Critical dynamics of the directed percolation with Lévy-driven temporally quenched disorder

Abstract

Quenched disorder in absorbing phase transitions can disrupt the structure and symmetry of reaction-diffusion processes, offering a more accurate mapping to real physical systems. We developed a temporally quenched disorder method in the (1+1)-dimensional direct percolation (DP) model, where the increment of conditional probability is determined by the cumulative distribution function (CDF) of the Lévy distribution. Monte Carlo (MC) simulations reveal that the model has a critical region governing the transition between absorbing and active states, and this region changes as the parameter $β$, which influences distribution properties. Guided by dynamic scaling laws, we observe that significant variations in the Lévy distribution parameter $β$ lead to notable changes in the particle density decay exponent $α$, total particle number exponent $θ$, and spreading exponent $\tilde{z}$. The quenching mechanism we introduced has broad potential applications in various theoretical and experimental studies of absorbing phase transitions.

Figures (12)

• Figure 1: The evolution of the temporally quenched disorder DP system, compared to the standard DP process, is illustrated with examples. Here, $S_i$ represents the evolution of the standard DP process, $S_j$ represents the long-duration, high-percolation local multi-particle structures that may arise under temporally quenched disorder, and $S_k$ represents another possible effect, where large vacancy gaps appear under low conditional probabilities.
• Figure 2: Cluster diagrams of the Lévy-driven temporally quenched disorder DP system are shown. (a) and (b) illustrate that, for the same $\beta$ value but different conditional probabilities $p$, the cluster diagrams exhibit a transition from the absorbing state to the active state, along with the appearance of large vacancy gaps and edge "avalanche" phenomena. (c) and (d) illustrates the phase transition behavior as the conditional probability changes, with different control parameters than those in (a) and (b). These phenomena indicate that the phase transition characteristics of the temporally quenched disorder DP system are influenced by the properties of the Lévy distribution.
• Figure 3: (a) With reference to the dashed line, the critical region is gradually narrowed using the bisection method. Within the interval $[0.2568, 0.2600]$, the system’s average particle density still does not exhibit a global power-law behavior. (b) As the critical region is further narrowed, the critical point is determined using the goodness of fit. The value $p=0.2584$ corresponds to the optimal goodness of fit, making it a reliable reference for determining the critical point.
• Figure 4: Supplementary analysis of the power-law fitting for the temporal evolution of particle density at $\beta = 1.2$ and $p=0.2584$. (a) Results from a single simulation run comprising $200$ independent clusters on a $10^4 \times 10^5$ lattice. The error bars represent confidence intervals defined by the absolute values of the fitting residuals. (b) The average of five independent measurements. Compared to the single-run outcome in (a), the averaged confidence intervals exhibit greater stability and a significant narrowing. This improvement effectively enhances the reliability of the critical point determination.
• Figure 5: (a)(b) Data collapse of the particle density $\rho(t)$ following the scaling law in (\ref{['e_13']}) for various system sizes at $\beta=1.2$. The excellent overlap of data across different sizes at $p_c=0.2584$ validates the accuracy of the critical point, allowing for the determination of the dynamic exponent $z=1.58(6)$. With increasing system size, the particle density exhibits more pronounced power-law characteristics. This trend justifies the validity of selecting $L=10^4$ for the simulations.