We study the long-time behavior of stochastic models with an absorbing state, conditioned on survival. For a large class of processes, in which saturation prevents unlimited growth, statistical properties of the surviving sample attain time-independent limiting values. We may then define a quasi-stationary probability distribution as one in which the ratios p n (t)/p m (t) (for any pair of nonabsorbing states n and m), are time-independent. This is not a true stationary distribution, since the overall normalization decays as probability flows irreversibly to the absorbing state. We construct quasi-stationary solutions for the contact process on a complete graph, the Malthus-Verhulst process, Schlögl's second model, and the voter model on a complete graph. We also construct the master equation and quasi-stationary state in a twosite approximation for the contact process, and for a pair of competing Malthus-Verhulst processes.

No abstract

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.

We derive the first four terms in a series for the order paramater (the stationary activity density ρ) in the supercritical regime of a onedimensional 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 ψ k=0 → p, when N → ∞, and so is not a random variable. Isolating this mode, we obtain a new effective action leading to an expansion for ρ in the parameter κ ≡ 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-κ (large-p) limit.

We study diffusion of particles in large-scale simulations of one-dimensional stochastic sandpiles, in both the restricted and unrestricted versions. The results indicate that the diffusion constant scales in the same manner as the activity density, so that it represents an alternative definition of an order parameter. The critical behavior of the unrestricted sandpile is very similar to that of its restricted counterpart, including the fact that a data collapse of the order parameter as a function of the particle density is only possible over a very narrow interval near the critical point. We also develop a series expansion, in inverse powers of the density. for the collective diffusion coefficient in a variant of the stochastic sandpile in which the toppling rate at a site with n particles is n(n − 1), and compare the theoretical prediction with simulation results. §

We study driven particle systems with excluded volume interactions on a two-lane ladder with periodic boundaries, using Monte Carlo simulation, cluster mean-field theory, and numerical solution of the master equation. Particles in one lane are subject to a drive that forbids motion along one direction, while in the other lane the motion is unbiased; particles may jump between lanes.Despite the symmetry of the rates for transitions between lanes, the associated particle densities are unequal: at low densities there is an excess of particles in the undriven lane, while at higher densities the tendency is reversed. Similar results are found for an off-lattice model. We quantify the reduction in the stationary entropy caused by the drive. The stationary two-point correlation functions are found to decay algebraically, both on-and off-lattice. In the latter case the exponent governing the decay varies continuously with the density.

Using operator algebra, we extend the series for the activity density in a one-dimensional stochastic sandpile with fixed particle density p, the first terms of which were obtained via perturbation theory [R. Dickman and R. Vidigal, J. Phys. A 35, 7269 (2002)]. The expansion is in powers of the time; the coefficients are polynomials in p. We devise an algorithm for evaluating expectations of operator products and extend the series to O(t 16 ). Constructing Padé approximants to a suitably transformed series, we obtain predictions for the activity that compare well against simulations, in the supercritical regime.