Asymptotic behavior of the order parameter in a stochastic sandpile

We derive the first four terms in a series for the order paramater (the stationary activity density rho) in the supercritical regime of a one-dimensional stochastic sandpile; in the two-dimensional case the first three terms are reported. We reorganize the pertubation theory for the model, recently derived using a path-integral formalism [R. Dickman e R. Vidigal, J. Phys. A 35, 7269 (2002)], to obtain an expansion for stationary properties. Since the process has a strictly conserved particle density p, the Fourier mode N^{-1} psi_{k=0} -> p, when the number of sites N -> infinity, and so is not a random variable. Isolating this mode, we obtain a new effective action leading to an expansion for rho in the parameter kappa = 1/(1+4p). This requires enumeration and numerical evaluation of more than 200 000 diagrams, for which task we develop a computational algorithm. Predictions derived from this series are in good accord with simulation results. We also discuss the nature of correlation functions and one-site reduced densities in the small-kappa (large-p) limit.Comment: 18 pages, 5 figure

Path-integral representation for a stochastic sandpile

We introduce an operator description for a stochastic sandpile model with a conserved particle density, and develop a path-integral representation for its evolution. The resulting (exact) expression for the effective action highlights certain interesting features of the model, for example, that it is nominally massless, and that the dynamics is via cooperative diffusion. Using the path-integral formalism, we construct a diagrammatic perturbation theory, yielding a series expansion for the activity density in powers of the time.Comment: 22 pages, 6 figure

Sandpiles with height restrictions

We study stochastic sandpile models with a height restriction in one and two dimensions. A site can topple if it has a height of two, as in Manna's model, but, in contrast to previously studied sandpiles, here the height (or number of particles per site), cannot exceed two. This yields a considerable simplification over the unrestricted case, in which the number of states per site is unbounded. Two toppling rules are considered: in one, the particles are redistributed independently, while the other involves some cooperativity. We study the fixed-energy system (no input or loss of particles) using cluster approximations and extensive simulations, and find that it exhibits a continuous phase transition to an absorbing state at a critical value zeta_c of the particle density. The critical exponents agree with those of the unrestricted Manna sandpile.Comment: 10 pages, 14 figure

Activated Random Walkers: Facts, Conjectures and Challenges

We study a particle system with hopping (random walk) dynamics on the integer lattice $\mathbb Z^d$. The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate $\lambda > 0$, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density $\zeta$ and the sleeping rate $\lambda$. We discuss the general case, and then, for the one-dimensional totally asymmetric case, study the phase transition between an active phase (for sufficiently large particle densities and/or small $\lambda$) and an absorbing one. We also present arguments regarding the asymptotic mean hopping velocity in the active phase, the rate of fixation in the absorbing phase, and survival of the infinite system at criticality. Using mean-field theory and Monte Carlo simulation, we locate the phase boundary. The phase transition appears to be continuous in both the symmetric and asymmetric versions of the process, but the critical behavior is very different. The former case is characterized by simple integer or rational values for critical exponents ($\beta = 1$, for example), and the phase diagram is in accord with the prediction of mean-field theory. We present evidence that the symmetric version belongs to the universality class of conserved stochastic sandpiles, also known as conserved directed percolation. Simulations also reveal an interesting transient phenomenon of damped oscillations in the activity density

Abrupt transition in a sandpile model

We present a fixed energy sandpile (FES) model which, by increasing the initial energy,undergoes, at the level of individual configurations, a discontinuous transition.The model is obtained by modifying the toppling procedure in the BTW rules: the energy transfer from a toppling site takes place only to neighbouring sites with less energy (negative gradient constraint) and with a time ordering (asynchronous). The model is minimal in the sense that removing either of the two above mentioned constraints (negative gradient or time ordering) the abrupt transition goes over to a continuous transition as in the usual BTW case. Therefore the proposed model offers an unique possibility to explore at the microscopic level the basic mechanisms underlying discontinuous transitions.Comment: 7 pages, 5 figure

Theory of the NO+CO surface reaction model

We derive a pair approximation (PA) for the NO+CO model with instantaneous reactions. For both the triangular and square lattices, the PA, derived here using a simpler approach, yields a phase diagram with an active state for CO-fractions y in the interval y_1 < y < y_2, with a continuous (discontinuous) phase transition to a poisoned state at y_1 (y_2). This is in qualitative agreement with simulation for the triangular lattice, where our theory gives a rather accurate prediction for y_2. To obtain the correct phase diagram for the square lattice, i.e., no active state, we reformulate the PA using sublattices. The (formerly) active regime is then replaced by a poisoned state with broken symmetry (unequal sub- lattice coverages), as observed recently by Kortluke et al. [Chem. Phys. Lett. 275, 85 (1997)]. In contrast with their approach, in which the active state persists, although reduced in extent, we report here the first qualitatively correct theory of the NO+CO model on the square lattice. Surface diffusion of nitrogen can lead to an active state in this case. In one dimension, the PA predicts that diffusion is required for the existence of an active state.Comment: 15 pages, 9 figure

Nonuniversal Critical Spreading in Two Dimensions

Continuous phase transitions are studied in a two dimensional nonequilibrium model with an infinite number of absorbing configurations. Spreading from a localized source is characterized by nonuniversal critical exponents, which vary continuously with the density phi in the surrounding region. The exponent delta changes by more than an order of magnitude, and eta changes sign. The location of the critical point also depends on phi, which has important implications for scaling. As expected on the basis of universality, the static critical behavior belongs to the directed percolation class.Comment: 21 pages, REVTeX, figures available upon reques

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 neighbour is occupied and the right neighbour 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 non-equilibrium phase transitions belong to a new universality class different from the generic active-absorbing-state phase transition, namely directed percolation.Comment: 5 pages, revtex4, 1 eps fi

Renormalization group of probabilistic cellular automata with one absorbing state

We apply a recently proposed dynamically driven renormalization group scheme to probabilistic cellular automata having one absorbing state. We have found just one unstable fixed point with one relevant direction. In the limit of small transition probability one of the cellular automata reduces to the contact process revealing that the cellular automata are in the same universality class as that process, as expected. Better numerical results are obtained as the approximations for the stationary distribution are improved.Comment: Errors in some formulas have been corrected. Additional material available at http://mestre.if.usp.br/~javie