Log-periodic route to fractal functions

Details Log-periodic route to fractal functions Log-periodic route to fractal functions

2001-06-16

arxiv.org/abs/cond-mat/0106316v3

art. no. 036142, Phys. Rev. E V6503 N3 PT2A:U418-U436 (2002)

Physics

Condensed Matter

Statistical Mechanics

Latex document of 29 pages + 20 ps figures, addition of a new demonstration of the source of strong log-periodicity and of a j

Scientific paper

10.1103/PhysRevE.65.036142

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 (DSI) symmetry. We classify the `Weierstrass-type'' solutions of the renormalization group equation F(x)= g(x)+(1/m)F(g x) 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 seem to be characterized by oscillatory or bounded regular microscopic functions g(x) that lead to a slow power law decay of A(n), giving strong log-periodic amplitudes. In contrast, the regular function g(x) of statistical physics models with ``ferromagnetic''-type interactions at equibrium 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. These two classes of behavior can be traced back to the existence or abscence of ``antiferromagnetic'' or ``dipolar''-type interactions which, when present, make the Green functions non-monotonous oscillatory and favor spatial modulated patterns.

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