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Log-periodic route to fractal functions.
1Institute of Geophysics and Planetary Physics, University of California Los Angeles, Los Angeles, California 90095-1567, USA.
Summary
Log-periodic oscillations in critical phenomena differ by four orders of magnitude between equilibrium statistical physics models and natural systems like DLA, earthquakes, and financial crashes. This study explains this difference via the microscopic "regular function" in renormalization group approaches.
Area of Science:
- Statistical physics
- Complex systems
- Non-linear dynamics
Background:
- Power-law behavior describes critical phenomena, often accompanied by log-periodic oscillations when scale invariance is broken.
- Log-periodic corrections are typically small (10^-5) in equilibrium statistical physics models (e.g., Ising, Potts spins).
- Significantly larger log-periodic oscillations (around 10%) are observed in growth processes (DLA), rupture, earthquakes, and financial crashes.
Purpose of the Study:
- To provide a technical explanation for the four orders of magnitude difference in log-periodic oscillation amplitudes.
- To classify solutions of renormalization group (RG) equations based on the properties of the microscopic 'regular function' g(x).
- To link observable properties (e.g., self-affinity, differentiability) to the characteristics of the regular function and RG solutions.
Main Methods:
- Analysis of the renormalization group (RG) equation F(x)=g(x)+mu(-1)F(gamma x), where g(x) is the microscopic regular function.
- Interpretation of the RG equation as a Jackson q-integral for systems lacking a derived RG equation.
- Classification of 'Weierstrass-type' RG solutions into two classes based on the amplitudes A(n) of their power-law series expansion.
Main Results:
- The difference in amplitudes is attributed to the nature of the regular function g(x): unbound logarithms in equilibrium models lead to weak oscillations, while oscillatory/bounded functions in growth/crash systems lead to strong oscillations.
- A novel critical point separates two classes of RG solutions, characterized by the decay rate of A(n).
- Systems with ergodic and mixing phases of A(n) exhibit self-affine, non-differentiable properties.
Conclusions:
- Growth processes, rupture, earthquakes, and financial crashes are characterized by specific microscopic functions leading to slow A(n) decay and strong log-periodic amplitudes.
- Equilibrium statistical physics models with ferromagnetic interactions exhibit fast A(n) decay and weak log-periodic amplitudes due to unbound logarithmic regular functions.
- The nature of the microscopic regular function is key to understanding the magnitude of log-periodic oscillations and the resulting observable properties.