Generic sandpile models have directed percolation exponents

Anomalous relaxation and hyperuniform fluctuations in center-of-mass conserving systems with broken time-reversal symmetry

We study the Oslo model, a paradigm for absorbing-phase transition, on a one-dimensional ring of L sites with a fixed global density ρ[over ¯]; we consider the system strictly above critical density ρ_{c}. Notably, microscopic dynamics conserve both mass and center of mass (CoM), but lack time-reversal symmetry. We show that, despite having highly constrained dynamics due to CoM conservation, the system exhibits diffusive relaxation away from criticality and superdiffusive relaxation near criticality. Furthermore, the CoM conservation severely restricts particle movement, causing the mobility-a transport coefficient analogous to the conductivity for charged particles-to vanish exactly. Indeed, the steady-state temporal growth of current fluctuation is qualitatively different from that observed in diffusive systems with a single conservation law. Remarkably, far from criticality where the relative density Δ=ρ[over ¯]-ρ_{c}≫ρ_{c}, the second cumulant, or the variance, 〈Q_{i}^{2}(T,Δ)〉_{c}, of current Q_{i} across the ith bond up to time T in the steady-state saturates as 〈Q_{i}^{2}〉_{c}≃Σ_{Q}^{2}(Δ)-constT^{-1/2}; near criticality, it grows subdiffusively as 〈Q_{i}^{2}〉_{c}∼T^{α}, with 0<α<1/2, and eventually saturates to Σ_{Q}^{2}(Δ). Interestingly, the asymptotic current fluctuation Σ_{Q}^{2}(Δ) is a nonmonotonic function of Δ: It diverges as Σ_{Q}^{2}(Δ)∼Δ^{2} for Δ≫ρ_{c} and Σ_{Q}^{2}(Δ)∼Δ^{-δ}, with δ>0, for Δ→0^{+}. Using a mass-conservation principle, we exactly determine the exponents δ=2(1-1/ν_{⊥})/ν_{⊥} and α=δ/zν_{⊥} via the correlation-length and dynamic exponents, ν_{⊥} and z, respectively. Finally, we show that in the steady state the self-diffusion coefficient D_{s}(ρ[over ¯]) of tagged particles is connected to activity through the relation D_{s}(ρ[over ¯])=a(ρ[over ¯])/ρ[over ¯].

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.

Local smoothing in sandpiles: Spanning avalanches, bifurcation, and temporal oscillations

The manipulation of the self-organized critical systems by repeatedly deliberate local relaxations (local smoothing) is considered. During a local smoothing, the grains diffuse to the neighboring regions, causing a smoothening of the height filed over the system. The local smoothings are controlled by a parameter ζ which is related to the number of local smoothening events in an avalanche. The system shows some new (mass and time) scales, leading to some oscillatory behaviors. A bifurcation occurs at some ζ value, above which some oscillations are observed for the mean number of grains, and also in the autocorrelation functions. These oscillations are associated with spanning avalanches which are due to the accumulation of grains in the smoothed system. The analysis of the rare event waiting time confirms also the appearance of a new time scale.

Hydrodynamics, density fluctuations, and universality in conserved stochastic sandpiles

We study conserved stochastic sandpiles (CSSs), which exhibit an active-absorbing phase transition upon tuning density ρ. We demonstrate that a broad class of CSSs possesses a remarkable hydrodynamic structure: There is an Einstein relation σ^{2}(ρ)=χ(ρ)/D(ρ), which connects bulk-diffusion coefficient D(ρ), conductivity χ(ρ), and mass fluctuation, or scaled variance of subsystem mass, σ^{2}(ρ). Consequently, density large-deviations are governed by an equilibrium-like chemical potential μ(ρ)∼lna(ρ), where a(ρ) is the activity in the system. By using the above hydrodynamics, we derive two scaling relations: As Δ=(ρ-ρ_{c})→0^{+}, ρ_{c} being the critical density, (i) the mass fluctuation σ^{2}(ρ)∼Δ^{1-δ} with δ=0 and (ii) the dynamical exponent z=2+(β-1)/ν_{⊥}, expressed in terms of two static exponents β and ν_{⊥} for activity a(ρ)∼Δ^{β} and correlation length ξ∼Δ^{-ν_{⊥}}, respectively. Our results imply that conserved Manna sandpile, a well studied variant of the CSS, belongs to a distinct universality-not that of directed percolation (DP), which, without any conservation law as such, does not obey scaling relation (ii).

Multichain models of conserved lattice gas

Conserved lattice-gas models in one dimension exhibit absorbing state phase transition (APT) with simple integer exponents β=1=ν=η, whereas the same on a ladder belong to directed percolation (DP) universality. We conjecture that additional stochasticity in particle transfer is a relevant perturbation and its presence on a ladder forces the APT to be in the DP class. To substantiate this we introduce a class of restricted conserved lattice-gas models on a multichain system (M×L square lattice with periodic boundary condition in both directions), where particles which have exactly one vacant neighbor are active and they move deterministically to the neighboring vacant site. We show that for odd number of chains, in the thermodynamic limit L→∞, these models exhibit APT at ρ_{c}=1/2(1+1/M) with β=1. On the other hand, for even-chain systems transition occurs at ρ_{c}=1/2 with β=1,2 for M=2,4, respectively, and β=3 for M≥6. We illustrate this unusual critical behavior analytically using a transfer-matrix method.

Classification of universality classes for quasideterministic sandpile models

The critical behavior of the two-state rotational sandpile model proposed by Santra et al. [Phys. Rev. E 75, 041122 (2007)PLEEE81539-375510.1103/PhysRevE.75.041122] and the locally deterministic and globally stochastic three-state sandpile model are investigated via Monte Carlo simulations. Through these simulations, we are able to estimate critical exponents that characterize the avalanche properties, i.e., the probability distributions of the avalanche size, area, lifetime, and gyration radius, and the expectation values of the avalanche size and area against time and of the size against area. The results are compared with those of the known universality classes. The two models are found to yield consistent results within the range of statistical error, and appear to be consistent with the stochastic two-state Manna sandpile model; therefore, both models appear to belong to the Manna universality class. Our results contradict the earlier conclusion of Santra et al., which we attribute to the slow convergence of the probability distribution to the asymptotic power-law behavior, particularly for the size and lifetime of avalanches.

Phase separation transition in a nonconserved two-species model

A one-dimensional stochastic exclusion process with two species of particles, + and -, is studied where density of each species can fluctuate but the total particle density is conserved. From the exact stationary state weights we show that, in the limiting case where density of negative particles vanishes, the system undergoes a phase separation transition where a macroscopic domain of vacancies form in front of a single surviving negative particle. We also show that the phase-separated state is associated with a diverging correlation length for any density and that the critical exponents characterizing the behavior in this region are different from those at the transition line. The static and the dynamical critical exponents are obtained from the exact solution and numerical simulations, respectively.

Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model

We present simulations of the one-dimensional Oslo rice pile model in which the critical height at each site is randomly reset after each toppling. We use the fact that the stationary state of this sand-pile model is hyperuniform to reach system of sizes >10^{7}. Most previous simulations were seriously flawed by important finite-size corrections. We find that all critical exponents have values consistent with simple rationals: ν=4/3 for the correlation length exponent, D=9/4 for the fractal dimension of avalanche clusters, and z=10/7 for the dynamical exponent. In addition, we relate the hyperuniformity exponent to the correlation length exponent ν. Finally, we discuss the relationship with the quenched Edwards-Wilkinson model, where we find in particular that the local roughness exponent is α_{loc}=1.

Additivity property and emergence of power laws in nonequilibrium steady states

We show that an equilibriumlike additivity property can remarkably lead to power-law distributions observed frequently in a wide class of out-of-equilibrium systems. The additivity property can determine the full scaling form of the distribution functions and the associated exponents. The asymptotic behavior of these distributions is solely governed by branch-cut singularity in the variance of subsystem mass. To substantiate these claims, we explicitly calculate, using the additivity property, subsystem mass distributions in a wide class of previously studied mass aggregation models as well as in their variants. These results could help in the thermodynamic characterization of nonequilibrium critical phenomena.

Crossover from rotational to stochastic sandpile universality in the random rotational sandpile model

In the rotational sandpile model, either the clockwise or the anticlockwise toppling rule is assigned to all the lattice sites. It has all the features of a stochastic sandpile model but belongs to a different universality class than the Manna class. A crossover from rotational to Manna universality class is studied by constructing a random rotational sandpile model and assigning randomly clockwise and anticlockwise rotational toppling rules to the lattice sites. The steady state and the respective critical behavior of the present model are found to have a strong and continuous dependence on the fraction of the lattice sites having the anticlockwise (or clockwise) rotational toppling rule. As the anticlockwise and clockwise toppling rules exist in equal proportions, it is found that the model reproduces critical behavior of the Manna model. It is then further evidence of the existence of the Manna class, in contradiction with some recent observations of the nonexistence of the Manna class.

Fixed-energy sandpiles belong generically to directed percolation

Fixed-energy sandpiles with stochastic update rules are known to exhibit a nonequilibrium phase transition from an active phase into infinitely many absorbing states. Examples include the conserved Manna model, the conserved lattice gas, and the conserved threshold transfer process. It is believed that the transitions in these models belong to an autonomous universality class of nonequilibrium phase transitions, the so-called Manna class. Contrarily, the present numerical study of selected (1+1)-dimensional models in this class suggests that their critical behavior converges to directed percolation after very long time, questioning the existence of an independent Manna class.

Asymptotic shape of the region visited by an Eulerian walker

We study an Eulerian walker on a square lattice, starting from an initial randomly oriented background using Monte Carlo simulations. We present evidence that, for a large number of steps N , the asymptotic shape of the set of sites visited by the walker is a perfect circle. The radius of the circle increases as N1/3, for large N , and the width of the boundary region grows as Nalpha/3, with alpha=0.40+/-0.06 . If we introduce stochasticity in the evolution rules, the mean-square displacement of the walker, <RN2> approximately <RN2> approximately N2nu, shows a crossover from the Eulerian (nu=1/3) to a simple random-walk (nu=1/2) behavior.

Active-absorbing-state phase transition beyond directed percolation: a class of exactly solvable models

We introduce and solve a model of hardcore particles on a one-dimensional periodic lattice which undergoes an active-absorbing-state phase transition at finite density. In this model, an occupied site is defined to be active if its left neighbor is occupied and the right neighbor is vacant. Particles from such active sites hop stochastically to their right. We show that both the density of active sites and the survival probability vanish as the particle density is decreased below half. The critical exponents and spatial correlations of the model are calculated exactly using the matrix product ansatz. Exact analytical study of several variations of the model reveals that these nonequilibrium phase transitions belong to a new universality class different from the generic active-absorbing-state phase transition, namely, directed percolation.

Confirming and extending the hypothesis of universality in sandpiles

Stochastic sandpiles self-organize to an absorbing-state critical point with scaling behavior different from directed percolation (DP) and characterized by the presence of an additional conservation law. This is usually called the C-DP or Manna universality class. There remains, however, an exception to this universality principle: a sandpile automaton introduced by Maslov and Zhang, which was claimed to be in the DP class despite the existence of a conservation law. We show, by means of careful numerical simulations as well as by constructing and analyzing a field theory, that (contrarily to what was previously thought) this sandpile is also in the C-DP or Manna class. This confirms the hypothesis of universality for stochastic sandpiles and gives rise to a fully coherent picture of self-organized criticality in systems with conservation. In passing, we obtain a number of results for the C-DP class and introduce a strategy to easily discriminate between DP and C-DP scaling.

Absorbing states and elastic interfaces in random media: two equivalent descriptions of self-organized criticality

We elucidate a long-standing puzzle about the nonequilibrium universality classes describing self-organized criticality in sandpile models. We show that depinning transitions of linear interfaces in random media and absorbing phase transitions (with a conserved nondiffusive field) are two equivalent languages to describe sandpile criticality. This is so despite the fact that local roughening properties can be radically different in the two pictures, as explained here. Experimental implications of our work as well as promising paths for future theoretical investigations are also discussed.

Sticky grains do not change the universality class of isotropic sandpiles

We revisit the sandpile model with "sticky" grains introduced by Mohanty and Dhar [Phys. Rev. Lett. 89, 104303 (2002)] whose scaling properties were claimed to be generically in the universality class of directed percolation for both isotropic and directed models. While for directed models this conclusion is unquestionable, for isotropic models we present strong evidence that the asymptotic scaling in the self-organized regime (in which a stationary critical state exists in the limit of slow driving and vanishing dissipation rate) is, like other stochastic sandpiles, generically in the Manna universality class. This conclusion is drawn from extensive Monte Carlo simulations, and is strengthened by the analysis of the Langevin equations (proposed by the same authors to account for this problem), argued to converge upon coarse-graining to the well-established set of Langevin equations for the Manna class.

Directed percolation with incubation times

We introduce a model for directed percolation with a long-range temporal diffusion, while the spatial diffusion is kept short ranged. In an interpretation of directed percolation as an epidemic process, this non-Markovian modification can be understood as incubation times, which are distributed accordingly to a Lévy distribution. We argue that the best approach to find the effective action for this problem is through a generalization of the Cardy-Sugar method, adding the non-Markovian features into the geometrical properties of the lattice. We formulate a field theory for this problem and renormalize it up to one loop in a perturbative expansion. We solve the various technical difficulties that the integrations possess by means of an asymptotic analysis of the divergences. We show the absence of field renormalization at one-loop order, and we argue that this would be the case to all orders in perturbation theory. Consequently, in addition to the characteristic scaling relations of directed percolation, we find a scaling relation valid for the critical exponents of this theory. In this universality class, the critical exponents vary continuously with the Lévy parameter.

Critical exponents for the restricted sandpile

I report large-scale Monte Carlo studies of a one-dimensional height-restricted stochastic sandpile using the quasistationary simulation method. Results for systems of up to 50 000 sites yield estimates for critical exponents that differ significantly from those obtained using smaller systems, but are consistent with recent predictions derived from a Langevin equation for stochastic sandpiles [Ramasco, Phys. Rev. E 69, 045105(R) (2004)]. This suggests that apparent violations of universality in one-dimensional sandpiles are due to strong corrections to scaling and finite-size effects.

Universality class of one-dimensional directed sandpile models

A general n-state directed "sandpile" model is introduced. The stationary properties of the n-state model are derived for n<infinity, and analytical arguments based on a central limit theorem show that the model belongs to the universality class of the totally asymmetric Oslo model, with a crossover to uncorrelated branching process behavior for small system sizes. Hence, the central limit theorem allows us to identify the existence of a large universality class of one-dimensional directed sandpile models.

Simple sandpile model of active-absorbing state transitions

We study a simple sandpile model of active-absorbing state transitions in which a particle can hop out of a site only if the number of particles at that site is above a certain threshold. We show that the active phase has product measure whereas nontrivial correlations are found numerically in the absorbing phase. It is argued that the system relaxes to the latter phase slower than exponentially. The critical behavior of this model is found to be different from that of the other known universality classes.

Directed fixed energy sandpile model

We numerically study the directed version of the fixed energy sandpile. On a closed square lattice, the dynamical evolution of a fixed density of sand grains is studied. The activity of the system shows a continuous phase transition around a critical density. While the deterministic version has the set of nontrivial exponents, the stochastic model is characterized by mean field like exponents.

Pattern formation in a metastable, gradient-driven sandpile

With a toppling rule which generates metastable sites, we explore the properties of a gradient-driven sandpile that is minimally perturbed at one boundary. In two dimensions we find that the transport of grains takes place along deep valleys, generating a set of patterns as the system approaches the stationary state. We use two versions of the toppling rule to analyze the time behavior and the geometric properties of clusters of valleys, also discussing the relation between this model and the general properties of models displaying self-organized criticality.

Epidemic processes with immunization

We study a model of directed percolation (DP) with immunization, i.e., with different probabilities for the first infection and subsequent infections. The immunization effect leads to an additional non-Markovian term in the corresponding field theoretical action. We consider immunization as a small perturbation around the DP fixed point in d<6, where the non-Markovian term is relevant. The immunization causes the system to be driven away from the neighborhood of the DP critical point. In order to investigate the dynamical critical behavior of the model, we consider the limits of low and high first-infection rate, while the second-infection rate remains constant at the DP critical value. Scaling arguments are applied to obtain an expression for the survival probability in both limits. The corresponding exponents are written in terms of the critical exponents for ordinary DP and DP with a wall. We find that the survival probability does not obey a power-law behavior, decaying instead as a stretched exponential in the low first-infection probability limit and to a constant in the high first-infection probability limit. The theoretical predictions are confirmed by optimized numerical simulations in 1+1 dimensions.