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# Fractals, Multifractals, and Thermodynamics

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## Abstract

The basic concept of fractals and multifractals are introduced for pedagogical purposes, and the present status is reviewed. The emphasis is put on illustrative examples with simple mathematical structures rather than on numerical methods or experimental techniques. As a general characteriza­tion of fractals and multifractals a thermodynamical formalism is introduced, establishing a connec­tion between fractal properties and the statistical mechanics of spin chains.
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... where D q refers to the order q generalized dimension (Tel 1988), I q e ð Þ is called the generalized entropy (R enyi 1961;Appleby 1996), and P i e ð Þ q is the spatial probability of the ith fractal units in the linear scale of e: In general, this parameter can characterize the scale invariance of the generalized entropy, and the order q plays a role in adjusting the contribution of each fractal unit to the overall spatial probability. ...
... For a strictly fractal system, the singularity exponents of its subsystems should remain constant across all scales. For random fractals in urban systems, however, these are generated by nondeterministic rules and accordingly show more randomness and anomalies (Tel 1988), with the singularity exponents changing with the scales. Consequently, they become unlikely to estimate. ...
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... Here, we focus on a more detailed description by reconstructing the distribution of the different words for different noise amplitudes. This is essentially an application of the so-called multifractal formalism [8]. We consider a single m value, namely, m = 10: this choice, while being still computationally manageable, provides a sufficiently large number of possible symbolic sequences (≈3.6 ×10 6 ) so that the related set can be considered as being continuous. ...
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We analyze the permutation entropy of deterministic chaotic signals affected by a weak observational noise. We investigate the scaling dependence of the entropy increase on both the noise amplitude and the window length used to encode the time series. In order to shed light on the scenario, we perform a multifractal analysis, which allows highlighting the emergence of many poorly populated symbolic sequences generated by the stochastic fluctuations. We finally make use of this information to reconstruct the noiseless permutation entropy. While this approach works quite well for Hénon and tent maps, it is much less effective in the case of hyperchaos. We argue about the underlying motivations.
... Additionally, significant differences in the long-term crosscorrelation were accompanied by changes in the degree of multifractality, in most cases. A possible explanation could be that multifractality results from more complex dynamics (Tel, 1988) which tend to vary more from region to region. On the other hand, this contradicts the findings of our previous resting-state study, where H(2) values varied the most [cf. ...
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The human brain consists of anatomically distant neuronal assemblies that are interconnected via a myriad of synapses. This anatomical network provides the neurophysiological wiring framework for functional connectivity (FC), which is essential for higher-order brain functions. While several studies have explored the scale-specific FC, the scale-free (i.e., multifractal) aspect of brain connectivity remains largely neglected. Here we examined the brain reorganization during a visual pattern recognition paradigm, using bivariate focus-based multifractal (BFMF) analysis. For this study, 58 young, healthy volunteers were recruited. Before the task, 3-3 min of resting EEG was recorded in eyes-closed (EC) and eyes-open (EO) states, respectively. The subsequent part of the measurement protocol consisted of 30 visual pattern recognition trials of 3 difficulty levels graded as Easy, Medium, and Hard. Multifractal FC was estimated with BFMF analysis of preprocessed EEG signals yielding two generalized Hurst exponent-based multifractal connectivity endpoint parameters, H (2) and Δ H 15 ; with the former indicating the long-term cross-correlation between two brain regions, while the latter captures the degree of multifractality of their functional coupling. Accordingly, H (2) and Δ H 15 networks were constructed for every participant and state, and they were characterized by their weighted local and global node degrees. Then, we investigated the between- and within-state variability of multifractal FC, as well as the relationship between global node degree and task performance captured in average success rate and reaction time. Multifractal FC increased when visual pattern recognition was administered with no differences regarding difficulty level. The observed regional heterogeneity was greater for Δ H 15 networks compared to H (2) networks. These results show that reorganization of scale-free coupled dynamics takes place during visual pattern recognition independent of difficulty level. Additionally, the observed regional variability illustrates that multifractal FC is region-specific both during rest and task. Our findings indicate that investigating multifractal FC under various conditions – such as mental workload in healthy and potentially in diseased populations – is a promising direction for future research.
... Maybe the reader has expected a more quantitative description of the transient chaos encountered. Unfortunately this was not possible because of the following reasons: the usual measures of chaos can be calculated best by the thermodynamical formalism [24,25]. This formalism (and other ones equally) and also the mathematical definitions of these measures are based on a hierarchical structure of the chaotic set and on an (at least approximate) exponential scaling behaviour of this hierarchy. ...
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The evolution of cracks in reinforced concrete (RC) structures may trigger critical failure modes of the entire structure. In order to adequately assess the damage in RC structures, it is essential to characterize the distribution and evolution of cracks. Compared with existing quantitative methods of describing concrete cracks, multifractal analysis (MUTFA) is an emerging and sophisticated tool for characterizing the complexity and irregularity of crack distributions in RC structures. Despite the fact that MUTFA embodies a greater capability than monofractal analysis (MONFA), the uncertainty of the mechanism of the multifractal spectrum for representing concrete cracks is a major limitation in using MUTFA to depict cracks. To address this drawback, this study illustrates the difference between the features of MUTFA and MONFA as well as proposes a multifractal-spectrum shape parameter to characterize the complexity and irregularity of fractal-like cracks. The advantages of MUTFA in portraying concrete cracks are demonstrated in two typical cracking scenarios of concrete structures, namely, crack distributions in shear-controlled concrete beams and crack patterns in thin and lightly-reinforced concrete walls. The results of the study show that the proposed multifractal-spectrum shape parameter offsets the deficiency of traditional multifractal parameters in revealing the heterogeneity of crack distributions. The results demonstrate that MUTFA is competent in distinguishing the subtle differences between two similar distributions of concrete cracks, and it provides a path to assess damage in concrete structures.
Thesis
En dimension trois, un système quantique désordonné peut présenter une transition entre un état métallique/diffusif à faible désordre et un état isolant/localisé à fort désordre. Au voisinage de cette transition appelée transition d'Anderson, il est connu que les fonctions d'onde des états propres présentent des fluctuations géantes et un caractère multifractal. Dans ce manuscrit, nous utilisons un système spécifique, le rotateur pulsé --- également appelé kicked rotor --- quasi périodique pour étudier les propriétés de multifractalité de paquets d'onde. C'est un système unidimensionnel, donc facile à étudier expérimentalement et à simuler numériquement, mais sa dépendance temporelle est telle qu'il présente une transition d'Anderson aisément contrôlable. Nous étudions numériquement et interprétons théoriquement les propriétés de multifractalité des paquets d'onde au voisinage de la transition d'Anderson. Nous montrons que celles-ci permettent de remonter partiellement aux propriétés de multifractalité des états propres en dimension trois, ouvrant ainsi des perspectives pour une étude expérimentale future.
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Preface Introduction Notation 1. Measure and dimension 2. Basic density properties 3. Structure of sets of integral dimension 4. Structure of sets of non-integral dimension 5. Comparable net measures 6. Projection properties 7. Besicovitch and Kakeya sets 8. Miscellaneous examples of fractal sets References Index.
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Anomalous scaling laws appear in a wide class of phenomena where global dilation invariance fails. In this case, the description of scaling properties requires the introduction of an infinite set of exponents.Numerical and experimental evidence indicates that this description is relevant in the theory of dynamical systems, of fully developed turbulence, in the statistical mechanics of disordered systems, and in some condensed matter problems.We describe anomalous scaling in terms of multifractal objects. They are defined by a measure whose scaling properties are characterized by a family of singularities, which are identified by a scaling exponent. Singularities corresponding to the same exponent are distributed on fractal set. The multifractal object arises as the superposition of these sets, whose fractal dimensions are related to the anomalous scaling exponents via a Legendre transformation. It is thus possible to reconstruct the probability distribution of the singularity exponents.We review the application of this formalism to the description of chaotic attractors in dissipative systems, of the energy dissipating set in fully developed turbulence, of some probability distributions in condensed matter problems. Moreover, a simple extension of the method allows us to treat from the same point of view temporal intermittency in chaotic systems and sample to sample fluctuations in disordered systems.We stress the phenomenological nature of the approach and discuss the few cases in which it was possible to reach a more fundamental understanding of anomalous scaling. We point out the need of a theory which should explain its origin and pave the way to a microscopic calculation of the probability distribution of the singularities.
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We introduce the definitions of generalized dimensions for fractal sets characterized by logarithmic and additive corrections to the measure. The definitions are simple to compute. This framework is used to analyze several classes of sets with partial scaling symmetry. This includes an important class of sets having scaling symmetry with respect to a fixed point. We give two important physical examples of the above class where our analysis is useful.
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Following an analogy to the formalism of statistical mechanics, an entropy function and a free energy are introduced for multifractals. These functions give a full description of the scaling behaviors of multifractals. The method of Halsey et al. (1986) for characterizing multifractals can naturally be interpreted by the use of these functions. For the invariant set of a dynamical system, these functions are furthermore related to the measure-theoretic (Kolmogolov-Sinai) entropy, the topological entropy, and the Lyapunov exponent.
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