Critical behavior in reaction-diffusion systems exhibiting absorbing phase transitions

Abstract

Phase transitions of reaction-diffusion systems with a site occupation restriction, particle creation requiring n > 2 parents, and in which explicit diffusion of single particles (A) is possible, are reviewed. Arguments based on mean-field approximation and simulations are given which support novel kind of nonequilibrium criticality. These are in contradiction with the implications of a suggested phenomenological, multiplicative noise Langevin equation approach and with some recent numerical analyses. Simulation results for one-and twodimensional binary spreading model, 2A → 4A, 4A → 2A, reveal a new type of mean-field criticality characterized by the critical exponents α = 1/3 and β = 1/2, as suggested in a recent preprint [cond-mat/0210615]. I IntroductionThe classification of universality classes of second order phase transitions is still one of the most important uncompleted tasks of statistical physics. Recently phase transitions of genuine nonequilibrium systems have been investigated intensively in reaction-diffusion (RD) type models exhibiting absorbing states [1][2][3]. There has been a hope that in such homogeneous systems symmetries and spatial dimensions are the most significant factors like in equilibrium ones, but gradually it turned out that there may be other relevant factors as well. The best known example is the parity conserving class (PC), which differs from the robust universality class of directed percolation (DP). The DP hypothesis stated by Janssen and Grassberger [4,5], according to which in one component systems exhibiting continuous phase transitions to a single absorbing state (without extra symmetry and inhomogeneity or disorder) short ranged interactions can generate DP class transitions only. However parity conservation itself proved to be an insufficient condition in many cases [6][7][8][9] and rather an underlying A → 3A, 2A → ∅ (BARW2) process [10] of particles and the Z 2 symmetry of absorbing states is necessary for this class [11]. On the other hand parity conservation in N-component branching and annihilating random walk (N-BARW) systems [10], or by triplet production models [12] was found to be responsible for novel classes again. In one dimensional, multicomponent reaction-diffusion systems site restriction turned out to be a relevant, new factor [13,14]. Global conservation laws by directed percolation and lattice gas models were shown to be irrelevant [15][16][17][18], while systems with multiple absorbing states [19] or with multi-components also exhibit DP class scaling behavior [20].Another important puzzle has been investigated intensively during the past three years that emerges at phase transitions of binary production reaction-diffusion systems [8,9,[20][21][22][23][24][25][26][27][28][29] (PCPD). In these systems particle production by pairs competes with pair annihilation and single particle diffusion. If production wins steady states with finite particle density appear in (site restricted), while in unrestricted (bosonic) models the density diverges. By lowerin...

A one-parameter family of stationary solutions in the Susceptible-Infected-Susceptible epidemic model

“…When the infection rate exceeds the network's epidemic threshold there is a phase transition and a strictly positive fraction of the population is infected in the long term. Similar critical behavior phenomena are observed in many other physical systems including pest control using impulses of biological pathogens [9], percolation [10], Ising-Potts models [11,12], synchronization [13], reaction-diffusion processes [14], sandpiles [15] and avalanches [16], making the SIS model an important paradigm for these more complicated systems. To better understand how network topology effects the long term distribution of infected and susceptible populations in the SIS model, we use a one parameter family of networks all having the same average connectivity.…”

Phase transitions of the binary production2A→3A,4A→∅

“…The upper critical dimension for such systems is debated [6,21,23,24] but should be quite low (d c = 1 − 2) allowing a few anomalous critical transitions only. For example d c < 1 was confirmed by simulations in case of the asymmetric, binary production 2A → 4A, 4A → 2A model [25]. It was also pointed out there that N > 1 cluster mean-field approximation, that takes into account the diffusion of particles would provide a more adequate description of such models.…”

“…The fluctuations in the absorbing state are so small, that systems cannot escape from it, hence such phase transitions may emerge in one dimension already. Several systems with binary, triplet or quadruplet, particle reactions have been investigated numerically and unclassified type of critical phase transitions were found [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Solid field theoretical treatment exists for bosonic, binary production systems only [26], but this is not applicable for the active and critical states of site restricted models, since it cannot describe a steady state with finite density.…”

Classification of phase transitions in reaction-diffusion models

“…The other mean-field exponents can be also deduced in a straightforward way 70 . There are two important exceptions where the µ = 0 transitions in the reaction set Eq.…”