Some recent developments in models with absorbing states

Abstract

We describe some of the recent results obtained for models with absorbing states. First, we present the nonequilibrium absorbing-state Potts model and discuss some of the factors that might affect the critical behaviour of such models. In particular we show that in two dimensions the further neighbour interactions might split the voter critical point into two critical points. We also describe some of the results obtained in the context of synchronization of chaotic dynamical systems. Moreover, we discuss the relation of the synchronization transition with some interfacial models. I IntroductionNonequilibrium statistical mechanics is nowadays a very active research field [1] and there are several reasons for that. First, the most interesting phenomena in Nature take place out of equilibrium. The best example is provided by living matter. More generally, systems which are open (traversed by fluxes of energy, entropy or matter) may reach stationary states which cannot be described by equilibrium statistical physics. Second, the properties of systems in equilibrium are by now rather well understood as equilibrium statistical physics is a well established theory. The puzzling problem of universal behavior observed in the vicinity of second order phase transitions is beautifully explained by the renormalization group approach [2]. It is thus natural to try to extend our understanding of equilibrium systems to nonequilibrium ones.Indeed, the situation is not so clear for nonequilibrium statistical mechanics for which no general theory has been developed yet. This is particularly true for the case of nonequilibrium phase transitions where, according to the values of some control parameters, the system can change, continuously or not, from one stationary state to another one [1].At the microscopic level, models for nonequilibrium phase transitions are usually defined in terms of a master equation [3]. Most of the physics is contained in the transition rates. One of the key differences between equilibrium and nonequilibrium systems is related with the detailedbalance condition which is generally not obeyed in nonequilibrium. As a result, it is often impossible to find an analytical solution to the master equation, even for the stationary state. This is why a lot of results in this field are obtained by numerical simulations. At a coarse-grained level, the physics is often described in terms of a generalized Langevin equation. Unfortunately, there is no general method which allows us to perform, in a controlled way, the coarse-graining process and usually the determination of the form taken by the Langevin equation is based only on general symmetry arguments [4][5][6][7]. This procedure is particularly ambiguous for systems with noise [8]. When the noise is multiplicative different interpretations of the stochastic process described by the master equation are possible, leading to further confusion [9,10].There are many examples of nonequilibrium phase transitions. One of the simplest examples is provided by an Ising like m...

Dynamic critical properties of non-equilibrium Potts models with absorbing states

“…In Figure 2 we show that the LD model displays the same scaling behavior as the KBB model, with the same two exponents b = 0.120(5) and λ/z = 1.00 (2). This simple aging scaling is encountered for both the two-time autocorrelation (8) and the two-time autoresponse function (9). We note that the autoresponse in the KBB model also displays the same scaling behavior (not shown).…”

“…Starting from a disordered initial state, we follow the standard protocol for calculating this response by applying for the first s time steps a spatially random field with amplitude h. The random field at site i is given by the expression h i = h r i where the quenched random variable r i indicates the direction of the field and takes on one of the values 0, 1, · · · , q −1 [19]. After s time steps the field is removed and the relaxation of the system to the steady state is monitored for times t > s with the help of the two-time autoresponse function (9). In order to remain in the linear response regime, we choose the small value h = 0.05 for the field amplitude.…”

“…. After s time steps the field is removed and the relaxation of the system to the steady state is monitored for times t > s with the help of the two-time autoresponse function (9). In order to remain in the linear response regime, we choose the small value h = 0.05 for the field amplitude.…”

Hidden percolation transition in kinetic replication process

“…Along with the trivial absorbing state, we find two active phases. The examples of multiple active phases and transitions between them are actively discussed in the current literature in case of models in spatial dimensions d ⩾ 2 [6][7][8][9][10] and in 1d models with n-valued (n > 2) variables [11][12][13][14]. However, the short-range models with Ising-like (two-valued) variables which have a single absorbing state and no conservation laws are known to exhibit only one transition into a single active phase belonging to the directed percolation (DP) universality class [1].…”

Estudo de transições de fase em sistemas com simetria "up-down" e estados absorventes

“…O objeto de estudo deste trabalhoé um modelo contextualizado na dinâmica estocástica de não-equilíbrio [2] A motivação do nosso trabalho foi verificar e entender a natureza da transição descontínua representada pela transição do modelo do Votante através de um modelo estocástico na rede idealizado pelo nosso grupo de pesquisa. Diversos estudos descreveram e caracterizaram as propriedades da transição do modelo do Votante [10] [11] [12]. Uma característica a ser confirmadaé que essa transição seja uma justaposição de duas outras transições:…”