This review addresses recent developments in nonequilibrium statistical physics. Focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail. The survey gives a general introduction to various lattice models of directed percolation and studies their scaling properties, fieldtheoretic aspects, numerical techniques, as well as possible experimental realizations. In addition, several examples of absorbing-state transitions which do not belong to the directed percolation universality class will be discussed. As a closely related technique, we investigate the concept of damage spreading. It is shown that this technique is ambiguous to some extent, making it impossible to define chaotic and regular phases in stochastic nonequilibrium systems. Finally, we discuss various classes of depinning transitions in models for interface growth which are related to phase transitions into absorbing states.
We show that the two-dimensional voter model, usually considered to only be a marginal coarsening system, represents a broad class of models for which phase-ordering takes place without surface tension. We argue that voter-like growth is generically observed at order-disorder nonequilibrium transitions solely driven by interfacial noise between dynamically symmetric absorbing states.PACS numbers: 02.50. Ey, 05.70.Fh, 64.60.C, 64.60.Ht Coarsening phenomena occur in a large variety of situations in and out of physics, ranging from the demixion of alloys [1] to population dynamics [2]. At a fundamental level, phase-ordering challenges our capacity to deal with nonequilibrium systems and our understanding of the mechanisms determining different universality classes. In many cases, phase competition is driven by surface tension, leading to 'curvature-driven' growth. Coarsening patterns are then characterized by a single length scale L(t) ∼ t 1/z , where the exponent z only depends on general symmetry and conservation properties of the system [1]. For instance, z = 2 for the common case of a non-conserved scalar order parameter (NCOP), a large class including the Ising model. In this context, the twodimensional voter model (VM) [3], a caricatural process in which sites on a square lattice adopt the opinion of a randomly-chosen neighbor, stands out as an exception. Its coarsening process, which gives rise to patterns with clusters of all sizes between 1 and √ t ( Fig. 1) [4,5], is characterized by a slow, logarithmic decay of the density of interfaces ρ ∼ 1/ ln t (as opposed to the algebraic decay ρ ∼ 1/L(t) ∼ t −1/z of curvature-driven growth). The marginality of the VM is usually attributed to the exceptional character of its analytic properties [4][5][6].In this Letter, we show that, in fact, large classes of models exhibit the same type of domain growth as the simple VM, without being endowed with any of its peculiar symmetry and integrability properties. We argue that voter-like coarsening is best defined by the absence of surface tension and that it is generically observed at the transitions between disordered and fully-ordered phases in the absence of bulk fluctuations, when these nonequilibrium transitions are driven by interfacial noise only. Finally, we discuss the universality of the scaling properties associated with voter-like critical points.We first review the properties of the usual two-state VM, emphasizing those of importance for defining generalized models. A 'voter' (or spin) residing on site x of a hypercubic lattice can have two different opinions s x = ±1. In any space dimension d, an elementary move consists in randomly choosing one site and assigning to it the opinion of one of its randomly chosen nearest neighbors (n.n.). This ensures that the two homogeneous configurations (where all spins are either + or −) are absorbing states, and that the model is Z 2 -symmetric under global inversion (s x → −s x ). Recasting the dynamic rule in terms of n.n. pair updating, a pair of opposite spins +− ...
We study two models with n equivalent absorbing states that generalize the Domany-Kinzel cellular automaton and the contact process. Numerical investigations show that for n = 2 both models belong to the same universality class as branching annihilating walks with an even number of offspring. Unlike previously known models, these models have no explicit parity conservation law.
Small corrections to the uncertainty relations, with effects in the ultraviolet and/or infrared, have been discussed in the context of string theory and quantum gravity. Such corrections lead to small but finite minimal uncertainties in position and/or momentum measurements. It has been shown that these effects could indeed provide natural cutoffs in quantum field theory. The corresponding underlying quantum theoretical framework includes small 'noncommutative geometric' corrections to the canonical commutation relations. In order to study the full implications on the concept of locality it is crucial to find the physical states of then maximal localisation. These states and their properties have been calculated for the case with minimal uncertainties in positions only. Here we extend this treatment, though still in one dimension, to the general situation with minimal uncertainties both in positions and in momenta.
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