Effect of diffusion in one-dimensional discontinuous absorbing phase transitions

It is known that diffusion provokes substantial changes in continuous absorbing phase transitions. Conversely, its effect on discontinuous transitions is much less understood. In order to shed light in this direction, we study the inclusion of diffusion in the simplest one-dimensional model with a discontinuous absorbing phase transition, namely, the long-range contact process ($\sigma$-CP). Particles interact as in the usual CP, but the transition rate depends on the length $\ell$ of inactive sites according to $1+a{\ell }^{-\sigma }$, where $a$ and $\sigma$ are control parameters. The inclusion of diffusion in this model has been investigated by numerical simulations and mean-field calculations. Results show that there exists three distinct regimes. For sufficiently low and large $\sigma$'s the transition is, respectively, always discontinuous or continuous, independently of the strength of the diffusion. On the other hand, in an intermediate range of $\sigma$'s, the diffusion causes a suppression of the phase coexistence leading to a continuous transition belonging to the directed percolation universality class.

Figure 1

Mean-field results. (a) and (b) $\rho$ vs $\alpha$ for $a=2,D=0.15$, and $\sigma =1.5$ and $2.4$. The dashed curve in (a) correspond to the unstable solution. (c) Phase diagram showing the value ${\sigma }_t\left(D\right)$ where the transition changes from continuous to discontinuous. For $D>0.5$ the transition is always discontinuous.

Figure 2

Log-log plot of the maximum distance $R$ vs the particle number $n$ for distinct values of $\sigma$ and for $D=0.1$ (a), $0.5$ (b), and $0.95$ (c). The upper and lower straight lines have slopes $1/d_F=1.337...$ and 1, respectively [cf. Eq. (10)]. In (d) we plot the cluster density ${\rho }_{\text{cl}}$ vs $D$, at the phase coexistence, for distinct values of $\sigma$.

Figure 3

The tricritical line ${\sigma }_t\left(D\right)$ obtained from numerical simulations (cf. Fig. 2).

Figure 4

The time evolution of $P_s\left(t\right)$ and $N\left(t\right)$ in the constant rate ensemble for distinct values of $\alpha$ and $D=0.1$. (a) and (b) $\sigma =0.25$; (c) and (d) $\sigma =0.8$; (e) and (f) $\sigma =1.2$. The slopes of the black straight lines are given by Eq. (13) for (a)–(d) and Eq. (12) for (e) and (f). The inset in image (f) shows the decay of the density $\rho$ and the black line has slope $\theta =0.159464\left(6\right)$.

Figure 5

Same as Fig. 4 but for $D=0.5$. The slopes of the black straight lines are given by Eq. (13) for (a) and (b) and Eq. (12) for (c)–(f). The insets show the time decay of $\rho$ for distinct $\alpha$'s and the black lines have slope $\theta =0.159464\left(6\right)$.

Figure 6

Same as Fig. 5 but for $D=0.95$. The slopes of the black straight lines are given by Eq. (13) for (a) and (b) and Eq. (12) for (c)–(f). The insets show the time decay of $\rho$ for distinct $\alpha$'s and the black lines have slope $\theta =0.159464\left(6\right)$.