Absorbing-state phase transitions: exact solutions of small systems

I derive precise results for absorbing-state phase transitions using exact (numerically determined) quasistationary probability distributions for small systems. Analysis of the contact process on rings of 23 or fewer sites yields critical properties (control parameter, order-parameter ratios, and critical exponents z and beta/nu_perp) with an accuracy of better than 0.1%; for the exponent nu_perp the accuracy is about 0.5%. Good results are also obtained for the pair contact process

Diffusive epidemic process: theory and simulation

We study the continuous absorbing-state phase transition in the one-dimensional diffusive epidemic process via mean-field theory and Monte Carlo simulation. In this model, particles of two species (A and B) hop on a lattice and undergo reactions B -> A and A + B -> 2B; the total particle number is conserved. We formulate the model as a continuous-time Markov process described by a master equation. A phase transition between the (absorbing) B-free state and an active state is observed as the parameters (reaction and diffusion rates, and total particle density) are varied. Mean-field theory reveals a surprising, nonmonotonic dependence of the critical recovery rate on the diffusion rate of B particles. A computational realization of the process that is faithful to the transition rates defining the model is devised, allowing for direct comparison with theory. Using the quasi-stationary simulation method we determine the order parameter and the survival time in systems of up to 4000 sites. Due to strong finite-size effects, the results converge only for large system sizes. We find no evidence for a discontinuous transition. Our results are consistent with the existence of three distinct universality classes, depending on whether A particles diffusive more rapidly, less rapidly, or at the same rate as B particles.Comment: 19 pages, 5 figure

A field theoretic approach to master equations and a variational method beyond the Poisson ansatz

We develop a variational scheme in a field theoretic approach to a stochastic process. While various stochastic processes can be expressed using master equations, in general it is difficult to solve the master equations exactly, and it is also hard to solve the master equations numerically because of the curse of dimensionality. The field theoretic approach has been used in order to study such complicated master equations, and the variational scheme achieves tremendous reduction in the dimensionality of master equations. For the variational method, only the Poisson ansatz has been used, in which one restricts the variational function to a Poisson distribution. Hence, one has dealt with only restricted fluctuation effects. We develop the variational method further, which enables us to treat an arbitrary variational function. It is shown that the variational scheme developed gives a quantitatively good approximation for master equations which describe a stochastic gene regulatory network.Comment: 13 pages, 2 figure

Series expansion for a stochastic sandpile

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. A35, 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 Pade approximants to a suitably transformed series, we obtain predictions for the activity that compare well against simulations, in the supercritical regime.Comment: Extended series and improved analysi

A globally accurate theory for a class of binary mixture models

Using the self-consistent Ornstein-Zernike approximation (SCOZA) results for the 3D Ising model, we obtain phase diagrams for binary mixtures described by decorated models. We obtain the plait point, binodals, and closed-loop coexistence curves for the models proposed by Widom, Clark, Neece, and Wheeler. The results are in good agreement with series expansions and experiments.Comment: 16 pages, 10 figure

Contact process on a Voronoi triangulation

We study the continuous absorbing-state phase transition in the contact process on the Voronoi-Delaunay lattice. The Voronoi construction is a natural way to introduce quenched coordination disorder in lattice models. We simulate the disordered system using the quasistationary simulation method and determine its critical exponents and moment ratios. Our results suggest that the critical behavior of the disordered system is unchanged with respect to that on a regular lattice, i.e., that of directed percolation

Wang-Landau sampling in three-dimensional polymers

Monte Carlo simulations using Wang-Landau sampling are performed to study three-dimensional chains of homopolymers on a lattice. We confirm the accuracy of the method by calculating the thermodynamic properties of this system. Our results are in good agreement with those obtained using Metropolis importance sampling. This algorithm enables one to accurately simulate the usually hardly accessible low-temperature regions since it determines the density of states in a single simulation.Comment: 5 pages, 9 figures arch-ive/Brazilian Journal of Physic

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

N-Site approximations and CAM analysis for a stochastic sandpile

I develop n-site cluster approximations for a stochastic sandpile in one dimension. A height restriction is imposed to limit the number of states: each site can harbor at most two particles (height z_i \leq 2). (This yields a considerable simplification over the unrestricted case, in which the number of states per site is unbounded.) On the basis of results for n \leq 11 sites, I estimate the critical particle density as zeta_c = 0.930(1), in good agreement with simulations. A coherent anomaly analysis yields estimates for the order parameter exponent [beta = 0.41(1)] and the relaxation time exponent (nu_|| \simeq 2.5).Comment: 12 pages, 7 figure

On the absorbing-state phase transition in the one-dimensional triplet creation model

We study the lattice reaction diffusion model 3A -> 4A, A -> 0 (``triplet creation") using numerical simulations and n-site approximations. The simulation results provide evidence of a discontinuous phase transition at high diffusion rates. In this regime the order parameter appears to be a discontinuous function of the creation rate; no evidence of a stable interface between active and absorbing phases is found. Based on an effective mapping to a modified compact directed percolation process, shall nevertheless argue that the transition is continuous, despite the seemingly discontinuous phase transition suggested by studies of finite systems.Comment: 23 pages, 11 figure