Phase transition classes in triplet and quadruplet reaction diffusion models

Phase transitions of reaction-diffusion systems with site occupation restriction and with particle creation that requires n=3,4 parents, whereas explicit diffusion of single particles (A) is present are investigated in low dimensions by mean-field approximation and simulations. The mean-field approximation of general nA -> (n+k)A, mA -> (m-l)A type of lattice models is solved and novel kind of critical behavior is pointed out. In d=2 dimensions the 3A -> 4A, 3A -> 2A model exhibits a continuous mean-field type of phase transition, that implies d_c<2 upper critical dimension. For this model in d=1 extensive simulations support a mean-field type of phase transition with logarithmic corrections unlike the Park et al.'s recent study (Phys. Rev E {\bf 66}, 025101 (2002)). On the other hand the 4A -> 5A, 4A -> 3A quadruplet model exhibits a mean-field type of phase transition with logarithmic corrections in d=2, while quadruplet models in 1d show robust, non-trivial transitions suggesting d_c=2. Furthermore I show that a parity conserving model 3A -> 5A, 2A->0 in d=1 has a continuous phase transition with novel kind of exponents. These results are in contradiction with the recently suggested implications of a phenomenological, multiplicative noise Langevin equation approach and with the simulations on suppressed bosonic systems by Kockelkoren and Chat\'e (cond-mat/0208497).Comment: 8 pages, 10 figures included, Updated with new data, figures, table, to be published in PR

Phase transition of the one-dimensional coagulation-production process

Recently an exact solution has been found (M.Henkel and H.Hinrichsen, cond-mat/0010062) for the 1d coagulation production process: 2A ->A, A0A->3A with equal diffusion and coagulation rates. This model evolves into the inactive phase independently of the production rate with $t^{-1/2}$ density decay law. Here I show that cluster mean-field approximations and Monte Carlo simulations predict a continuous phase transition for higher diffusion/coagulation rates as considered in cond-mat/0010062. Numerical evidence is given that the phase transition universality agrees with that of the annihilation-fission model with low diffusions.Comment: 4 pages, 4 figures include

Rare regions of the susceptible-infected-susceptible model on Barabási-Albert networks

I extend a previous work to susceptible-infected-susceptible (SIS) models on weighted Barabási-Albert scale-free networks. Numerical evidence is provided that phases with slow, power-law dynamics emerge as the consequence of quenched disorder and tree topologies studied previously with the contact process. I compare simulation results with spectral analysis of the networks and show that the quenched mean-field (QMF) approximation provides a reliable, relatively fast method to explore activity clustering. This suggests that QMF can be used for describing rare-region effects due to network inhomogeneities. Finite-size study of the QMF shows the expected disappearance of the epidemic threshold λc in the thermodynamic limit and an inverse participation ratio ∼0.25, meaning localization in case of disassortative weight scheme. Contrarily, for the multiplicative weights and the unweighted trees, this value vanishes in the thermodynamic limit, suggesting only weak rare-region effects in agreement with the dynamical simulations. Strong corrections to the mean-field behavior in case of disassortative weights explains the concave shape of the order parameter ρ(λ) at the transition point. Application of this method to other models may reveal interesting rare-region effects, Griffiths phases as the consequence of quenched topological heterogeneities

One-dimensional Nonequilibrium Kinetic Ising Models with local spin-symmetry breaking: N-component branching annihilation transition at zero branching rate

The effects of locally broken spin symmetry are investigated in one dimensional nonequilibrium kinetic Ising systems via computer simulations and cluster mean field calculations. Besides a line of directed percolation transitions, a line of transitions belonging to N-component, two-offspring branching annihilating random-walk class (N-BARW2) is revealed in the phase diagram at zero branching rate. In this way a spin model for N-BARW2 transitions is proposed for the first time.Comment: 6 pages, 5 figures included, 2 new tables added, to appear in PR

Phase transition of a two dimensional binary spreading model

We investigated the phase transition behavior of a binary spreading process in two dimensions for different particle diffusion strengths ($D$). We found that $N>2$ cluster mean-field approximations must be considered to get consistent singular behavior. The $N=3,4$ approximations result in a continuous phase transition belonging to a single universality class along the $D\in (0,1)$ phase transition line. Large scale simulations of the particle density confirmed mean-field scaling behavior with logarithmic corrections. This is interpreted as numerical evidence supporting that the upper critical dimension in this model is $d_c=2$.The pair density scales in a similar way but with an additional logarithmic factor to the order parameter. At the D=0 endpoint of the transition line we found DP criticality.Comment: 8 pages, 10 figure

Probability distribution of the order parameter in the directed percolation universality class

The probability distributions of the order parameter for two models in the directed percolation universality class were evaluated. Monte Carlo simulations have been performed for the one-dimensional generalized contact process and the Domany-Kinzel cellular automaton. In both cases, the density of active sites was chosen as the order parameter. The criticality of those models was obtained by solely using the corresponding probability distribution function. It has been shown that the present method, which has been successfully employed in treating equilibrium systems, is indeed also useful in the study of nonequilibrium phase transitions.Comment: 6 pages, 4 figure

Multicomponent binary spreading process

I investigate numerically the phase transitions of two-component generalizations of binary spreading processes in one dimension. In these models pair annihilation: AA->0, BB->0, explicit particle diffusion and binary pair production processes compete with each other. Several versions with spatially different productions have been explored and shown that for the cases: 2A->3A, 2B->3B and 2A->2AB, 2B->2BA a phase transition occurs at zero production rate ($\sigma=0$), that belongs to the class of N-component, asymmetric branching and annihilating random walks, characterized by the order parameter exponent $\beta=2$. In the model with particle production: AB->ABA, BA-> BAB a phase transition point can be located at $\sigma_c=0.3253$ that belongs to the class of the one-component binary spreading processes.Comment: 5 pages, 5 figure

Spectral analysis and slow spreading dynamics on complex networks

The susceptible-infected-susceptible (SIS) model is one of the simplest memoryless systems for describing information or epidemic spreading phenomena with competing creation and spontaneous annihilation reactions. The effect of quenched disorder on the dynamical behavior has recently been compared to quenched mean-field (QMF) approximations in scale-free networks. QMF can take into account topological heterogeneity and clustering effects of the activity in the steady state by spectral decomposition analysis of the adjacency matrix. Therefore, it can provide predictions on possible rare-region effects, thus on the occurrence of slow dynamics. I compare QMF results of SIS with simulations on various large dimensional graphs. In particular, I show that for Erdős-Rényi graphs this method predicts correctly the occurrence of rare-region effects. It also provides a good estimate for the epidemic threshold in case of percolating graphs. Griffiths Phases emerge if the graph is fragmented or if we apply a strong, exponentially suppressing weighting scheme on the edges. The latter model describes the connection time distributions in the face-to-face experiments. In case of a generalized Barabási-Albert type of network with aging connections, strong rare-region effects and numerical evidence for Griffiths Phase dynamics are shown. The dynamical simulation results agree well with the predictions of the spectral analysis applied for the weighted adjacency matrices

Absorbing Phase Transitions of Branching-Annihilating Random Walks

The phase transitions to absorbing states of the branching-annihilating reaction-diffusion processes mA --> (m+k)A, nA --> (n-l)A are studied systematically in one space dimension within a new family of models. Four universality classes of non-trivial critical behavior are found. This provides, in particular, the first evidence of universal scaling laws for pair and triplet processes.Comment: 4 pages, 4 figure

The universal behavior of one-dimensional, multi-species branching and annihilating random walks with exclusion

A directed percolation process with two symmetric particle species exhibiting exclusion in one dimension is investigated numerically. It is shown that if the species are coupled by branching ($A\to AB$, $B\to BA$) a continuous phase transition will appear at zero branching rate limit belonging to the same universality class as that of the dynamical two-offspring (2-BARW2) model. This class persists even if the branching is biased towards one of the species. If the two systems are not coupled by branching but hard-core interaction is allowed only the transition will occur at finite branching rate belonging to the usual 1+1 dimensional directed percolation class.Comment: 3 pages, 3 figures include