bottom).