Contact process with long-range interactions: a study in the ensemble of constant particle number

Phase transitions of the random-bond Potts chain with long-range interactions

We study phase transitions of the ferromagnetic q-state Potts chain with random nearest-neighbor couplings having a variance Δ^{2} and with homogeneous long-range interactions, which decay with distance as a power r^{-(1+σ)}, σ>0. In the large-q limit the free-energy of random samples of length L≤2048 is calculated exactly by a combinatorial optimization algorithm. The phase transition stays first order for σ<σ_{c}(Δ)≤0.5, while the correlation length becomes divergent at the transition point for σ_{c}(Δ)<σ<1. In the latter regime the average magnetization is continuous for small enough Δ, but for larger Δ-according to the numerical results-it becomes discontinuous at the transition point, thus the phase transition is expected of mixed order.

Mean-field theory for the long-range contact process with diffusion

The effect of diffusion in the one-dimensional long-range contact process is investigated by mean-field calculations. Recent works have shown that diffusion decreases the effectiveness of long-range interactions, affecting the character of the phase transition: for higher values of the diffusion coefficient, stronger long-range interactions are required to enable phase coexistence and first-order behavior. Here we apply a generalized mean-field approximation for the master equation of the model that considers states of an aggregate of L lattice sites. The phase diagram of the model for values of L up to 10 is obtained, and for some values of the diffusion rate extrapolations to infinite-sized systems are given. For low-diffusive systems, approximations with L≥3 are able to reveal the suppression of the phase coexistence induced by diffusion, however, in the high-diffusion regime, larger values of L are necessary to correctly account for the higher range of correlations. We present a very efficient method to study the mean-field equations and determine the nature of the phase transitions that may be of general utility.

Long-range epidemic spreading in a random environment

Modeling long-range epidemic spreading in a random environment, we consider a quenched, disordered, d-dimensional contact process with infection rates decaying with distance as 1/rd+σ. We study the dynamical behavior of the model at and below the epidemic threshold by a variant of the strong-disorder renormalization-group method and by Monte Carlo simulations in one and two spatial dimensions. Starting from a single infected site, the average survival probability is found to decay as P(t)∼t-d/z up to multiplicative logarithmic corrections. Below the epidemic threshold, a Griffiths phase emerges, where the dynamical exponent z varies continuously with the control parameter and tends to zc=d+σ as the threshold is approached. At the threshold, the spatial extension of the infected cluster (in surviving trials) is found to grow as R(t)∼t1/zc with a multiplicative logarithmic correction and the average number of infected sites in surviving trials is found to increase as Ns(t)∼(lnt)χ with χ=2 in one dimension.

Effect of diffusion in one-dimensional discontinuous absorbing phase transitions

It is known that diffusion provokes substantial changes in continuous absorbing phase transitions. Conversely, its effect on discontinuous transitions is much less understood. In order to shed light in this direction, we study the inclusion of diffusion in the simplest one-dimensional model with a discontinuous absorbing phase transition, namely, the long-range contact process (σ-CP). Particles interact as in the usual CP, but the transition rate depends on the length ℓ of inactive sites according to 1+aℓ(-σ), where a and σ are control parameters. The inclusion of diffusion in this model has been investigated by numerical simulations and mean-field calculations. Results show that there exists three distinct regimes. For sufficiently low and large σ's the transition is, respectively, always discontinuous or continuous, independently of the strength of the diffusion. On the other hand, in an intermediate range of σ's, the diffusion causes a suppression of the phase coexistence leading to a continuous transition belonging to the directed percolation universality class.

Quenched disorder forbids discontinuous transitions in nonequilibrium low-dimensional systems

Quenched disorder affects significantly the behavior of phase transitions. The Imry-Ma-Aizenman-Wehr-Berker argument prohibits first-order or discontinuous transitions and their concomitant phase coexistence in low-dimensional equilibrium systems in the presence of random fields. Instead, discontinuous transitions become rounded or even continuous once disorder is introduced. Here we show that phase coexistence and first-order phase transitions are also precluded in nonequilibrium low-dimensional systems with quenched disorder: discontinuous transitions in two-dimensional systems with absorbing states become continuous in the presence of quenched disorder. We also study the universal features of this disorder-induced criticality and find them to be compatible with the universality class of the directed percolation with quenched disorder. Thus, we conclude that first-order transitions do not exist in low-dimensional disordered systems, not even in genuinely nonequilibrium systems with absorbing states.

Scaling functions for systems with finite range of interaction

We present a numerical determination of the scaling functions of the magnetization, the susceptibility, and the Binder's cumulant for two nonequilibrium model systems with varying range of interactions. We consider Monte Carlo simulations of the block voter model (BVM) on square lattices and of the majority-vote model (MVM) on random graphs. In both cases, the satisfactory data collapse obtained for several system sizes and interaction ranges supports the hypothesis that these functions are universal. Our analysis yields an accurate estimation of the long-range exponents, which govern the decay of the critical amplitudes with the range of interaction, and is consistent with the assumption that the static exponents are Ising-like for the BVM and classical for the MVM.

Robustness of first-order phase transitions in one-dimensional long-range contact processes

It has been proposed [Ginelli et al., Phys. Rev. E 71, 026121 (2005)] that, unlike the short-range contact process, the long-range counterpart may lead to the existence of a discontinuous phase transition in one dimension. Aiming to explore such a link, here we investigate thoroughly a family of long-range contact processes. They are introduced through the transition rate 1+aℓ(-σ), where ℓ is the length of inactive islands surrounding particles. In the former approach we reconsider the original model (called the σ-contact process) by considering distinct mechanisms of weakening the long-range interaction toward the short-range limit. In addition, we study the effect of different rules, including creation and annihilation by clusters of particles and distinct versions with infinitely many absorbing states. Our results show that for all examples presenting a single absorbing state, a discontinuous transition is possible for small σ. On the other hand, the presence of infinite absorbing states leads to a distinct scenario depending on the interactions at the perimeter of inactive sites.

Explaining the high number of infected people by dengue in Rio de Janeiro in 2008 using a susceptible-infective-recovered model

An epidemiological model for dengue propagation using cellular automata is constructed. Dependence on temperature and rainfall index are taken into account. Numerical results fit pretty well with the registered cases of dengue for the city of Rio de Janeiro for the period from 2006 to 2008. In particular, our approach explains very well an abnormally high number of cases registered in 2008. A phase transition from endemic to epidemic regimes is discussed.