On generalized scaling laws with continuously varying exponents

Sittler, Lionel; Hinrichsen, Haye

doi:10.1088/0305-4470/35/49/303

Cited by 13 publications

(18 citation statements)

References 11 publications

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“…22 According to Ref. 27, the physical origin of logarithmic scaling is associated with multifractality and local scaling invariance. To the best of our knowledge, the results presented above are the first numerical confirmation of the scaling relation (4), in which the scaling variable is given by the ratio of the logarithm of the system width L to the logarithm of the correlation length ξ, instead of the ratio L/ξ as applied usually.…”

Section: Discussionmentioning

confidence: 99%

Logarithmic scaling of Lyapunov exponents in disordered chiral two-dimensional lattices

Markoš

Schweitzer

2010

Phys. Rev. B

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We analyze the scaling behavior of the two smallest Lyapunov exponents for electrons propagating on two-dimensional lattices with energies within a very narrow interval around the chiral critical point at E = 0 in the presence of a perpendicular random magnetic flux. By a numerical analysis of the energy and size dependence we confirm that the two smallest Lyapunov exponents are functions of a single parameter. The latter is given by ln L/ ln ξ(E), which is the ratio of the logarithm of the system width L to the logarithm of the correlation length ξ(E). Close to the chiral critical point and energy |E| ≪ E0, we find a logarithmically divergent energy dependence ln ξ(E) ∝ | ln(E0/|E|)| 1/2 , where E0 is a characteristic energy scale. Our data are in agreement with the theoretical prediction of M. Fabrizio and C. Castelliani [Nucl. Phys. B 583, 542 (2000)] and resolve an inconsistency of previous numerical work.

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Section: Discussionmentioning

confidence: 99%

Logarithmic scaling of Lyapunov exponents in disordered chiral two-dimensional lattices

Markoš

Schweitzer

2010

Phys. Rev. B

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“…[5] it was suggested that for PLD this quantity follows an unusual logarithmic scaling. Later this type of scaling behavior was explained in the framework of scaling laws with continuously varying exponents [14]. Opposed to these early conjectures, we present numerical evidence that such logarithmic scaling laws do not hold asymptotically, although they may be used as a good approximation.…”

Section: Introductionmentioning

confidence: 52%

“…8 shows a data collapse of the normalized nucleation density according to this scaling form, which at first glance seems to be convincing. Initially it was speculated that the unusual type of scaling behavior, which can be explained in terms of continuously varying critical exponents [14], may be related to the fractal structure of the islands, which become more and more compact as the first monolayer is filled up. However, later it was shown [7] that the same logarithmic scaling form can be used in a 1+1-dimensional model for pulsed deposition, where the (one-dimensional) islands are always compact.…”

Section: Scaling Properties Of Pulsed Depositionmentioning

confidence: 99%

Rate equations and scaling in pulsed laser deposition

2008

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We study a simplified model for pulsed laser deposition [Hinnemann, Phys. Rev. Lett. 87, 135701 (2001)] by rate equations. We consider a set of equations where islands are assumed to be pointlike, as well as an improved one that takes the size of the islands into account. The first set of equations is solved exactly but its predictive power is restricted to a few pulses. The improved set of equations is integrated numerically, is in excellent agreement with simulations, and fully accounts for the crossover from continuous to pulsed deposition. Moreover, we analyze the scaling of the nucleation density and show numerical results indicating that a previously observed logarithmic scaling does not apply.

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“…Such intrinsic properties in the systems' equations can lead to intriguing and interesting dynamical consequences (Lai, 1997). Many physical systems share the property of scale invariance, most of them show ordinary power-law scaling, where quantities can be expressed as a leading powerlaw scaling function which depends on scalinginvariant ratios of the parameters (Sittler and Hinrichsen, 2002). For example, earthquakes, proteins, the wealth of nations and stock market cracks.…”

Section: Fractal Mathematicsmentioning

confidence: 99%

“…In other words, the question is how much isomorphism in there between the real systems and its fractal model. Such systems do not obey power-law scaling; instead there is numerical evidence for a logarithmic scaling form, in which the scaling function depends on ratios of the parameters' logarithms (Sittler and Hinrichsen, 2002). Perhaps the most important practical aspect of fractal analysis may be the use of fractal dimension as a quantitative variable that morphologists can study as a dependent variable in the context of many independent variables (Klonowsk, 2000).…”

Section: Fractal Mathematicsmentioning

confidence: 99%

Fractal behaviour of complex systems

2010

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One of the most important properties of systems is complexity. In a simple way, we can define the complexity of a system in terms of the number of elements that it contains, the nature and number of interrelations and the number of levels of embeddedness. When a high level of complexity exists in a system, it is considered a complex system. Complex systems can be soft systems and hard systems. In hard systems, when their elements are interrelated in a non-linear way, they are considered complex systems when they contain a great number of elements interacting in a non-linear way. To try to understand the behaviour of this type of system diverse mathematical tools have been developed. A new scientific discipline with great impact in the analysis of the complex systems has been developed in recent years, called fractal analysis. The study of the complex systems in the framework of fractal theory has been recognized as a new scientific discipline, being sustained by advances that have been made in diverse fields ranging from physics to economics. In this paper the history and the basic concepts of the fractal analysis of complex systems are discussed briefly and an oil crude price volatility fractal analysis is provided.

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On generalized scaling laws with continuously varying exponents

Cited by 13 publications

References 11 publications

Logarithmic scaling of Lyapunov exponents in disordered chiral two-dimensional lattices

Logarithmic scaling of Lyapunov exponents in disordered chiral two-dimensional lattices

Rate equations and scaling in pulsed laser deposition

Fractal behaviour of complex systems

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