**Abstract**

## Journal Menu

► Journal Menu# Topical Collection "Advances in Applied Statistical Mechanics"

A topical collection in *Entropy* (ISSN 1099-4300). This collection belongs to the section "Statistical Mechanics".

## Editor

Collection Editor
Dr. Antonio M. Scarfone
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
Interests: nonextensive statistical mechanics; nonlinear fokker-planck equations; geometry information; nonlinear schroedinger equation; quantum groups and quantum algebras; complex systems |

## Topical Collection Information

Dear Colleagues,

There is a diffuse belief that statistical properties of physical systems are well described by BoltzmannGibbs statistical mechanics. However, a constantly increasing amount of situations are known to violate the predictions of orthodox statistical mechanics. Systems where these emerging features are observed seem do not fulfill the standard ergodic and mixing properties on which the Boltzmann-Gibbs formalism are founded. In general, these systems are governed by nonlinear dynamics which establishes a deep relation among the parts. As a consequence, they reach a dynamical equilibrium in which the equilibrium probability distribution can differ deeply from the exponential shape typical of the Gibbs distribution.

In the last decades, we assisted to an intense research activity that has modified our understanding of statistical physics, extending and renewing its applicability considerably. Important developments, relating equilibrium and nonequilibrium statistical physics, kinetic theory, information theory and others, have produced a new understanding of the properties of complex systems that requires, in many

cases, the extension of the theory beyond the Boltzmann-Gibbs formalism.

The aim of this collection, is to collect papers in both the foundations and the applications of Statistical Mechanics going outside its traditional application. In particular, foundations regard classical and quantum aspects of statistical physics including generalized entropies, free-scale distributions, information theory, geometry information, nonextensive statistical mechanics, kinetic theory, long-range interactions and small systems. Applications are different and may include biophysics, seismology, econophysics, social systems, physics of networks, physics of risk, traffic flow, complex systems, fractal systems and others.

Specific topics of interest include (but are not limited to): * Generalized entropies * Boltzmann entropy * Renyi entropy * Non linear kinetic * Fokker-Planck equations * Quantum information * Geometry information * Fractal systems * Complex systems * Networks * Econophysics * Sociophysics * Biophysics

Dr. Antonio Maria Scarfone*Collection Editor*

**Manuscript Submission Information**

Manuscripts for the topical collection can be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on this website. The topical collection considers regular research articles, short communications and review articles. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1500 CHF (Swiss Francs).

## Keywords

- non extensive systems
- generalized entropies
- boltzmann entropy
- renyi entropy
- non linear kinetic
- fokker-planck equations
- quantum information
- geometry information
- fractal systems
- complex systems
- networks
- econophysics
- sociophysics
- biophysics

*Xiphias gladius Linnaeus*)

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*Xiphias gladius Linnaeus*) length dataset. Full article

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*X*—ordinary or defective, according to whether the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of the order

*s*,

*s*

*defective*lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types. It is shown that no labeled Fano plane can have all points of zero-th order, or feature just one point of order two. A connection with prominent configurations in Steiner triple systems is also pointed out. Full article

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*q*-th quantile of the lifetime distribution at the use condition is minimized. In addition, compromise plans are developed to provide means to check the adequacy of the assumed acceleration model. Finally, sensitivity analysis procedures for assessing the effects of the uncertainties in the pre-estimates of unknown parameters are illustrated with an example. Full article

**Abstract**

*χ*

*-superstatistics, inverse*

^{2}*χ*-superstatistics, and lognormal superstatistics, all maximizing different effective entropy measures. We investigate how the three classes of extreme value theory are related to the three classes of superstatistics. We show that for any superstatistical process whose local equilibrium distribution does not live on a finite support, the Weibull distribution cannot occur. Under the above mild asymptotic independence assumptions, we also show that

^{2 }*χ*-superstatistics generally leads an extreme value statistics described by a Fréchet distribution, whereas inverse

^{2}*χ*

*-superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution. Full article*

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_{1}of -Δ in ${H}_{0}^{1}\left(\Omega \right)$.Translating mathematics into cryptographic applications will be relevant in everyday life, where in there are situations in which two parts that communicate need a third part to confirm this process. For example, if two persons want to agree on something they need an impartial person to confirm this agreement, like a notary. This third part does not influence in anyway the communication process. It is just a witness to the agreement. We present a system where the communicating parties do not authenticate one another. Each party authenticates itself to a third part who also sends the keys for the encryption/decryption process. Another advantage of such a system is that if someone (sender) wants to transmit messages to more than one person (receivers), he needs only one authentication, unlike the classic systems where he would need to authenticate himself to each receiver. We propose an authentication method based on zero-knowledge and elliptic curves. Full article

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*H*Theorem and Second Law of Thermodynamics: a Pathway by Path Probability

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*H*theorem. We argue that the gap between the regular Newtonian dynamics and the random dynamics was not considered in the criticisms of the

*H*theorem. Full article

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*n*of species, a phenomenon which have puzzled ecologists for decades. An interesting point is that this derivation uses results obtained from a statistical mechanics model for ferromagnets. Second, going beyond the mean field approximation, I study the spatial version of a popular ecological model involving just one species representing vegetation. The goal is to address the phenomena of catastrophic shifts—gradual cumulative variations in some control parameter that suddenly lead to an abrupt change in the system—illustrating it by means of the process of desertification of arid lands. The focus is on the aggregation processes and the effects of diffusion that combined lead to the formation of non trivial spatial vegetation patterns. It is shown that different quantities—like the variance, the two-point correlation function and the patchiness—may serve as early warnings for the desertification of arid lands. Remarkably, in the onset of a desertification transition the distribution of vegetation patches exhibits scale invariance typical of many physical systems in the vicinity a phase transition. I comment on similarities of and differences between these catastrophic shifts and paradigmatic thermodynamic phase transitions like the liquid-vapor change of state for a fluid. Third, I analyze the case of many species interacting in space. I choose tropical forests, which are mega-diverse ecosystems that exhibit remarkable dynamics. Therefore these ecosystems represent a research paradigm both for studies of complex systems dynamics as well as to unveil the mechanisms responsible for the assembly of species-rich communities. The more classical equilibrium approaches are compared versus non-equilibrium ones and in particular I discuss a recently introduced cellular automaton model in which species compete both locally in physical space and along a niche axis. Full article

**Abstract**Stirling approximation of the factorials and multinominal coefficients are generalized based on the κ-generalized functions introduced by Kaniadakis. We have related the κ-generalized multinominal coefficients to the κ-entropy by introducing a new κ-product operation, which exists only when κ ≠ 0. Full article

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*q*-Gaussian Distributions Induced by Beta-Divergence

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*q*-exponential family particularly admits natural geometrical structures among deformed exponential families. The property is the invariance of structures with respect to a general linear group, which transitively acts on the space of positive definite matrices. We prove this

*q*-exponential family particularly admits natural geometrical structures among deformed exponential families. The property is the invariance of structures with respect to a general linear group, which transitively acts on the space of positive definite matrices. We prove this property via the correspondence between information geometry induced by a deformed potential on the space and the one induced by what we call

*β-divergence*defined on the q-exponential family with

*q*=

*β*+ 1. The results are fundamental in robust multivariate analysis using the q-Gaussian family. Full article

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*κ*-deformed exponential function ${\mathrm{exp}}_{k}\left(x\right)\text{}=\text{}{(\sqrt{1\text{}+\text{}{k}^{2}{x}^{2}}\text{}+\text{}kx)}^{\frac{1}{k}}$, with 0

*≤*

*κ <*1, developed

*κ*-deformed exponential function ${\mathrm{exp}}_{k}\left(x\right)\text{}=\text{}{(\sqrt{1\text{}+\text{}{k}^{2}{x}^{2}}\text{}+\text{}kx)}^{\frac{1}{k}}$, with 0

*≤*

*κ <*1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The

*κ*-mathematics has its roots in special relativity and furnishes the theoretical foundations of the

*κ*-statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the

*κ*-algebra, we present the associated

*κ*-differential and

*κ*-integral calculus. Then, we obtain the corresponding

*κ*-exponential and

*κ*-logarithm functions and give the

*κ*-version of the main functions of the ordinary mathematics. Full article

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*q*-exponential

*q*-exponential functions present in the Tsallis framework. In the case of the long-tailed behavior, in the asymptotic limit, these solutions can also be connected with the L´evy distributions. In addition, from the results presented here, a rich class of diffusive processes, including normal and anomalous ones, can be obtained. Full article

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