Random Fields at a Nonequilibrium Phase Transition

We study nonequilibrium phase transitions in the presence of disorder that locally breaks the symmetry between two equivalent macroscopic states. In low-dimensional equilibrium systems, such random-field disorder is known to have dramatic effects: it prevents spontaneous symmetry breaking and completely destroys the phase transition. In contrast, we show that the phase transition of the one-dimensional generalized contact process persists in the presence of random-field disorder. The ultraslow dynamics in the symmetry-broken phase is described by a Sinai walk of the domain walls between two different absorbing states. We discuss the generality and limitations of our theory, and we illustrate our results by large-scale Monte Carlo simulations.

Figure 1

Time evolution of the generalized contact process in the inactive phase (a) without ($\mu =5/6$) and (b) with random-field disorder (${\mu }_h=1$, ${\mu }_l=2/3$). $I_1$ and $I_2$ are shown in yellow and blue (light and dark grey). Active sites between the domains are marked in red (middle grey). The difference between the diffusive domain wall motion (a) and the much slower Sinai walk (b) is clearly visible (part of a system of ${10}^5$ sites for times up to ${10}^8$).

Figure 2

Density $\rho$ vs time $t$ for several values of the healing rate $\mu$. The data are averages over 60 to 200 disorder configurations. Inset: The log-log plot shows that the density decay is slower than a power law for all $\mu$.

Figure 3

${\rho }^{-1/2}$ vs $\ln (t)$ for several values of the healing rate $\mu$. The solid straight lines are fits to the predicted behavior $\rho \sim {ln}^{-2}(t/t_0)$.

Figure 4

Density vs time at the critical healing rate ${\mu }_c=0.8$, plotted as ${\rho }^{-1/x}$ vs $\ln (t)$ with $x=0.5$. The solid line is a fit to $\rho (t)\sim {ln}^{-x}(t/t_0)$. Inset: Identifying ${\mu }_c$ from the stationary density ${\rho }_{\mathrm{st}}$ in the active phase and the prefactor of the $\rho =B{ln}^{-2}(t/t_0)$ decay in the inactive phase.