Boltzmann and the Statistical Multifractals

Durán-Meza G,López-García J,Del Río-Correa J.L

doi:10.2478/amns.2019.1.00007

Durán-Meza G, López-García J + Show 1 more

Open Access

https://doi.org/10.2478/amns.2019.1.00007

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Abstract

Abstract We extend the Boltzmann’s ideas that describe the evolution to the equilibrium of many body systems to the multifractal decomposition of the unitary interval 𝕀, in terms of sets Jα conformed by points with the same pointwise dimension, and obtain the D(α) singularity spectrum.

Highlights

The multifractal formalism introduced by Halsey et al [1] can be understood in a simple way by applying a similar reasoning that was used by Boltzmann for obtaining the thermodynamics of an ideal gas using statistical arguments instead of the microscopic description of a system conformed by 1023 particles
We introduce a Bernoulli measure with a probability vector p; make the multifractal decomposition in terms of sets Jα, conformed by points with the same pointwise dimension α, and show that they are conformed by an infinite number of sets M(φ)
The statistical multifractal decomposition of I consists of grouping the points x in subsets with the same value of the pointwise dimension, and each subset Jα is characterized by its Hausdorff dimension D(α)

Summary

Introduction

The multifractal formalism introduced by Halsey et al [1] can be understood in a simple way by applying a similar reasoning that was used by Boltzmann for obtaining the thermodynamics of an ideal gas using statistical arguments instead of the microscopic description of a system conformed by 1023 particles. We obtain the Eggleston’s theorem, which relates the Hausdorff dimension with the Shannon entropy; this theorem plays a similar role in fractals like that the relation between entropy and probability in the Boltzmann treatment. The theorem is showed using a multiplicative process to decompose the unitary interval in fractals M(φ), conformed by points with the same frequency of digits φ; evaluate the Hausdorff dimension of M(φ) and obtain that it is related to the Shannon entropy. We introduce a Bernoulli measure with a probability vector p; make the multifractal decomposition in terms of sets Jα , conformed by points with the same pointwise dimension α, and show that they are conformed by an infinite number of sets M(φ). The explicit values α∗ = α(q) and the Hausdorff dimension D(α(q)) are the functions obtained in the Boltzmann procedure to determine the D(α) singularity spectrum

Fundamental Boltzmann ideas

The geometric multifractal decomposition of the unitary interval I

Statistical multifractal decomposition of the unitary interval I

Boltzmann scheme for Multifractals

Statistical q-measures of Jα

Conclusions