arXiv subjects

G. Kaniadakis

Publications and source records attributed to G. Kaniadakis.

51 records · Page 3Linked to original sources

arXiv2001

A new one parameter deformation of the exponential function

Recently, in the ref. Physica A \bfm{296} 405 (2001), a new one parameter deformation for the exponential function $\exp_{_{\{{\scriptstyle κ}\}}}(x)= (\sqrt{1+κ^2x^2}+κx)^{1/κ}; \exp_{_{\{{\scriptstyle 0}\}}}(x)=\exp (x)$, which presents a power law asymptotic behaviour, has been proposed. The statistical distribution $f=Z^{-1}\exp_{_{\{{\scriptstyle κ}\}}}[-β(E-μ)]$, has been obtained both as stable stationary state of a proper non linear kinetics and as the state which maximizes a new entropic form. In the present contribution, starting from the $κ$-algebra and after introducing the $κ$-analysis, we obtain the $κ$-exponential $\exp_{_{\{{\scriptstyle κ}\}}}(x)$ as the eigenstate of the $κ$-derivative and study its main mathematical properties.

cond-mat.stat-mech↗

arXiv2001

H-Theorem and Generalized Entropies Within the Framework of Non Linear Kinetics

In the present effort we consider the most general non linear particle kinetics within the framework of the Fokker-Planck picture. We show that the kinetics imposes the form of the generalized entropy and subsequently we demonstrate the H-theorem. The particle statistical distribution is obtained, both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy. The present approach allows to treat the statistical distributions already known in the literature in a unifying scheme. As a working example we consider the kinetics, constructed by using the $κ$-exponential $\exp_{_{\{κ\}}}(x)= (\sqrt{1+κ^2x^2}+κx)^{1/κ}$ recently proposed which reduces to the standard exponential as the deformation parameter $κ$ approaches to zero and presents the relevant power law asymptotic behaviour $\exp_{_{\{κ\}}}(x){\atop\stackrel\sim x\to \pm \infty}|2κx|^{\pm 1/|κ|}$. The $κ$-kinetics obeys the H-theorem and in the case of Brownian particles, admits as stationary state the distribution $f=Z^{-1}\exp_{_{\{κ\}}}[-(βmv^2/2-μ)]$ which can be obtained also by maximizing the entropy $S_κ=\int d^n v [ c(κ)f^{1+κ}+c(-κ)f^{1-κ}]$ with $c(κ)=-Z^κ/ [2κ(1+κ)]$ after properly constrained.

cond-mat.stat-mech↗

arXiv2001

Non Linear Kinetics underlying Generalized Statistics

The purpose of the present effort is threefold. Firstly, it is shown that there exists a principle, that we call Kinetical Interaction Principle (KIP), underlying the non linear kinetics in particle systems, independently on the picture (Kramers, Boltzmann) used to describe their time evolution. Secondly, the KIP imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution, both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy. Thirdly, the KIP allows, on one hand, to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and, on the other hand, suggests how we can introduce naturally new distributions. Finally, as a working example of the approach to the non linear kinetics here presented, a new non extensive statistics is constructed and studied starting from a one-parameter deformation of the exponential function holding the relation $f(-x)f(x)=1$.

cond-mat.stat-mech↗

arXiv2000

Microscopic dynamics underlying the anomalous diffusion

The time dependent Tsallis statistical distribution describing anomalous diffusion is usually obtained in the literature as the solution of a non-linear Fokker-Planck (FP) equation [A.R. Plastino and A. Plastino, Physica A, 222, 347 (1995)]. The scope of the present paper is twofold. Firstly we show that this distribution can be obtained also as solution of the non-linear porous media equation. Secondly we prove that the time dependent Tsallis distribution can be obtained also as solution of a linear FP equation [G. Kaniadakis and P. Quarati, Physica A, 237, 229 (1997)] with coefficients depending on the velocity, that describes a generalized Brownian motion. This linear FP equation is shown to arise from a microscopic dynamics governed by a standard Langevin equation in presence of multiplicative noise.

cond-mat.stat-mech↗

arXiv1998

Nonextensive statistical effects in nuclear physics problems

Recent progresses in statistical mechanics indicate the Tsallis nonextensive thermostatistics as the natural generalization of the standard classical and quantum statistics, when memory effects and long-range forces are not negligible. In this framework, weakly nonextensive statistical deviations can strongly reduce the puzzling discrepancies between experimental data and theoretical previsions for solar neutrinos and for pion transverse-momentum correlations in Pb-Pb high-energy nuclear collisions.

nucl-th↗

arXiv1998

Thermal distributions in stellar plasmas, nuclear reactions and solar neutrinos

The physics of nuclear reactions in stellar plasma is reviewed with special emphasis on the importance of the velocity distribution of ions. Then the properties (density and temperature) of the weak-coupled solar plasma are analysed, showing that the ion velocities should deviate from the Maxwellian distribution and could be better described by a weakly-nonexstensive (|q-1|<0.02) Tsallis' distribution. We discuss concrete physical frameworks for calculating this deviation: the introduction of higher-order corrections to the diffusion and friction coefficients in the Fokker-Plank equation, the influence of the electric-microfield stochastic distribution on the particle dynamics, a velocity correlation function with long-time memory arising from the coupling of the collective and individual degrees of freedom. Finally, we study the effects of such deviations on stellar nuclear rates, on the solar neutrino fluxes, and on the pp neutrino energy spectrum, and analyse the consequences for the solar neutrino problem.

nucl-th↗

arXiv1998

Non-Markovian effects in the solar neutrino problem

The solar core, because of its density and temperature, is not a weakly-interacting or a high-temperature plasma. Collective effects have time scales comparable to the average time between collisions, and the microfield distribution influences the particle dynamics. In this conditions ion and electron diffusion is a non-Markovian process, memory effects are present and the equilibrium statistical distribution function differs from the Maxwellian one. We show that, even if the deviations from the standard velocity distribution that are compatible with our present knowledge of the solar interior are small, they are sufficient to sensibly modify the sub-barrier nuclear reaction rates. The consequent changes of the neutrino fluxes are comparable to the flux deficits that constitute the solar neutrino problem.

physics.plasm-ph↗

arXiv1998

Anomalous diffusion modifies solar neutrino fluxes

Density and temperature conditions in the solar core suggest that the microscopic diffusion of electrons and ions could be nonstandard: Diffusion and friction coefficients are energy dependent, collisions are not two-body processes and retain memory beyond the single scattering event. A direct consequence of nonstandard diffusion is that the equilibrium energy distribution of particles departs from the Maxwellian one (tails goes to zero more slowly or faster than exponentially) modifying the reaction rates. This effect is qualitatively different from temperature and/or composition modification: Small changes in the number of particles in the distribution tails can strongly modify the rates without affecting bulk properties, such as the sound speed or hydrostatic equilibrium, which depend on the mean values from the distribution. This mechanism can considerably increase the range of predictions for the neutrino fluxes allowed by the current experimental values (cross sections and solar properties) and can be used to reduce the discrepancy between these predictions and the solar neutrino experiments.

astro-ph↗

arXiv1997

Kinetic model for q-deformed bosons and fermions

We have studied the kinetics of $q$-deformed bosons and fermions, within a semiclassical approach. This investigation is realized by introducing a generalized exclusion-inclusion principle, intrinsically connected with the quantum $q$-algebra by means of the creation and annihilation operators matrix elements. In this framework, we have derived a non-linear Fokker-Planck equation for $q$-deformed bosons and fermions which can be seen as a time evolution equation, appropriate to consider non-equilibrium or near-equilibrium systems in a semiclassical approximation. The steady state of this equation reproduces in a simple mode the $q$-oscillators equilibrium statistics.

hep-th↗

arXiv1997

Non-Extensive Statistics and Solar Neutrinos

In this paper we will show that, because of the long-range microscopic memory of the random force, acting in the solar core, mainly on the electrons and the protons than on the light and heavy ions (or, equally, because of anomalous diffusion of solar core constituents of light mass and of normal diffusion of heavy ions), the equilibrium statistical distribution that these particles must obey, is that of generalized Boltzmann-Gibbs statistics (or the Tsallis non-extensive statistics), the distribution differing very slightly from the usual Maxwellian distribution. Due to the high-energy depleted tail of the distribution, the nuclear rates are reduced and, using earlier results on the standard solar model neutrino fluxes, calculated by Clayton and collaborators, we can evaluate fluxes in good agreement with the experimental data. While proton distribution is only very slightly different from Maxwellian there is a little more difference with electron distribution. We can define one central electron temperature as a few percent higher than the ion central temperature nearly equal to the standard solar model temperature. The difference is related to the different reductions with respect to the standard solar model values needed for $B$ and $CNO$ neutrinos and for $Be$ neutrinos.

astro-ph↗

arXiv1996

Non-Extensive Thermostatistical Approach of the Peculiar Velocity Function of Galaxy Clusters

We show that the observational data recently provided by Giovanelli et al. (1996 a, b) and discussed by Bahcall and Oh (1996) concerning the velocity distribution of clusters of galaxies can be naturally fitted by a statistical distribution which generalizes the Maxwell-Boltzmann one (herein recovered for the entropic index $q=1$). Indeed a recent generalization of the Boltzmann-Gibbs thermostatistics suggests for this problem that the probability function is given, within a simple phenomenological model, by \begin{eqnarray} P(>v) & = & \frac{ \int_{v}^{v_{max}} dv \left[ 1-(1-q)(v/v_0)^2 \right]^\frac{q}{1-q} } {\int_{0}^{v_{max}} dv \left[ 1-(1-q)(v/v_0)^2 \right]^\frac{q}{1-q}}, \nonumber \\ v_{max} & \equiv & \left\{ \begin{array}{ll} v_0 (1-q)^{-1/2} & \mbox{if $q<1$} \\ \infty & \mbox{if $q \geq 1$.} \end{array}\right. \nonumber \end{eqnarray} A remarkably good fitting with the data is obtained for $q=0.23^{+0.07}_{-0.05}$ and $v_0=490\pm 5$ km s$^{-1}$.

astro-ph↗

arXiv1996

Photofission and Quasi-Deuteron-Nuclear State as Mixing of Bosons and Fermions

The empirical-phenomenological quasi-deuteron photofission description is theoretically justified within the semiclassical, intermediate statistics model. The transmutational fermion (nucleon) - boson (quasi-deuteron) potential plays an essential role in the present context and is expressed in terms of thermodynamical and of microscopical quantities, analogous to those commonly used in the superfluid nuclear model.

nucl-th↗

arXiv1996

Generalized Statistics and Solar Neutrinos

The generalized Tsallis statistics produces a distribution function appropriate to describe the interior solar plasma, thought as a stellar polytrope, showing a tail depleted respect to the Maxwell-Boltzmann distribution and reduces to zero at energies greater than about $20 \, k_{_B} T$. The Tsallis statistics can theoretically support the distribution suggested in the past by Clayton and collaborators, which shows also a depleted tail, to explain the solar neutrino counting rate.

astro-ph↗

arXiv1996

Generalized Fractional Statistics

We link, by means of a semiclassical approach, the fractional statistics of particles obeying the Haldane exclusion principle to the Tsallis statistics and derive a generalized quantum entropy and its associated statistics.

hep-th↗

arXiv1995

Kinetic Approach to Fractional Exclusion Statistics

We show that the kinetic approach to statistical mechanics permits an elegant and efficient treatment of fractional exclusion statistics. By using the exclusion-inclusion principle recently proposed [Phys. Rev. E49, 5103 (1994)] as a generalization of the Pauli exclusion principle, which is based on a proper definition of the transition probability between two states, we derive a variety of different statistical distributions interpolating between bosons and fermions. The Haldane exclusion principle and the Haldane-Wu fractional exclusion statistics are obtained in a natural way as particular cases. The thermodynamic properties of the statistical systems obeying the generalized exclusion-inclusion principle are discussed.

hep-th↗