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• W2024659360 abstract "Log-periodic oscillations have been found to decorate the usual power-law behavior found to describe the approach to a critical point, when the continuous scale-invariance symmetry is partially broken into a discrete-scale invariance symmetry. For Ising or Potts spins with ferromagnetic interactions on hierarchical systems, the relative magnitude of the log-periodic corrections are usually very small, of order 10(-5). In growth processes [diffusion limited aggregation (DLA)], rupture, earthquake, and financial crashes, log-periodic oscillations with amplitudes of the order of 10% have been reported. We suggest a technical explanation for this 4 order-of-magnitude difference based on the property of the regular function g(x) embodying the effect of the microscopic degrees of freedom summed over in a renormalization group (RG) approach F(x)=g(x)+mu(-1)F(gamma x) of an observable F as a function of a control parameter x. For systems for which the RG equation has not been derived, the previous equation can be understood as a Jackson q integral, which is the natural tool for describing discrete-scale invariance. We classify the Weierstrass-type solutions of the RG into two classes characterized by the amplitudes A(n) of the power-law series expansion. These two classes are separated by a novel critical point. Growth processes (DLA), rupture, earthquake, and financial crashes thus seem to be characterized by oscillatory or bounded regular microscopic functions that lead to a slow power-law decay of A(n), giving strong log-periodic amplitudes. If in addition, the phases of A(n) are ergodic and mixing, the observable presents self-affine nondifferentiable properties. In contrast, the regular function of statistical physics models with ferromagnetic-type interactions at equilibrium involves unbound logarithms of polynomials of the control variable that lead to a fast exponential decay of A(n) giving weak log-periodic amplitudes and smoothed observables." @default.
• W2024659360 created "2016-06-24" @default.
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• W2024659360 date "2002-03-07" @default.
• W2024659360 modified "2023-10-14" @default.
• W2024659360 title "Log-periodic route to fractal functions" @default.
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• W2024659360 doi "https://doi.org/10.1103/physreve.65.036142" @default.
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