Dynamical scaling behavior of the one-dimensional conserved directed-percolation universality class

Alternative method for measuring characteristic lengths in absorbing phase transitions

We applied an alternative method for measuring characteristic lengths reported recently by one of us [J. M. Kim, J. Stat. Mech. (2021) 03321310.1088/1742-5468/abe599] to the models in the Manna universality class, i.e., the stochastic Manna sandpile and conserved lattice gas models in various dimensions. The universality of the Manna model has been under long debate particularly in one dimension since the work of M. Basu et al. [Phys. Rev. Lett. 109, 015702 (2012)10.1103/PhysRevLett.109.015702], who claimed that the Manna model belongs to the directed percolation (DP) universality class and that the independent Manna universality class does not exist. We carried out Monte Carlo simulations for the stochastic Manna sandpile model in one, two, and three dimensions and the conserved lattice gas model in three dimensions, using both the natural initial states (NISs) and uniform initial states (UISs). In two and three dimensions, the results for R(t), defined by R(t)=L[〈ρ_{a}^{2}〉/〈ρ_{a}〉^{2}-1]^{1/d}, L and ρ_{a} being, respectively, the system size and activity density, yielded consistent results for the two initial states. R(t) is proportional to the correlation length following R(t)∼t^{1/z} at the critical point. In one dimension, the data of R(t) for the Manna model using NISs yielded anomalous behavior, suggesting that NISs require much longer prerun time steps to homogenize the distribution of particles and larger systems to eliminate the finite-size effect than those employed in the literature. On the other hand, data from UISs yielded a power-law behavior, and the estimated critical exponents differed from the values in the DP class.

Hyperuniformity of initial conditions and critical decay of a diffusive epidemic process belonging to the Manna class

For a fixed-energy Manna sandpile model belonging to a Manna class in one dimension (d=1), we recently showed that the critical decay is different for random and regular initial conditions (ICs). Compared with previous results of natural IC for several models, we suggested for the Manna class that the critical decay depends on the characteristics of the three ICs. But the dependence on the random and regular ICs was shown only for a single model. In this work, we study the critical decay for the random and regular ICs for another model of the Manna class in d=1, a diffusive epidemic process. It is shown that the critical decay exponent agrees with the previous result for each IC, which verifies that IC dependence is a common feature of the Manna class. In addition, for the random and regular ICs, we measure the variance σ^{2}(r) of total particle density in a region of size r by increasing r up to system size and investigate its temporal evolution toward the value σ_{q}^{2}(r) of the quasisteady state at criticality. In d=1,σ^{2}(r) scales as σ^{2}(r)∼r^{-ψ} with ψ=1 for random distributions and 1<ψ≤2 for hyperuniform ones. The temporal evolution shows that σ^{2}(r) of the two ICs differently relax toward σ_{q}^{2}(r) and the regular IC becomes a hyperuniform distribution of ψ=2 in the beginning of the evolution. We estimate ψ=1.45(3) for both the quasisteady state and absorbing states, so the quasisteady state is also as hyperuniform as absorbing states. The hyperuniformity of the quasisteady state shows that the natural IC also should be hyperuniform as much as the quasisteady state, because the natural IC is obtained from particle configurations close to the quasisteady state. Consequently, the different ψ of the three ICs suggest that σ^{2}(r) can classify the characteristics of the three ICs in a unified way and the different degree of hyperuniformity of the ICs provides another explanation for the observed IC-dependent critical decay in a point of view of initial fluctuations and correlations.

Critical behavior for random initial conditions in the one-dimensional fixed-energy Manna sandpile model

A fixed-energy Manna sandpile model undergoes an absorbing phase transition at a critical ρ_{c}, where an order parameter ϕ(t) decays as t^{-α} in time t. As the prototype of the Manna class, the model has been extensively studied in one dimension. However, the previous estimates of ρ_{c} and some critical exponents are different, depending on the types of initial conditions; random, natural, and regular conditions. The estimates of ρ_{c} for the random and the regular conditions are the lower and the upper bound among currently known estimates, respectively. In this work, for the random conditions, ρ_{c} and α are measured by taking into account finite-size (FS) effects. At the previous estimate of ρ_{c}, simulation results show that the temporal decay of ϕ(t) is strongly affected by the FS effects up to much larger system size (∼10^{6}). For the sizes for which ϕ(t) is independent up to t=2×10^{7}, we estimate ρ_{c}=0.8925(1) and α=0.110(5), which clearly differ from the previous results for the random conditions, ρ_{c}=0.89199(5) and α=0.141(24). Instead, the present ρ_{c} agrees with ρ_{c}=0.89255(2) of the regular conditions. In addition, the present α is substantially distinguishable from the results of the other types of initial conditions, α=0.159(3) and 0.146(2) for the natural and the regular conditions, respectively, which supports the claim of the initial condition dependence of dynamical exponents in the Manna class.

Absence of absorbing phase transitions in a conserved lattice-gas model in one dimension

A one-dimensional conserved lattice-gas model is known to undergo continuous absorbing phase transitions where some of the critical exponents are exactly known. In one dimension, we recently showed that the model is mapped onto a two species reaction A+B→0 with diffusion rate of D_{A}>0 and D_{B}=0. In this work, it is explicitly shown from the scaling theory for A+B→0 that the observed scaling behavior of the conserved lattice-gas model is not associated with the absorbing phase transitions. Instead, the model indeed undergoes a crossover between two different scaling behaviors of A+B→0, the scaling behaviors of equal and unequal initial densities of two species. The crossover is similar to the absorbing transitions in many respects but some important features of continuous transitions such as the diverging fluctuations of an order parameter are absent.

Critical behavior of a fixed-energy Manna sandpile model for regular initial conditions in one dimension

For a fixed-energy (FE) Manna sandpile model in one dimension, we investigate the critical behavior for regular initial conditions in which activities are distributed at regular intervals on average. The FE Manna model conserves the density ρ of total particles and undergoes an absorbing phase transition at a critical ρ(c). For the regular initial conditions, we show via extensive simulations that the dynamical scaling behaviors differ from those of the random and the natural initial conditions. Off-critical scaling exponents β and ν(⊥) are also measured and shown to agree well with the values of the directed percolation (DP) class as reported by Basu et al. [Phys. Rev. Lett. 109, 015702 (2012)]. Our results suggest that the dynamical scaling behaviors depend on the characteristics of initial conditions, but the off-critical scaling behaviors in the steady state are independent of initial conditions and belong to the DP class.

Effects of random initial conditions on the dynamical scaling behaviors of a fixed-energy Manna sandpile model in one dimension

For a fixed-energy (FE) Manna sandpile model in one dimension, we investigate the effects of random initial conditions on the dynamical scaling behavior of an order parameter. In the FE Manna model, the density ρ of total particles is conserved, and an absorbing phase transition occurs at ρ(c) as ρ varies. In this work, we show that, for a given ρ, random initial distributions of particles lead to the domain structure in which domains with particle densities higher and lower than ρ(c) alternate with each other. In the domain structure, the dominant length scale is the average domain length, which increases via the coalescence of adjacent domains. At ρ(c), the domain structure slows down the decay of an order parameter and also causes anomalous finite-size effects, i.e., power-law decay followed by an exponential one before the quasisteady state. As a result, the interplay of particle conservation and random initial conditions causes the domain structure, which is the origin of the anomalous dynamical scaling behaviors for random initial conditions.