Universality class of absorbing phase transitions with a conserved field

Non-equilibrium dynamic hyperuniform states

Disordered hyperuniform structures are an exotic state of matter having suppressed density fluctuations at large length-scale similar to perfect crystals and quasicrystals but without any long range orientational order. In the past decade, an increasing number of non-equilibrium systems were found to have dynamic hyperuniform states, which have emerged as a new research direction coupling both non-equilibrium physics and hyperuniformity. Here we review the recent progress in understanding dynamic hyperuniform states found in various non-equilibrium systems, including the critical hyperuniformity in absorbing phase transitions, non-equilibrium hyperuniform fluids and the hyperuniform structures in phase separating systems via spinodal decomposition.

Hydrodynamic behavior near dynamical criticality of a facilitated conservative lattice gas

We investigate a 2d-conservative lattice gas exhibiting a dynamical active-absorbing phase transition with critical density ρ_{c}. We derive the hydrodynamic equation for this model, showing that all critical exponents governing the large scale behavior near criticality can be obtained from two independent ones. We show that as the supercritical density approaches criticality, distinct length scales naturally appear. Remarkably, this behavior is different from the subcritical one. Numerical simulations support the critical relations and the scale separation.

Yielding Is an Absorbing Phase Transition with Vanishing Critical Fluctuations

The yielding transition in athermal complex fluids can be interpreted as an absorbing phase transition between an elastic, absorbing state with high mesoscopic degeneracy and a flowing, active state. We characterize quantitatively this phase transition in an elastoplastic model under fixed applied shear stress, using a finite-size scaling analysis. We find vanishing critical fluctuations of the order parameter (i.e., the shear rate), and relate this property to the convex character of the phase transition (β>1). We locate yielding within a family of models akin to fixed-energy sandpile (FES) models, only with long-range redistribution kernels with zero modes that result from mechanical equilibrium. For redistribution kernels with sufficiently fast decay, this family of models belongs to a short-range universality class distinct from the conserved directed percolation class of usual FES, which is induced by zero modes.

Interplay between an Absorbing Phase Transition and Synchronization in a Driven Granular System

Absorbing phase transitions (APTs) are widespread in nonequilibrium systems, spanning condensed matter, epidemics, earthquakes, ecology, and chemical reactions. APTs feature an absorbing state in which the system becomes entrapped, along with a transition, either continuous or discontinuous, to an active state. Understanding which physical mechanisms determine the order of these transitions represents a challenging open problem in nonequilibrium statistical mechanics. Here, by numerical simulations and mean-field analysis, we show that a quasi-2D vibrofluidized granular system exhibits a novel form of APT. The absorbing phase is observed in the horizontal dynamics below a critical packing fraction, and can be continuous or discontinuous based on the emergent degree of synchronization in the vertical motion. Our results provide a direct representation of a feasible experimental scenario, showcasing a surprising interplay between dynamic phase transition and synchronization.

Approach to hyperuniformity of steady states of facilitated exclusion processes

We consider the fluctuations in the number of particles in a box of sizeLdinZd,d⩾1, in the (infinite volume) translation invariant stationary states of the facilitated exclusion process, also called the conserved lattice gas model. When started in a Bernoulli (product) measure at densityρ, these systems approach, ast→∞, a 'frozen' state forρ⩽ρc, withρc=1/2ford = 1 andρc<1/2ford⩾2. Atρ=ρcthe limiting state is, as observed by Hexner and Levine, hyperuniform, that is, the variance of the number of particles in the box grows slower thanLd. We give a general description of how the variances at different scales ofLbehave asρ↗ρc. On the largest scale,L≫L2, the fluctuations are normal (in fact the same as in the original product measure), while in a regionL1≪L≪L2, with bothL1andL2going to infinity asρ↗ρc, the variance grows faster than normal. For1≪L≪L1the variance is the same as in the hyperuniform system. (All results discussed are rigorous ford = 1 and based on simulations ford⩾2.).

Quest for the golden ratio universality class

Using mode-coupling theory, the conditions for all allowed dynamical universality classes for the conserved modes in one-dimensional driven systems are presented in closed form as a function of the stationary currents and their derivatives. With an eye on the search for the golden ratio universality class, the existence of some families of microscopic models is ruled out a priori by using an Onsager-type macroscopic current symmetry. In particular, if the currents are symmetric or antisymmetric under the interchange of the conserved densities, then at equal mean densities the golden modes can only appear in the antisymmetric case and if the conserved quantities are correlated, but not in the symmetric case where at equal densities one mode is always diffusive and the second may be either Kardar-Parisi-Zhang (KPZ), modified KPZ, 3/2-Lévy, or also diffusive. We also show that the predictions of mode-coupling theory for a noisy chain of harmonic oscillators are exact.

Quantifying spatiotemporal patterns in classical and quantum systems out of equilibrium

A rich variety of nonequilibrium dynamical phenomena and processes unambiguously calls for the development of general numerical techniques to probe and estimate a complex interplay between spatial and temporal degrees of freedom in many-body systems of completely different nature. In this work we provide a solution to this problem by adopting a structural complexity measure to quantify spatiotemporal patterns in the time-dependent digital representation of a system. On the basis of very limited amount of data our approach allows us to distinguish different dynamical regimes and define critical parameters in both classical and quantum systems. By the example of the discrete time crystal realized in nonequilibrium quantum systems we provide a complete low-level characterization of this nontrivial dynamical phase with only processing bitstrings, which can be considered as a valuable alternative to previous studies based on the calculations of qubit correlation functions.

Continuum contact process and influence of impurity on the critical behavior in absorbing-state phase transitions in two dimensions

We study via Monte Carlo simulations the influence of quenched and mobile impurities in the contact process (CP) on two-dimensional lattice and continuum systems. In the lattice system, the effect of mobile impurity was studied for the density n_{i}=0.2 and two selected values of hopping probability for impurity particles, w=0.5 and 1. In the continuum system, the CP was defined by distributing spherical impurity particles of diameter σ_{i} and number density n_{i}=0.2 and active particles of diameter unity and number density 1-n_{i} on a square substrate with periodic boundaries. In each dynamic process, a particle is selected at random; the active particle either creates with a rate λ an offspring at a distance r (1≤r≤1.5) from the active particle or annihilates with a unit rate, and the impurity particle hops a distance r (0≤r≤1), both along randomly selected directions. We found that the lattice CP shows power-law behaviors with varying critical exponents depending on the values of w. For the continuum CP with quenched impurity, the critical behavior followed the activated scaling scenario, whereas with mobile impurity usual power-law behaviors were observed but the critical exponents varied depending on the values of σ_{i}.

Transfer learning of phase transitions in percolation and directed percolation

The latest advances of statistical physics have shown remarkable performance of machine learning in identifying phase transitions. In this paper, we apply domain adversarial neural network (DANN) based on transfer learning to studying nonequilibrium and equilibrium phase transition models, which are percolation model and directed percolation (DP) model, respectively. With the DANN, only a small fraction of input configurations (two-dimensional images) needs to be labeled, which is automatically chosen, to capture the critical point. To learn the DP model, the method is refined by an iterative procedure in determining the critical point, which is a prerequisite for the data collapse in calculating the critical exponent ν_{⊥}. We then apply the DANN to a two-dimensional site percolation with configurations filtered to include only the largest cluster which may contain the information related to the order parameter. The DANN learning of both models yields reliable results which are comparable to the ones from Monte Carlo simulations. Our study also shows that the DANN can achieve quite high accuracy at much lower cost, compared to the supervised learning.

Absorbing phase transitions in systems with mediated interactions

Experiments of periodically sheared colloidal suspensions or soft amorphous solids display a transition from reversible to irreversible particle motion that, when analyzed stroboscopically in time, is interpreted as an absorbing phase transition with infinitely many absorbing states. In these systems, interactions mediated by hydrodynamics or elasticity are present, causing passive regions to be affected by nearby active ones. We show that mediated interactions induce a universality class of absorbing phase transitions distinct from conserved directed percolation, and we obtain the corresponding critical exponents. We do so with large-scale numerical simulations of a minimal model for the stroboscopic dynamics of sheared soft materials and we derive the minimal field theoretical description.

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.

Activity waves and freestanding vortices in populations of subcritical Quincke rollers

Virtually all of the many active matter systems studied so far are made of units (biofilaments, cells, colloidal particles, robots, animals, etc.) that move even when they are alone or isolated. Their collective properties continue to fascinate, and we now understand better how they are unique to the bulk transduction of energy into work. Here we demonstrate that systems in which isolated but potentially active particles do not move can exhibit specific and remarkable collective properties. Combining experiments, theory, and numerical simulations, we show that such subcritical active matter can be realized with Quincke rollers, that is, dielectric colloidal particles immersed in a conducting fluid subjected to a vertical DC electric field. Working below the threshold field value marking the onset of motion for a single colloid, we find fast activity waves, reminiscent of excitable systems, and stable, arbitrarily large self-standing vortices made of thousands of particles moving at the same speed. Our theoretical model accounts for these phenomena and shows how they can arise in the absence of confining boundaries and individual chirality. We argue that our findings imply that a faithful description of the collective properties of Quincke rollers need to consider the fluid surrounding particles.

Barrier-controlled nonequilibrium criticality in reactive particle systems

Nonequilibrium critical phenomena generally exist in many dynamic systems, like chemical reactions and some driven-dissipative reactive particle systems. Here, by using computer simulation and theoretical analysis, we demonstrate the crucial role of the activation barrier on the criticality of dynamic phase transitions in a minimal reactive hard-sphere model. We find that at zero thermal noise, with increasing the activation barrier, the type of transition changes from a continuous conserved directed percolation into a discontinuous dynamic transition by crossing a tricritical point. A mean-field theory combined with field simulation is proposed to explain this phenomenon. The possibility of Ising-type criticality in the nonequilibrium system at finite thermal noise is also discussed.

Hidden Order Beyond Hyperuniformity in Critical Absorbing States

Disordered hyperuniformity is a description of hidden correlations in point distributions revealed by an anomalous suppression in fluctuations of local density at various coarse-graining length scales. In the absorbing phase of models exhibiting an active-absorbing state transition, this suppression extends up to a hyperuniform length scale that diverges at the critical point. Here, we demonstrate the existence of additional many-body correlations beyond hyperuniformity. These correlations are hidden in the higher moments of the probability distribution of the local density and extend up to a longer length scale with a faster divergence than the hyperuniform length on approaching the critical point. Our results suggest that a hidden order beyond hyperuniformity may generically be present in complex disordered systems.

Geometry-induced nonequilibrium phase transition in sandpiles

We study the sandpile model on three-dimensional spanning Ising clusters with the temperature T treated as the control parameter. By analyzing the three-dimensional avalanches and their two-dimensional projections (which show scale-invariant behavior for all temperatures), we uncover two universality classes with different exponents (an ordinary BTW class, and SOC_{T=∞}), along with a tricritical point (at T_{c}, the critical temperature of the host) between them. The transition between these two criticalities is induced by the transition in the support. The SOC_{T=∞} universality class is characterized by the exponent of the avalanche size distribution τ^{T=∞}=1.27±0.03, consistent with the exponent of the size distribution of the Barkhausen avalanches in amorphous ferromagnets Durin and Zapperi [Phys. Rev. Lett. 84, 4705 (2000)PRLTAO0031-900710.1103/PhysRevLett.84.4705]. The tricritical point is characterized by its own critical exponents. In addition to the avalanche exponents, some other quantities like the average height, the spanning avalanche probability (SAP), and the average coordination number of the Ising clusters change significantly the behavior at this point, and also exhibit power-law behavior in terms of ε≡T-T_{c}/T_{c}, defining further critical exponents. Importantly, the finite-size analysis for the activity (number of topplings) per site shows the scaling behavior with exponents β=0.19±0.02 and ν=0.75±0.05. A similar behavior is also seen for the SAP and the average avalanche height. The fractal dimension of the external perimeter of the two-dimensional projections of avalanches is shown to be robust against T with the numerical value D_{f}=1.25±0.01.

Absorbing-State Transitions in Granular Materials Close to Jamming

We consider a model for driven particulate matter in which absorbing states can be reached both by particle isolation and by particle caging. The model predicts a nonequilibrium phase diagram in which analogs of hydrodynamic and elastic reversibility emerge at low and high volume fractions respectively, partially separated by a diffusive, nonabsorbing region. We thus find a single phase boundary that spans the onset of chaos in sheared suspensions to the onset of yielding in jammed packings. This boundary has the properties of a nonequilibrium second order phase transition, leading us to write a Manna-like mean field description that captures the model predictions. Dependent on contact details, jamming marks either a direct transition between the two absorbing states, or occurs within the diffusive region.

Hydrodynamics of random-organizing hyperuniform fluids

Disordered hyperuniform structures are locally random while uniform like crystals at large length scales. Recently, an exotic hyperuniform fluid state was found in several nonequilibrium systems, while the underlying physics remains unknown. In this work, we propose a nonequilibrium (driven-dissipative) hard-sphere model and formulate a hydrodynamic theory based on Navier-Stokes equations to uncover the general mechanism of the fluidic hyperuniformity (HU). At a fixed density, this model system undergoes a smooth transition from an absorbing state to an active hyperuniform fluid and then, to the equilibrium fluid by changing the dissipation strength. We study the criticality of the absorbing-phase transition. We find that the origin of fluidic HU can be understood as the damping of a stochastic harmonic oscillator in q space, which indicates that the suppressed long-wavelength density fluctuation in the hyperuniform fluid can exhibit as either acoustic (resonance) mode or diffusive (overdamped) mode. Importantly, our theory reveals that the damping dissipation and active reciprocal interaction (driving) are the two ingredients for fluidic HU. Based on this principle, we further demonstrate how to realize the fluidic HU in an experimentally accessible active spinner system and discuss the possible realization in other systems.

Critical density of the Abelian Manna model via a multitype branching process

A multitype branching process is introduced to mimic the evolution of the avalanche activity and determine the critical density of the Abelian Manna model. This branching process incorporates partially the spatiotemporal correlations of the activity, which are essential for the dynamics, in particular in low dimensions. An analytical expression for the critical density in arbitrary dimensions is derived, which significantly improves the results over mean-field theories, as confirmed by comparison to the literature on numerical estimates from simulations. The method can easily be extended to lattices and dynamics other than those studied in the present work.

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).

Logarithmic speed-up of relaxation in A-B annihilation with exclusion

We show that the decay of the density of active particles in the reaction A+B→0 in one dimension, with exclusion interaction, results in logarithmic corrections to the expected power law decay, when the starting initial condition (i.c.) is periodic. It is well known that the late-time density of surviving particles goes as t^{-1/4} with random initial conditions, and as t^{-1/2} with alternating initial conditions (ABABAB⋯). We show that the decay for periodic i.c.'s made of longer blocks (A^{n}B^{n}A^{n}B^{n}⋯) do not show a pure power-law decay when n is even. By means of first-passage Monte Carlo simulations, and a mapping to a q-state coarsening model which can be solved in the independent interval approximation (IIA), we show that the late-time decay of the density of surviving particles goes as t^{-1/2}[ln(t)]^{-1} for n even, but as t^{-1/2} when n is odd. We relate this kinetic symmetry breaking in the Glauber Ising model. We also see a very slow crossover from a t^{-1/2}[ln(t)]^{-1} regime to eventual t^{-1/2} behavior for i.c.'s made of mixtures of odd- and even-length blocks.

A Unifying Theory of Branching Morphogenesis

The morphogenesis of branched organs remains a subject of abiding interest. Although much is known about the underlying signaling pathways, it remains unclear how macroscopic features of branched organs, including their size, network topology, and spatial patterning, are encoded. Here, we show that, in mouse mammary gland, kidney, and human prostate, these features can be explained quantitatively within a single unifying framework of branching and annihilating random walks. Based on quantitative analyses of large-scale organ reconstructions and proliferation kinetics measurements, we propose that morphogenesis follows from the proliferative activity of equipotent tips that stochastically branch and randomly explore their environment but compete neutrally for space, becoming proliferatively inactive when in proximity with neighboring ducts. These results show that complex branched epithelial structures develop as a self-organized process, reliant upon a strikingly simple but generic rule, without recourse to a rigid and deterministic sequence of genetically programmed events.

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Branching morphogenesis follows conserved statistical rules in multiple organs

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Ductal tips grow and branch as default state and stop dividing in high-density regions

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Model reproduces quantitatively organ properties in a parameter-free manner

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Shows that complex organ formation proceeds in a stochastic, self-organized manner

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.

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.

Multicritical absorbing phase transition in a class of exactly solvable models

We study diffusion of hard-core particles on a one-dimensional periodic lattice subjected to a constraint that the separation between any two consecutive particles does not increase beyond a fixed value n+1; an initial separation larger than n+1 can however decrease. These models undergo an absorbing state phase transition when the conserved particle density of the system falls below a critical threshold ρ_{c}=1/(n+1). We find that the ϕ_{k}, the density of 0-clusters (0 representing vacancies) of size 0≤k<n, vanish at the transition point along with activity density ρ_{a}. The steady state of these models can be written in matrix product form to obtain analytically the static exponents β_{k}=n-k and ν=1=η corresponding to each ϕ_{k}. We also show from numerical simulations that, starting from a natural condition, ϕ_{k}(t)s decay as t^{-α_{k}} with α_{k}=(n-k)/2 even though other dynamic exponents ν_{t}=2=z are independent of k; this ensures the validity of scaling laws β=αν_{t} and ν_{t}=zν.

Noise, Diffusion, and Hyperuniformity

We consider driven many-particle models which have a phase transition between an active and an absorbing phase. Like previously studied models, we have particle conservation, but here we introduce an additional symmetry-when two particles interact, we give them stochastic kicks which conserve the center of mass. We find that the density fluctuations in the active phase decay in the fastest manner possible for a disordered isotropic system, and we present arguments that the large scale fluctuations are determined by a competition between a noise term which generates fluctuations, and a deterministic term which reduces them. Our results may be relevant to shear experiments and may further the understanding of hyperuniformity which occurs at the critical point.

Out-of-equilibrium stationary states, percolation, and subcritical instabilities in a fully nonconservative system

The exploration of the phase diagram of a minimal model for barchan fields leads to the description of three distinct phases for the system: stationary, percolable, and unstable. In the stationary phase the system always reaches an out-of-equilibrium, fluctuating, stationary state, independent of its initial conditions. This state has a large and continuous range of dynamics, from dilute-where dunes do not interact-to dense, where the system exhibits both spatial structuring and collective behavior leading to the selection of a particular size for the dunes. In the percolable phase, the system presents a percolation threshold when the initial density increases. This percolation is unusual, as it happens on a continuous space for moving, interacting, finite lifetime dunes. For extreme parameters, the system exhibits a subcritical instability, where some of the dunes in the field grow without bound. We discuss the nature of the asymptotic states and their relations to well-known models of statistical physics.

Directed percolation with a conserved field and the depinning transition

Conserved directed percolation (C-DP) and the depinning transition of a disordered elastic interface belong to the same universality class, as has been proven very recently by Le Doussal and Wiese [Phys. Rev. Lett. 114, 110601 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.110601] through a mapping of the field theory for C-DP onto that of the quenched Edwards-Wilkinson model. Here, we present an alternative derivation of the C-DP field theoretic functional, starting with the coherent-state path integral formulation of the C-DP and then applying the Grassberger transformation, which avoids the disadvantages of the so-called Doi shift. We revisit the aforementioned mapping with focus on a specific term in the field theoretic functional that has been problematic in the past when it came to assessing its relevance. We show that this term is redundant in the sense of the renormalization group.

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.

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.

Absorbing phase transition in a conserved lattice gas model with next-nearest-neighbor hopping in one dimension

The absorbing phase transition of the modified conserved lattice gas (m-CLG) model was investigated in one dimension. The m-CLG model was modified from the conserved lattice gas (CLG) model in such a way that each active particle hops to one of the nearest-neighbor and next-nearest-neighbor empty sites. The order parameter exponent, the dynamic exponent, and the correlation length exponent were estimated from the power-law behavior and finite-size scaling of the active particle densities. The exponents were found to differ considerably from those of the ordinary CLG model and were also distinct from those of the Manna model, suggesting that next-nearest-neighbor hopping is a relevant factor that alters the critical behavior in the one-dimensional CLG model.

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.

Particle-density fluctuations and universality in the conserved stochastic sandpile

We examine fluctuations in particle density in the restricted-height, conserved stochastic sandpile (CSS). In this and related models, the global particle density is a temperaturelike control parameter. Thus local fluctuations in this density correspond to disorder; if this disorder is a relevant perturbation of directed percolation (DP), then the CSS should exhibit non-DP critical behavior. We analyze the scaling of the variance Vℓ of the number of particles in regions of ℓd sites in extensive simulations of the quasistationary state in one and two dimensions. Our results, combined with a Harris-like argument for the relevance of particle-density fluctuations, strongly suggest that conserved stochastic sandpiles belong to a universality class distinct from that of DP.

Computational model of a vector-mediated epidemic

We discuss a lattice model of vector-mediated transmission of a disease to illustrate how simulations can be applied in epidemiology. The population consists of two species, human hosts and vectors, which contract the disease from one another. Hosts are sedentary, while vectors (mosquitoes) diffuse in space. Examples of such diseases are malaria, dengue fever, and Pierce's disease in vineyards. The model exhibits a phase transition between an absorbing (infection free) phase and an active one as parameters such as infection rates and vector density are varied.

Exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media

We show that the stochastic field theory for directed percolation in the presence of an additional conservation law [the conserved directed-percolation (C-DP) class] can be mapped exactly to the continuum theory for the depinning of an elastic interface in short-range correlated quenched disorder. Along one line of the parameters commonly studied, this mapping leads to the simplest overdamped dynamics. Away from this line, an additional memory term arises in the interface dynamics; we argue that this does not change the universality class. Since C-DP is believed to describe the Manna class of self-organized criticality, this shows that Manna stochastic sandpiles and disordered elastic interfaces (i.e., the quenched Edwards-Wilkinson model) share the same universal large-scale behavior.

Hyperuniformity of critical absorbing states

The properties of the absorbing states of nonequilibrium models belonging to the conserved directed percolation universality class are studied. We find that, at the critical point, the absorbing states are hyperuniform, exhibiting anomalously small density fluctuations. The exponent characterizing the fluctuations is measured numerically, a scaling relation to other known exponents is suggested, and a new correlation length relating to this ordering is proposed. These results may have relevance to photonic band-gap materials.

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.

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.

Comment on "Dependence of asymptotic decay exponents on initial condition and the resulting scaling violation"

We present the mapping relation between a one-dimensional conserved lattice gas model and a two species reaction A+B→0. From the kinetics of A+B→0, we show that the anomalous critical decay of an order parameter in the conserved lattice gas model results from finite-size effects induced by the domain structure for random initial conditions where the scaling relation z=ν∥/ν⊥ is violated. A simple initial condition satisfying the scaling relation is also discussed.

Universality class of the conserved Manna model in one dimension

The nonequilibrium absorbing phase transition of the discrete conserved Manna model was studied via Monte Carlo simulations on a one-dimensional chain, using the natural initial states with a sequential update. The critical density of the particles was found to be smaller than the recently reported value, and the order-parameter exponent was considerably different from the directed percolation (DP) value. The influence of quenched disorder was also studied on a diluted strip of L_{x}×L_{y} lattice sites with L_{x}≫L_{y}, and the results were compared with those of the contact process (CP). It was found that the Manna model and the CP exhibited distinctly different behaviors; the CP exhibited nonuniversal power-law decreases of active-site densities in the Griffith phase, whereas the Manna model showed a standard critical behavior. These results consistently suggest that the Manna model belongs to a universality class that is different from the DP class.

Critical behavior of absorbing phase transitions for models in the Manna class with natural initial states

The critical behavior of absorbing phase transitions for two typical models in the Manna universality class, the conserved Manna model and the conserved lattice gas model, both on a square lattice, was investigated using the natural initial states. Various critical exponents were estimated using the static and dynamic simulations. The exponents characterizing dynamics of active particles differ considerably from the known exponents obtained using the random initial states, whereas those associated with the steady-state quantities remain the same. The critical exponents for both models were consistent with errors of less than 1% and satisfied the known scaling relations; thus, the known violation of scaling relations for models with a conserved field was resolved using the natural initial states. The results differed by 7%∼12% from the directed percolation values.

Connecting the random organization transition and jamming within a unifying model system

While the random organization transition describes the change from reversible to irreversible dynamics in a nonequilibrium system, the athermal jamming transition at zero shear rate occurs when particles can no longer avoid overlaps. Despite the obvious differences between these two transitions, we show that they both occur within the same model packing problem. In this unifying model system the particles are first randomly distributed and then displaced in each step if they overlap. For random displacements we obtain a random organization transition, while jamming occurs in the case of deterministic shifts. We also analyze the critical behavior of random organization. Our results show that random organization and jamming are opposite limits of random sphere packings, and we expect that various equilibrium and nonequilibrium transitions can be formulated as related intermediate packing problems.

Dependence of asymptotic decay exponents on initial condition and the resulting scaling violation

There are several examples which show that the critical exponents can be dependent on the initial condition of the system. In such situations, there are many systems where various issues related to the universal behavior, e.g., the existence of universality, the splitting of the universality class, scaling violations, whether the initial dependence should persist even after a sufficiently long time or is a transient effect, the reasons for such features, etc. are not yet quite clear. In this article, with the simple example of the conserved lattice gas model (CLG), we investigate such issues and clearly show that under certain situations the asymptotic decay exponents are, in fact, dependent on the initial condition of the system. We show that such an effect arises because of the existence of two competing time scales and identify the initial conditions which capture the universal features of the system.

Contact processes in crowded environments

Periodically sheared colloids at low densities demonstrate a dynamical phase transition from an inactive to active phase as the strain amplitude is increased. The inactive phase consists of no collisions (contacts) between particles in the steady state limit, while in the active phase collisions persist. To investigate this system at higher densities, we construct and study a conserved-particle-number contact process with three-body interactions, which are potentially more likely than two-body interactions at higher densities. For example, consider one active (diffusing) particle colliding with two inactive (nondiffusing) particles such that they become active and consider spontaneous inactivation. In mean field, this system exhibits a continuous dynamical phase transition. Simulations on square lattices also indicate a continuous transition with exponents similar to those measured for the conserved lattice gas (CLG) model. In contrast, the three-body interaction requiring two active particles to activate one inactive particle exhibits a discontinuous transition. Finally, inspired by kinetically constrained models of the glass transition, we investigate the "caging effect" at even higher particle densities to look for a second dynamical phase transition back to an inactive phase. Square lattice simulations suggest a continuous transition with a new set of exponents differing from both the CLG model and what is known as directed percolation, indicating a potentially new universality class for a contact process with a conserved particle number.

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.

Flooding transition in the topography of toppling surfaces of stochastic and rotational sandpile models

A continuous phase transition occurs in the topography of toppling surfaces of stochastic and rotational sandpile models when they are flooded with liquid, say water. The toppling surfaces are extracted from the sandpile avalanches that appear due to sudden burst of toppling activity in the steady state of these sandpile models. Though a wide distribution of critical flooding heights exists, a critical point is defined by merging the flooding thresholds of all the toppling surfaces. The criticality of the transition is characterized by power-law distribution of island area in the critical regime. A finite size scaling theory is developed and verified by calculating several new critical exponents. The flooding transition is found to be an interesting phase transition and does not belong to the percolation universality class. The universality class of this transition is found to depend on the degree of self-affinity of the toppling surfaces characterized by the Hurst exponent H and the fractal dimension D(f) of critical spanning islands. The toppling surfaces of different stochastic sandpile models are found to have a single Hurst exponent, whereas those of different rotational sandpile models have another Hurst exponent. As a consequence, the universality class of different sandpile models remains preserved within the same symmetry of the models.

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

We investigate the dynamical scaling behavior of the static diffusive epidemic process and a fixed-energy Manna sandpile model, undergoing nonequilibrium absorbing phase transitions in one dimension. These models belong to the so-called conserved directed-percolation or Manna universality class characterized by the conservation of the total particle number, activity coupled to a nondiffusive conserved field and infinitely many absorbing states. We measure the dynamical exponents of these models in one dimension by using the critical spreading simulation of a localized activity in absorbing configurations. In the spreading simulations, boundaries are never touched, so the results are free from the finite-size effects. In contrast to the scattered results for the different models from the previous finite-size scaling analyses, we obtain consistent estimates of the dynamical exponents for both models.

Determination of the critical exponents for absorbing phase transitions in the conserved lattice gas model in three dimensions

The critical exponents were measured for absorbing phase transitions in the conserved lattice gas (CLG) model on a simple cubic lattice. The correlation-length exponent calculated from the dynamic exponent obtained by finite-size scaling and from the mean spreading distance was consistently found to be ν(⊥)=0.631±0.02, which yields a positive specific heat exponent α=2-dν(⊥) on a pure system. The pure fixed point should, thus, be unstable if the Harris criterion established in equilibrium systems is applicable to the nonequilibrium absorbing phase transitions of the CLG model. However, this prediction is in contradiction with recent simulation results.