Temporal disorder in discontinuous nonequilibrium phase transitions: General results

We develop a general theory for discontinuous nonequilibrium phase transitions into an absorbing state in the presence of temporal disorder. We focus in two paradigmatic models for discontinuous transitions: the quadratic contact process (in which activation is only spread when two nearest-neighbor sites are both active) and the contact process with long-range interactions. Using simple stability arguments (supported by Monte Carlo simulations), we show that temporal disorder does not destroy the discontinuous transition in the former model. For the latter one, the first-order transition is turned into a continuous one only in the strong-disorder limit, with critical behavior belonging to the infinite-noise universality class of the contact process model. Finally, we have found that rare temporal fluctuations dramatically changes the behavior of metastable phase, turning it into a temporal Griffiths inactive phase characterized by an exponentially large decay time.

Figure 1

Mean-field phase diagram of the clean (a) QCP and (b) $\sigma \mathrm{CP}$ models (see main text). The dashed line denotes a first-order phase transition and the solid line denotes a second-order one belonging to the directed percolation universality class. The dotted line denotes the end of the bistability in the active phase. For the QCP model, ${\lambda }_c=\frac{4}{5}$. For the $\sigma \mathrm{CP}$ model, ${\lambda }^{*}=\frac{1}{2}$.

Figure 2

The mean-field density $\rho$ as a function of time $t$ for the QCP model for various activation rates $\lambda$ in the inactive phase $\lambda <{\lambda }_c$. The inset shows the time $T$ when $\rho =0.1$ (dotted line of the main panel). The dashed line is the analytical result Eq. (8).

Figure 3

The possible steady-state densities ${\rho }_{\infty }$ as a function of the activity spreading rate $\lambda$ for the values of the exponent $\sigma =0.5$ (top) and $\sigma =2.0$ (bottom panel) and various values of the parameter $a$ as indicated.

Figure 4

The mean-field density $\rho$ as a function of time for the QCP model. The temporal disorder parameters are $p=\frac{1}{2}, {\lambda }_{-}=0.79$, and ${\lambda }_{+}=0.95$. In the top panel, $\rho$ is shown for 20 disorder realizations by the data in various symbols and colors and thin lines for $\mathrm{\Delta }t=5$. The average density $〈\rho 〉$ (circles with thick lines) is obtained from ${10}^3$ disorder realizations. In the bottom panel, the density is averaged for ${10}^3$ disorder realizations for different time windows $\mathrm{\Delta }t$. In all cases, the lines connecting data symbols are guide to the eyes.

Figure 5

The mean-field phase diagram in the temporally disordered case for the (a) QCP and (b) $\sigma \mathrm{CP}$ models. The temporal disorder parameters are defined in Eq. (1) with $p=1/2$. In the $\sigma \mathrm{CP}$ model, we are considering that $a\zeta \left(\sigma \right)>1$; otherwise, the metastable phase vanishes. Dashed (solid) lines denote first- (second-) order phase transitions. Dotted lines represent crossovers. The parameter space outside the shaded triangle is unphysical. For the QCP model, ${\lambda }_c=\frac{4}{5}$ while it depends on $a$ and $\sigma$ for the $\sigma \mathrm{CP}$ model; and ${\lambda }^{*}=\frac{1}{2}$. (A) stands for active, (I) for inactive, (M) for metastable, and (TG) for temporal Griffiths.

Figure 6

The logarithm of the typical density as a function of the time $t$ for the $\sigma \mathrm{CP}$ model at the mean-field level. The long-range interaction parameters are $a=1$ and $\sigma =0.5$. The temporal disorder parameters are $\delta \lambda =0.4$ and $\mathrm{\Delta }t=10$ [see Eq. (1)]. Data points are averaged over $N=5\times {10}^5$ disorder realizations. Error bars are about the size of the symbols. Solid lines are the analytical predictions (no fitting parameters) based on the simple random-walk picture (see main text).

Figure 7

The average density as a function of the simulation time for the QCP. In the top panel, $\rho$ is shown for various $\lambda$ starting for ${\rho }_0=1$. The middle and bottom panels show $\rho$ for $\lambda =0.8613$ and $\lambda =0.98$ and various different initial densities ${\rho }_0$. Data are averaged over ${10}^2$ (for the cases when $\rho$ is large for large $t$) to ${10}^5$ (otherwise) different Monte Carlo runs for systems of linear size $L=200$.

Figure 8

Similar to Fig. 7 but for the $\sigma \mathrm{CP}$ model with $a=2.0, \sigma =0.5, L={10}^5$ and the data are averaged over ${10}^3–{10}^5$ different Monte Carlo runs.

Figure 9

The decay time $T$ to the system toward the absorbing state as a function of s $\lambda$ for the (top) QCP and (bottom) $\sigma \mathrm{CP}$ models. In both cases, we observe an algebraic behavior of type ${({\lambda }_c-\lambda )}^{-\varphi }$, where $\varphi =1.56\left(1\right)$ and 4.52(3), respectively.

Figure 10

The average density $\rho$ as a function of time $t$ for the 2D QCP model for systems of size $L=200$ averaged over $N_{\text{MC}}={10}^2–{10}^5$ disorder realizations. The disorder parameters [see Eq. (1)] ${\lambda }_{\pm }$ are indicated in the legends [the one in panel (a) also applies to (c)], $p=\frac{1}{2}$, and $\mathrm{\Delta }t={10}^2$ for panels (a) and (b) and $\mathrm{\Delta }t={10}^4$ for (c) and (d). In all panels the initial density is ${\rho }_0=1$ except in panel (b), where ${\rho }_0$ is varied.

Figure 11

The average density as a function of time for the disordered 1D $\sigma \mathrm{CP}$ with disorder parameters [see Eq. (1)] $p=\frac{1}{2}, \mathrm{\Delta }t$, and ${\lambda }_{\pm }$ being specified in the legends [with the ones in panel (a) applying to panel (c) as well]. The system size is $L={10}^5$ averaged over ${10}^2–{10}^4$ disorder realizations. In panels (a) and (c), ${\lambda }_{+}=0.65$ is in the metastable phase while the various ${\lambda }_{-}\le {\lambda }_c\approx 0.643$ are in the inactive phase. Panel (b) shows $\rho$ (starting from ${\rho }_0=5\times {10}^{-5}$) for various ${\lambda }_{-}$ in the inactive phase while ${\lambda }_{+}=0.700$ is in the active one. Panel (d) $\rho$ starting from various different initial conditions for ${\lambda }_{+}$ in the active phase while $\lambda$ is in the metastable one.

Figure 12

The average density as a function of time for the $\sigma \mathrm{CP}$ with ${\lambda }_{+}=0.7, p=\frac{1}{2}, \mathrm{\Delta }t={10}^2$, and various distinct values of ${\lambda }_{-}$. The straight (blue) line has slope $\rho \sim {(lnt)}^{-1}$ for more than two orders of magnitude in $t$. The system size is $L={10}^5$ averaged over ${10}^3–3\times {10}^3$ different disorder realizations.