The thermodynamics of fractals revisited with wavelets

doi:10.1016/0378-4371(94)00163-N

Physica A: Statistical Mechanics and its Applications

Volume 213, Issues 1–2, 1 January 1995, Pages 232-275

https://doi.org/10.1016/0378-4371(94)00163-N Get rights and content

Abstract

The multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that very much like thermodynamic functions, the generalized fractal dimensions D_q and the f(α) singularity spectrum can be readily determined from the scaling behavior of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of “generalized boxes”. We illustrate our theoretical considerations on pedagogical examples, e.g., devil's staircases and fractional Brownian motions. We also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data. That we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects. In particular, we show that a dynamical system which leaves invariant such an object can be uncovered form the space-scale arrangement of its wavelet transform modulus maxima. We elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem. This step towards a statistical mechanics of fractals is illustrated on discrete period-doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos. Finally, we apply our technique to analyze the fractal hierarchy of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off-lattice diffusion-limited aggregates (DLA) wit a circle. This study clearly lets out the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology.

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Citation Excerpt :
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The thermodynamics of fractals revisited with wavelets

Abstract

J. Stat. Phys.

S.I.A.M. Rev.

Phys. Rev. A

Phys. Rev. Lett.

Phys. Lett. A

Phys. Lett. A

Phys. Rev. A

Fractals: Form, Chance and Dimension

The Fractal Geometry of Nature

Fractals

Fractal Growth Phenomena

Dynamics of Fractal Surfaces

Fractals and Disordered Systems

Modern Theory of Critical Phenomena

Field Theory, the Renormalization Group and Critical Phenomena

Self-Organization in Non-Equilibrium Systems

Synergetics: An Introduction

Advanced Synergetics

Phys. Rev.

Dissipative Structures and Weak Turbulence

Rev. Mod. Phys.

Du Microscopique au Macroscopique

J. Russ. Math. Surveys

Thermodynamic Formalism

Phys. Rev. Lett.

Phys. Rev. A

Phys. Rev. A

Phys. Rev. A

J. Phys. A

J. Russ. Math. Surv.

J. Stat. Phys.

J. Stat. Phys.

J. Stat. Phys.

Ergod. Th. and Dyn. Sys.

J. Fluid Mech.

Turbulence: challenges for theory and experiments

Physics Today

J. Fluid Mech.