Universality classes in nonequilibrium lattice systems

Abstract

This work is designed to overview our present knowledge about universality classes occurring in nonequilibrium systems defined on regular lattices. In the first section I summarize the most important critical exponents, relations and the field theoretical formalism used in the text. In the second section I briefly address the question of scaling behavior at first order phase transitions. In section three I review dynamical extensions of basic static classes, show the effect of mixing dynamics and the percolation behavior. The main body of this work is given in section four where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. In section five I continue overviewing such nonequilibrium classes but in coupled, multi-component systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion type of systems. However by mapping they can be related to universal behavior of interface growth models, which I overview in section six. Finally in section seven I summarize families of absorbing state system classes, mean-field classes and give an outlook for further directions of research.

Renormalization Group: Applications in Statistical Physics

“…With the exponent identities (13) and (26), this is equivalent to the scaling hypothesis (7). In a similar manner, one readily derives the correlation function scaling law (25), employing the…”

Brain dynamics across levels of organization

“…These indices are independent of physical nature of the system's components, and are solely determined by the properties of the components' interactions (Binney, 1992;Yeomans, 2002). It is empirically established that nonlinear dynamic systems, including those operating far from equilibrium (Odor, 2004), can often be categorized by these critical indices into distinct classes. This means that having ascertained one or the other critical property for a system under study, it is then possible to predict all other critical properties of that system merely on the basis of its class membership.…”

“…The twin concepts of scaling and universality play an important role in description of dynamical systems for elimination of degrees of freedom and scale transformations at points near critical transition (Kadanoff et al, 1989;Kadanoff, 1990). The significance of this lies in the possibility of identifying universality classes (Odor, 2004) which will be pursued in Section 3. Although still lacking a comprehensive theory of SOC, it is now an established part of Dynamical Systems Theory by characterizing (specifically in some instances and in others, in principle) the critical state as the system's attractor, and its fractal structure (Blanchard et al, 2000 ).…”

Reaction–diffusion processes and non-perturbative renormalization group

“…Reaction-diffusion processes naturally lend themselves to Monte-Carlo simulations, which have indeed largely contributed to our understanding of these processes (see [6,9] for reviews). On the other hand, the simplest analytical approach is to device a rate equation for the time-dependent average density n(t), assuming the various reactions to occur proportionally to the concentration of reactants.…”