First-order nonequilibrium phase transition in a spatially extended system

Müller, Reinhard; Lippert, Karen; Kühnel, A.; Behn, Ulrich

doi:10.1103/physreve.56.2658

Cited by 84 publications

(63 citation statements)

References 20 publications

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“…First, noise can modify (shift) the threshold of pattern formation [159]. Second, owing to noise, the precursors of patterns can be seen below the pattern formation threshold [160][161][162]. While a noiseless pattern-forming system below the pattern formation thresh- old shows no pattern at all, since all perturbations decay, one observes in the presence of noise a particular form of spatially filtered noise, which in the field of nonlinear optics has been called "quantum patterns" when the noise is of quantum origin [163][164][165].…”

Section: Extended Patternsmentioning

confidence: 99%

Transverse Patterns in Nonlinear Optical Resonators

Staliūnas

Sánchez‐Morcillo

2003

Springer Tracts in Modern Physics

176

156

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This book is devoted to the formation and dynamics of localized structures (vortices, solitons) and of extended patterns (stripes, hexagons, tilted waves) in nonlinear optical resonators such as lasers, optical parametric oscillators, and photorefractive oscillators. Theoretical analysis is performed by deriving order parameter equations, and also through numerical integration of microscopic models of the systems under investigation. Experimental observations, and possible technological implementations of transverse optical patterns are also discussed. A comparison with patterns found in other nonlinear systems, i.e. chemical, biological, and hydrodynamical systems, is given.

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Section: Extended Patternsmentioning

confidence: 99%

Transverse Patterns in Nonlinear Optical Resonators

Staliūnas

Sánchez‐Morcillo

2003

Springer Tracts in Modern Physics

176

156

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“…which has appeared in the literature in the context of non-equilibrium wetting and depinning transitions [25,26]. This is a generalization of Eq.…”

mentioning

confidence: 99%

“…Therefore, for b > 0, the situation is completely analogous to ST in the MN class where Λ changes sign at the transition. On the other hand, for b < 0 [27] the transition can be easily shown not to occur at ḣ = 0 [25]. Instead, there is a finite interval of values of a, [a * , a c (b)], in which phase coexistence is found, ergodicity is broken, and the final state depends upon initial conditions [25].…”

mentioning

confidence: 99%

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Stochastic Theory of Synchronization Transitions in Extended Systems

Muñoz

Pastor-Satorras

2003

Phys. Rev. Lett.

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We propose a general Langevin equation describing the universal properties of synchronization transitions in extended systems. By means of theoretical arguments and numerical simulations we show that the proposed equation exhibits, depending on parameter values, either: i) a continuous transition in the bounded Kardar-Parisi-Zhang universality class, with a zero largest Lyapunov exponent at the critical point; ii) a continuous transition in the directed percolation class, with a negative Lyapunov exponent, or iii) a discontinuous transition (that is argued to be possibly just a transient effect). Cases ii) and iii) exhibit coexistence of synchronized and unsynchronized phases in a broad (fuzzy) region. This phenomenology reproduces almost all the reported features of synchronization transitions of coupled map lattices and other models, providing a unified theoretical framework for the analysis of synchronization transitions in extended systems. [7]. In particular, coupled map lattices (CMLs) [8], initially introduced as simple models of spatio-temporal chaos [11], have received a great deal of attention, as models of synchronization in spatially extended systems (another possibility is to analyze discrete cellular automata [9,10]). In this context, it was observed that distant patches of a given CML can oscillate in phase [11], a phenomenon related to the Kardar-Parisi-Zhang (KPZ) equation [12] describing the roughening of growing interfaces [13]. Afterwards, it was also realized that when two different replicas of a same CML are locally coupled, they become synchronized if the coupling strength is large enough [6,9,14]. Also globally coupled CML can achieve mutual synchronization [15] with interesting implications in neuro-science [16]. Last but not least, different replicas of a CML can be synchronized if they are coupled to a sufficiently large common external random noise, even if they are not directly coupled to each other [17,18,19].In all these examples, there is a transition from a chaotic or unsynchronized phase in which perturbations grow, and two replicas evolve independently, to a synchronized phase in which memory of the initial difference is asymptotically lost and replicas synchronize. This synchronization transition (ST) resembles very much other non-equilibrium critical phenomena, particularly transitions into absorbing states [20], and the determination of its universal properties has become the subject of many recent studies [9,19,22,23] as well as the main motivation of this Letter. Within this context, a major contribution was made by Pikovsky and Kurths (PK) who proposed a stochastic model for the dynamics of perturbations in locally coupled CML, in which, under very general conditions, the difference-between-two-replicas field can be described by the so-called multiplicative noise (MN) Langevin equation [22]:where φ(x, t) is the difference field (or "synchronization error"), a, b, σ 2 , and D > 0 are parameters, and η a delta correlated Gaussian white noise with zero average. It is worth ...

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“…Simulations were made on a square two-dimensional lattice of 100ϫ 100 cells. From Müller et al, 1997.…”

Section: ͑105͒mentioning

confidence: 99%

Spatiotemporal order out of noise

2007

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Natural systems are undeniably subject to random fluctuations, arising from either environmental variability or thermal effects. The consideration of those fluctuations supposes to deal with noisy quantities whose variance might at times be a sizable fraction of their mean levels. It is known that, under these conditions, noisy fluctuations can interact with the system's nonlinearities to render counterintuitive behavior, in which an increase in the noise level produces a more regular behavior. In systems with spatial degrees of freedom, this regularity takes the form of spatiotemporal order. An overview is presented of the mechanisms through which noise induces, enhances, and sustains ordered behavior in passive and active nonlinear media, and different spatiotemporal phenomena are described resulting from these effects. The general theoretical framework used in the analysis of these effects is reviewed, encompassing the theory of stochastic partial differential equations and coupled sets of ordinary stochastic differential equations. Experimental observations of self-organized behavior arising out of noise are also described, and details on the numerical algorithms needed in the computer simulation of these problems are given.

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First-order nonequilibrium phase transition in a spatially extended system

Cited by 84 publications

References 20 publications

Transverse Patterns in Nonlinear Optical Resonators

Transverse Patterns in Nonlinear Optical Resonators

Stochastic Theory of Synchronization Transitions in Extended Systems

Spatiotemporal order out of noise

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