Emergency of Tsallis statistics in fractal networks

Deppman, A.; Andrade-II, E.

doi:10.1371/journal.pone.0257855

Cited by 7 publications

(4 citation statements)

References 52 publications

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“…It is worth observing that generalized entropic forms consistent with power-law asymptotic distributions may also be related to certain fractional–differential equations or fractal–differential equations that differ substantially from the differential–difference equation used in this work. For instance, generalized entropies are derived within a kinetic approach based on a fractional diffusive equation [ 50 ] or in [ 58 ], where a study on fractal networks has been done within the Tsallis statistics.…”

Section: Discussionmentioning

confidence: 99%

Boltzmann Configurational Entropy Revisited in the Framework of Generalized Statistical Mechanics

Scarfone

2022

Entropy

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As known, a method to introduce non-conventional statistics may be realized by modifying the number of possible combinations to put particles in a collection of single-particle states. In this paper, we assume that the weight factor of the possible configurations of a system of interacting particles can be obtained by generalizing opportunely the combinatorics, according to a certain analytical function f{π}(n) of the actual number of particles present in every energy level. Following this approach, the configurational Boltzmann entropy is revisited in a very general manner starting from a continuous deformation of the multinomial coefficients depending on a set of deformation parameters {π}. It is shown that, when f{π}(n) is related to the solutions of a simple linear difference–differential equation, the emerging entropy is a scaled version, in the occupational number representation, of the entropy of degree (κ,r) known, in the framework of the information theory, as Sharma–Taneja–Mittal entropic form.

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Section: Discussionmentioning

confidence: 99%

Boltzmann Configurational Entropy Revisited in the Framework of Generalized Statistical Mechanics

Scarfone

2022

Entropy

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“…Zipf-Mandelbrot distributions [6][7][8]) instead of exponential distributions of energy. Tsallis statistical mechanics falls into this category and it has found many applications throughout the years [11][12][13][14][15][16][17][18][19][20]. On the other hand, not everybody is fully convinced about the validity of such approaches to statistical mechanics [21,22].…”

Section: Statement Of the Problemmentioning

confidence: 99%

Numerical studies for an ab initio investigation into the Boltzmann prescription in statistical mechanics of large systems

Dossetti¹,

Viswanathan²,

Kenkre³

2022

J. Stat. Mech.

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We present numerical investigations into the question of the validity of the Boltzmann prescription in statistical mechanics for large systems, addressing the issue of whether extensivity of energy implies the extensivity of the Boltzmann entropy. The importance of the question stems from the fact that it is currently considered open by some investigators but quite settled by others. We report ab initio results for gas-like Hamiltonian systems with long-range as well as short-range interactions, based on simulations that explicitly consider more than 230 ≈ 109 states of the full Hilbert space. The basis of the technique is Monte Carlo algorithms. Despite the largeness of the numbers used, careful inspection shows that the systems studied are still too small to settle uniquely the issues raised. Therefore, the new approach outlined represents a first step in addressing on first principles the question of non-extensive statistical mechanics. General theoretical comments are also supplied to supplement the numerical investigations.

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“…Fractional derivatives have been used in many applications [1,2], and advancing our understanding of their geometrical meaning and their relations with fractals is necessary. The q-calculus has been frequently applied to describe the statistical properties of fractal systems [3,4]. However, the relationship between q-calculus and fractal derivatives has not been fully understood yet.…”

Section: Introductionmentioning

confidence: 99%

Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus

Deppman

Megías

Pasechnik

2023

Entropy

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This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff’s concepts of fractional dimension geometry. The paper distinguishes between the derivative of a function on a fractal domain and the derivative of a fractal function, where the image is a fractal space. Different continuous approximations for the fractal derivative are discussed, and it is shown that the q-calculus derivative is a continuous approximation of the fractal derivative of a fractal function. A similar version can be obtained for the derivative of a function on a fractal space. Caputo’s derivative is also proportional to a continuous approximation of the fractal derivative, and the corresponding approximation of the derivative of a fractional function leads to a Caputo-like derivative. This work has implications for studies of fractional differential equations, anomalous diffusion, information and epidemic spread in fractal systems, and fractal geometry.

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Emergency of Tsallis statistics in fractal networks

Cited by 7 publications

References 52 publications

Boltzmann Configurational Entropy Revisited in the Framework of Generalized Statistical Mechanics

Boltzmann Configurational Entropy Revisited in the Framework of Generalized Statistical Mechanics

Numerical studies for an ab initio investigation into the Boltzmann prescription in statistical mechanics of large systems

Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus

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