Nonequilibrium critical behavior in unidirectionally coupled stochastic processes

Abstract

Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension d(c)=4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes A-->A+A, A+A-->A, and A-->0. We study a hierarchy of such DP processes for particle species A,B,..., unidirectionally coupled via the reactions A-->B, ...(with rates mu(AB),...). When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents beta(i) which are markedly reduced at each hierarchy level i> or =2. This scenario can be understood on the basis of the mean-field rate equations, which yield beta(i)=1/2(i-1) at the multicritical point. Using field-theoretic renormalization-group techniques in d=4-epsilon dimensions, we identify a new crossover exponent phi, and compute phi=1+O(epsilon(2)) in the multicritical regime (for small mu(AB)) of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, beta(2)=1/2-epsilon/8+O(epsilon(2)). Outside the multicritical region, we discuss the crossover to ordinary DP behavior, with the density exponent beta(1)=1-epsilon/6+O(epsilon(2)). Monte Carlo simulations are then employed to confirm the crossover scenario, and to determine the values for the new scaling exponents in dimensions d< or =3, including the critical initial slip exponent. Our theory is connected to specific classes of growth processes and to certain cellular automata, and the above ideas are also applied to unidirectionally coupled pair annihilation processes. We also discuss some technical as well as conceptual problems of the loop expansion, and suggest some possible interpretations of these difficulties.

Asymmetrically Coupled Directed Percolation Systems

“…The contact process was originally suggested as a model for epidemic spreading without immunization [4], where the particle represents an infected individual, who may recover spontaneously or infect its healthy neighbors. The vacuum state with all sites empty is the unique absorbing state, which is one of the key ingredients in the DP class.Recently, coupled DP systems have been studied extensively [5,6,7,8]. In particular, the unidirectionally coupled DP (UCDP) process introduced by Täuber et al [7] was found to display a series of new multicritical phenomena.…”

“…It turned out that their critical behaviors are described by the critical exponents distinct from those of the DP class [7,8]. This novel critical behavior is destroyed and the DP scaling is recovered as soon as we turn on the excitatory coupling in the other (downward) direction [6,8].It is noteworthy that the UCDP process was proposed for describing an interesting roughening transition in an onedimensional growth process [9], where the monomer-type restricted-solid-on-solid (RSOS) model is considered with the constraint that evaporation is allowed only at the edges (not terraces). It can be easily shown that the growth model is mapped to the UCDP process, except that the RSOS condition generates an extra inhibitory coupling in the downward direction, i.e.…”

Mirror symmetry breaking as a problem in dynamic critical phenomena

“…(17,18), reveals that the parameter g plays a special role [21]. Indeed, g < 1 leads to chiral amplification of an initial enantiomeric excess, whereas g > 1 leads to a racemic final state.…”

“…From this, an effective action S ef f can be straightforwardly derived which contains all the critical dynamics implied by the reaction scheme to be studied. The mapping of related kinetic schemes to continuum statistical path integrals is spelled out in [17,18] where the main steps can be found. Applying this procedure to the scheme in Eqs.…”

Mean-Field Theory of Meta-learning

“…This dynamical view of the consensus between various machine learning algorithms is especially useful for artificial intelligence, or robotic applications, where adaptive behavior given by the integration of results from ensemble of ML methods. My model of learning is based on nonlocal cellular automata (CA) approach known from physics [30][31][32][33][34], with a wide range of applications [35][36][37][38][39][40][41][42][43]. The first statistical mechanics of opinion formation in groups of individuals was proposed by Lewenstein et al [44] on the class of models that were based on probabilistic cellular automata and social impact theory introduced by Latane [45,46].…”