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G. Kaniadakis

Publications and source records attributed to G. Kaniadakis.

At least 37 records · Page 2Linked to original sources
arXiv2005

Statistical mechanics in the context of special relativity II

The special relativity laws emerge as one-parameter (light speed) generalizations of the corresponding laws of classical physics. These generalizations, imposed by the Lorentz transformations, affect both the definition of the various physical observables (e.g. momentum, energy, etc), as well as the mathematical apparatus of the theory. Here, following the general lines of [Phys. Rev. E {\bf 66}, 056125 (2002)], we show that the Lorentz transformations impose also a proper one-parameter generalization of the classical Boltzmann-Gibbs-Shannon entropy. The obtained relativistic entropy permits to construct a coherent and selfconsistent relativistic statistical theory, preserving the main features of the ordinary statistical theory, which recovers in the classical limit. The predicted distribution function is a one-parameter continuous deformation of the classical Maxwell-Boltzmann distribution and has a simple analytic form, showing power law tails in accordance with the experimental evidence. Furthermore the new statistical mechanics can be obtained as stationary case of a generalized kinetic theory governed by an evolution equation obeying the H-theorem and reproducing the Boltzmann equation of the ordinary kinetics in the classical limit.

cond-mat.stat-mech↗
arXiv2005

Two-parameter deformations of logarithm, exponential, and entropy: A consistent framework for generalized statistical mechanics

A consistent generalization of statistical mechanics is obtained by applying the maximum entropy principle to a trace-form entropy and by requiring that physically motivated mathematical properties are preserved. The emerging differential-functional equation yields a two-parameter class of generalized logarithms, from which entropies and power-law distributions follow: these distributions could be relevant in many anomalous systems. Within the specified range of parameters, these entropies possess positivity, continuity, symmetry, expansibility, decisivity, maximality, concavity, and are Lesche stable. The Boltzmann-Shannon entropy and some one parameter generalized entropies already known belong to this class. These entropies and their distribution functions are compared, and the corresponding deformed algebras are discussed.

cond-mat.stat-mech↗
arXiv2004

Deformed logarithms and entropies

By solving a differential-functional equation inposed by the MaxEnt principle we obtain a class of two-parameter deformed logarithms and construct the corresponding two-parameter generalized trace-form entropies. Generalized distributions follow from these generalized entropies in the same fashion as the Gaussian distribution follows from the Shannon entropy, which is a special limiting case of the family. We determine the region of parameters where the deformed logarithm conserves the most important properties of the logarithm, and show that important existing generalizations of the entropy are included as special cases in this two-parameter class.

cond-mat.stat-mech↗
arXiv2004

Lesche Stability of $κ$-Entropy

The Lesche stability condition for the Shannon entropy [B. Lesche, J. Stat. Phys. 27, 419 (1982)], represents a fundamental test, for its experimental robustness, for systems obeying the Maxwell-Boltzmann statistical mechanics. Of course, this stability condition must be satisfied by any entropic functional candidate to generate non-conventional statistical mechanics. In the present effort we show that the $κ$-entropy, recently introduced in literature [G. Kaniadakis, Phys. Rev. E 66, 056125 (2002)], satisfies the Lesche stability condition.

cond-mat.stat-mech↗
arXiv2004

Stabilities of generalized entropies

The generalized entropic measure, which is optimized by a given arbitrary distribution under the constraints on normalization of the distribution and the finite ordinary expectation value of a physical random quantity, is considered and its Lesche stability property (that is different from thermodynamic stability) is examined. A general condition, under which the generalized entropy becomes stable, is derived. Examples known in the literature, including the entropy for the stretched-exponential distribution, the quantum-group entropy, and the kappa-entropy are discussed.

cond-mat.stat-mech↗
arXiv2003

BBGKY Hierarchy Underlying Many Particle Quantum Mechanics

Recently, the one particle quantum mechanics has been obtained in the framework of an entirely classical subquantum kinetics. In the present paper we argue that, within the same scheme and without any additional assumption, it is possible to obtain also the $n$-particle non relativistic quantum mechanics. The main goal of the present effort is to show that the classical BBGKY hierarchical equation, for the $n$-particle reduced distribution function, is the ancestor of the $n$-particle Schrödinger equation. On the other hand we show that within the scenario of the subquantum structure of quantum particle, the Fisher information measure emerges naturally in quantum mechanics.

quant-ph↗
arXiv2003

Nonlinear Schroedinger Equations within the Nelson Quantization Picture

We present a class of nonlinear Schroedinger equations (NLSEs) describing, in the mean field approximation, systems of interacting particles. This class of NLSEs is obtained generalizing expediently the approach proposed in Ref. [G.K. Phys. Rev. A 55, 941 (1997)], where a classical system obeying to an exclusion-inclusion principle is quantized using the Nelson stochastic quantization. The new class of NLSEs is obtained starting from the most general nonlinear classical kinetics compatible with a constant diffusion coefficient D=\hbar/2m. Finally, in the case of s-stationary states, we propose a transformation which linearizes the NLSEs here proposed.

cond-mat.soft↗
arXiv2002

Statistical mechanics in the context of special relativity

In the present effort we show that $S_κ=-k_B \int d^3p (n^{1+κ}-n^{1-κ})/(2κ)$ is the unique existing entropy obtained by a continuous deformation of the Shannon-Boltzmann entropy $S_0=-k_B \int d^3p n \ln n$ and preserving unaltered its fundamental properties of concavity, additivity and extensivity. Subsequently, we explain the origin of the deformation mechanism introduced by $κ$ and show that this deformation emerges naturally within the Einstein special relativity. Furthermore, we extend the theory in order to treat statistical systems in a time dependent and relativistic context. Then, we show that it is possible to determine in a self consistent scheme within the special relativity the values of the free parameter $κ$ which results to depend on the light speed $c$ and reduces to zero as $c \to \infty$ recovering in this way the ordinary statistical mechanics and thermodynamics. The novel statistical mechanics constructed starting from the above entropy, preserves unaltered the mathematical and epistemological structure of the ordinary statistical mechanics and is suitable to describe a very large class of experimentally observed phenomena in low and high energy physics and in natural, economic and social sciences. Finally, in order to test the correctness and predictability of the theory, as working example we consider the cosmic rays spectrum, which spans 13 decades in energy and 33 decades in flux, finding a high quality agreement between our predictions and observed data. PACS number(s): 05.20.-y, 51.10.+y, 03.30.+p, 02.20.-a

cond-mat.stat-mech↗
arXiv2002

Nonrelativistic Quantum Mechanics with Spin in the Framework of a Classical Subquantum Kinetics

Recently it has been shown that the spinnless one particle quantum mechanics can be obtained in the framework of entirely classical subquantum kinetics. In the present paper we argue that, within the same scheme and without any extra assumption, it is possible to obtain both the non relativistic quantum mechanics with spin, in the presence of an arbitrary external electromagnetic field, as well as the nonlinear quantum mechanics. Pacs: 03.65.Ta, 05.20.Dd KEY WORDS: monads, subquantum physics, quantum potential, foundations of quantum mechanics

quant-ph↗
arXiv2002

Cole-Hopf Like Transformation for a Class of Coupled Nonlinear Schroedinger Equations

In this paper we consider a class of coupled nonlinear Schroedinger equations for the fields $ψ_i$ containing complex nonlinearities, that has been obtained by requiring that the norms $|ψ_i|^2$ are conserved densities. For this class of equations we introduce a Cole-Hopf like transformation, whose effect is to reduce the complex nonlinearities into real ones, this way reducing the continuity equations of the system to the standard bilinear form. Some examples are presented to illustrate the applicability of the method. Pacs numbers: 03.50.-z, 03.65.-w, 11.30.Na, 11.40.Dw Keywords: Coupled nonlinear Schroedinger equations, Cole-Hopf transformation, Nonlinear transformations.

quant-ph↗
arXiv2002

Nonlinear Transformation for a Class of Gauged Schroedinger Equations with Complex Nonlinearities

In the present contribution we consider a class of Schroedinger equations containing complex nonlinearities, describing systems with conserved norm $|ψ|^2$ and minimally coupled to an abelian gauge field. We introduce a nonlinear transformation which permits the linearization of the source term in the evolution equations for the gauge field, and transforms the nonlinear Schroedinger equations in another one with real nonlinearities. We show that this transformation can be performed either on the gauge field $A_μ$ or, equivalently, on the matter field $ψ$. Since the transformation does not change the quantities $|ψ|^2$ and $F_{μν}$, it can be considered a generalization of the gauge transformation of third kind introduced some years ago by other authors. Pacs numbers: 03.65.-w, 11.15.-q

quant-ph↗
arXiv2002

Nonlinear gauge transformation for a class of Schroedinger equations containing complex nonlinearities

We consider a wide class of nonlinear canonical quantum systems described by a one-particle Schroedinger equation containing a complex nonlinearity. We introduce a nonlinear unitary transformation which permits us to linearize the continuity equation. In this way we are able to obtain a new quantum system obeying to a nonlinear Schroedinger equation with a real nonlinearity. As an application of this theory we consider a few already studied Schroedinger equations as that containing the nonlinearity introduced by the exclusion-inclusion principle, the Doebner-Goldin equation and others. PACS numbers: 03.65.-w, 11.15.-q

quant-ph↗
arXiv2002

Bose-Einstein Condensation in the Framework of $κ$-Statistics

In the present work we study the main physical properties of a gas of $κ$-deformed bosons described through the statistical distribution function $f_κ=Z^{-1}[\exp_κ(β({1/2}m v^2-μ))-1]^{-1}$. The deformed $κ$-exponential $\exp_κ(x)$, recently proposed in Ref. [G.Kaniadakis, Physica A {\bf 296}, 405, (2001)], reduces to the standard exponential as the deformation parameter $κ\to 0$, so that $f_0$ reproduces the Bose-Einstein distribution. The condensation temperature $T_c^κ$ of this gas decreases with increasing $κ$ value, and approaches the $^{4}He(I)-^{4}He(II)$ transition temperature $T_λ=2.17K$, improving the result obtained in the standard case ($κ=0$). The heat capacity $C_V^κ(T)$ is a continuous function and behaves as $B_κT^{3/2}$ for $T T_c^κ$, in contrast with the standard case $κ=0$, it is always increasing. Pacs: 05.30.Jp, 05.70.-a Keywords: Generalized entropy; Boson gas; Phase transition.

cond-mat.stat-mech↗
arXiv2002

Quantum vortices in systems obeying a generalized exclusion principle

The paper deals with a planar particle system obeying a generalized exclusion principle (EP) and governed, in the mean field approximation, by a nonlinear Schroedinger equation. We show that the EP involves a mathematically simple and physically transparent mechanism, which allows the genesis of quantum vortices in the system. We obtain in a closed form the shape of the vortices and investigate its main physical properties. PACS numbers: 03.65.-w, 03.65.Ge, 05.45.Yv

cond-mat.soft↗
arXiv2002

Cole-Hopf Like Transformation for Schrödinger Equations Containing Complex Nonlinearities

We consider systems, which conserve the particle number and are described by Schrödinger equations containing complex nonlinearities. In the case of canonical systems, we study their main symmetries and conservation laws. We introduce a Cole-Hopf like transformation both for canonical and noncanonical systems, which changes the evolution equation into another one containing purely real nonlinearities, and reduces the continuity equation to the standard form of the linear theory. This approach allows us to treat, in a unifying scheme, a wide variety of canonical and noncanonical nonlinear systems, some of them already known in the literature. pacs{PACS number(s): 02.30.Jr, 03.50.-z, 03.65.-w, 05.45.-a, 11.30.Na, 11.40.Dw

quant-ph↗
arXiv2001

Statistical Origin of Quantum Mechanics

The one particle quantum mechanics is considered in the frame of a N-body classical kinetics in the phase space. Within this framework, the scenario of a subquantum structure for the quantum particle, emerges naturally, providing an ontological support to the orthodox quantum mechanics. This approach to quantum mechanics, constitutes a deductive and direct method which, in a self-consistent scheme of a classical kinetics, allows us: i) to obtain the probabilistic nature of the quantum description and to interpret the wave function $ψ$ according to the Copenhagen school; ii) to derive the quantum potential and then the Schrödinger equation; iii) to calculate the values of the physical observables as mean values of the associated quantum operators; iv) to obtain the Heisenberg uncertainty principle.

quant-ph↗
arXiv2001

Kinetical Foundations of Non Conventional Statistics

After considering the kinetical interaction principle (KIP) introduced in ref. Physica A {\bf296}, 405 (2001), we study in the Boltzmann picture, the evolution equation and the H-theorem for non extensive systems. The $q$-kinetics and the $κ$-kinetics are studied in detail starting from the most general non linear Boltzmann equation compatible with the KIP.

cond-mat.stat-mech↗