Universality classes of absorbing phase transitions in generic branching-annihilating particle systems with nearest-neighbor bias

We study absorbing phase transitions in systems of branching annihilating random walkers and pair contact process with diffusion on a one-dimensional ring, where the walkers hop to their nearest neighbor with a bias $\epsilon$. For $\epsilon =0$, three universality classes—directed percolation (DP), parity-conserving (PC), and pair contact process with diffusion (PCPD)—are typically observed in such systems. We find that the introduction of $\epsilon$ does not change the DP universality class but alters the other two universality classes. For nonzero $\epsilon$, the PCPD class crosses over to DP, and the PC class changes to a new universality class.

Figure 1

Log-log plot of $\rho \left(t\right)$ vs $t$ when $\epsilon =0$ in the PCPD model for various values of $q$, where the value of $q$ increases as one moves from top to bottom of the panel. For $q=0.13353$, the density decays with a power law as shown by the dashed line, thus giving an estimate of the critical point $q_c=0.13353$. The slope of the dashed line gives the estimate of the decay exponent $\alpha \sim 0.221$. Here $L=25000$.

Figure 2

Log-log plot of $\rho \left(t\right)$ vs $t$ when $\epsilon =1/2$ in PCPD model for different values of $q$, where the value of $q$ increases as one moves from top to bottom of the panel. For $q=0.08982$, the density decays with a power law as shown by the dashed line, thus giving an estimate of the critical point $q_c=0.08982$. The slope of the dashed line gives the estimate of the decay exponent $\alpha \sim 0.159$. Here $L=25000$.

Figure 3

Log-log plot of $\rho \left(t\right)$ vs $t$ in BARW ($m=4$) model at the estimated critical points $p_c=0.7215$ for $\epsilon =0$ (lower line in the panel) and $p_c=0.3369$ for $\epsilon =1/2$ (upper line), respectively. The density decays with a power law as shown by the dashed lines. The slope of the dashed line gives the estimate of the decay exponent $\alpha \sim 0.284$ for $\epsilon =0$, and $\alpha \sim 0.224$ for $\epsilon =1/2$.

Figure 4

Numerical estimate of the dynamical exponent $z$ in BARW ($m=4$) using finite-size scaling. The inset shows the unscaled data of $\rho \left(t\right)$ vs $t$ for increasing values of $L=200$, 400, 800, 1600, and 3200 as one moves from left to right. In the main panel, log-log plot of $\rho \left(t\right)t^{\alpha }$ vs $t/L^z$ at $p_c=0.7215$ for $\epsilon =0$, and all these $L$ values are plotted. A good data collapse is obtained for $z=1.7$.

Figure 5

Numerical estimate of the dynamical exponent $z$ in BARW ($m=4$) using finite=size scaling. The inset shows the unscaled data of $\rho \left(t\right)$ vs $t$ for increasing values of $L=200$, 400, 800, 1600, and 3200 as one moves from left to right. In the main panel, log-log plot of $\rho \left(t\right)t^{\alpha }$ vs $t/L^z$ at $p_c=0.3369$ for $\epsilon =1/2$, and these $L$ values are plotted. A good data collapse is obtained for $z=1.74$.

Figure 6

Numerical estimate of the dynamical exponent $z$ in PCPD using finite-size scaling. The inset shows the unscaled data of $\rho \left(t\right)$ vs $t$ for increasing values of $L=200$, 400, 800, 1600, and 3200 as one moves from left to right. In the main panel, log-log plot of $\rho \left(t\right)t^{\alpha }$ vs $t/L^z$ at $q_c=0.13353$ for $\epsilon =0$, and these $L$ values are plotted. A good data collapse is obtained for $z=1.72$.

Figure 7

Numerical estimate of the dynamical exponent $z$ in PCPD using finite-size scaling. The inset shows the unscaled data of $\rho \left(t\right)$ vs $t$ for increasing values of $L=200$, 400, 800, 1600, and 3200 as one moves from left to right. In the main panel, log-log plot of $\rho \left(t\right)t^{\alpha }$ vs $t/L^z$ at $q_c=0.08982$ for $\epsilon =1/2$, and these $L$ values are plotted. A good data collapse is obtained for $z=1.58$.

Figure 8

Log-log plot of ${\rho }_a$ vs $p_c-p$ in BARW (m = 4) for $\epsilon =0$ and $\epsilon =1/2$. The slopes of the lines (shown by dashed lines) near $p\to p_c$ give an estimate of the exponent $\beta$. For $\epsilon =0, \beta \sim 0.92$, and for $\epsilon =1/2, \beta \sim 0.51$.

Figure 9

Log-log plot of ${\rho }_a$ vs $q_c-q$ in PCPD for $\epsilon =0$ and $\epsilon =1/2$. The slopes of the lines (shown by dashed lines) near $q\to q_c$ give an estimate of the exponent $\beta$. For $\epsilon =0, \beta \sim 0.430$, and for $\epsilon =1/2, \beta \sim 0.275$.

Figure 10

Two-point correlation function $C\left(r\right)$ measured at the critical point $q_c$ in PCPD for $\epsilon =0$ and $\epsilon =1/2$ at $t={10}^5$ for a system of size $L=25\times {10}^3$. The slopes shown by dashed lines give the estimate of $\theta$. For $\epsilon =0$ (lower line in the panel), $\theta \sim 0.346$ and $\theta \sim 0.251$ for $\epsilon =1/2$ (upper line).

Figure 11

Numerical estimation of the exponent $\theta$ in BARW ($m=4$) using finite-size scaling and verification of the scaling relation $\theta =z\alpha$: Log-log plot of $\rho \left(t\right)L^{\theta }$ vs $t/L^z$ at $p_c=0.4182$ when $\epsilon =0.3$ for $L=200$, 400, 800, 1600, and 3200. A good data collapse is obtained for $\theta =z\alpha \sim 0.39$.

Figure 12

Space-time plots in PCPD for (a) $\epsilon =0$ and (b) $\epsilon =1/2$ at the critical points $q_c=0.13353$ and 0.08982 respectively. Clearly, the appearance of these plots looks different. This is because, unlike for $\epsilon =0$, the solitary walkers do not survive for large times when $\epsilon =1/2$.