Kaniadakis logistic distribution explained

The Kaniadakis Logistic distribution (also known as κ-Logisticdistribution) is a generalized version of the Logistic distribution associated with the Kaniadakis statistics. It is one example of a Kaniadakis distribution. The κ-Logistic probability distribution describes the population kinetics behavior of bosonic (

0 < λ < 1 ) or fermionic (λ > 1 ) character.

Definitions

Probability density function

The Kaniadakis κ-Logistic distribution is a four-parameter family of continuous statistical distributions, which is part of a class of statistical distributions emerging from the Kaniadakis κ-statistics. This distribution has the following probability density function:^[1]

f _{_{\kappa}} (x) = (λ \alpha \beta x ^{\alpha-1})/(\sqrt{ 1+ \kappa ^{2} \beta ^{2} x ^{2\alpha}) } \frac

valid for

x \geq 0 , where 0 \leq |\kappa| < 1 is the entropic index associated with the Kaniadakis entropy, \beta > 0 is the rate parameter, λ > 0 , and \alpha > 0 is the shape parameter.

The Logistic distribution is recovered as

\kappa → 0 .

Cumulative distribution function

The cumulative distribution function of κ-Logistic is given by

F _{\kappa(x)} = (1- \exp _{\kappa(-\beta} x ^{\alpha)})/(1+ ( λ-1 ) \exp _{\kappa(-\beta} x ^{\alpha)})

valid for

x \geq 0 . The cumulative Logistic distribution is recovered in the classical limit \kappa → 0 .

Survival and hazard functions

The survival distribution function of κ-Logistic distribution is given by

S _{\kappa(x)} = (λ)/(\exp _{\kappa(\beta} x ^{\alpha)} +λ-1)

valid for

x \geq 0 . The survival Logistic distribution is recovered in the classical limit \kappa → 0 .

The hazard function associated with the κ-Logistic distribution is obtained by the solution of the following evolution equation:

(S _{\kappa(x)})/(dx) = - h _{\kappa} S _{\kappa(x)} \left( 1 - (λ-1)/(λ) S _{\kappa(x)} \right)

with S _{\kappa(0)} = 1 , where h _{\kappa} is the hazard function:h _{\kappa} = (\alpha \beta x ^{\alpha-1})/(\sqrt{ 1+ \kappa ^{2} \beta ^{2} x ^{2\alpha}) }

The cumulative Kaniadakis κ-Logistic distribution is related to the hazard function by the following expression:

S _{\kappa} = e ^{- H _{\kappa(x)}}

where

H _{\kappa} (x) = \int ^{x} _{0} h _{\kappa(z)} dz is the cumulative hazard function. The cumulative hazard function of the Logistic distribution is recovered in the classical limit \kappa → 0 .

Related distributions

• The survival function of the κ-Logistic distribution represents the κ-deformation of the Fermi-Dirac function, and becomes a Fermi-Dirac distribution in the classical limit

\kappa → 0 .

• The κ-Logistic distribution is a generalization of the κ-Weibull distribution when

λ = 1 .

• A κ-Logistic distribution corresponds to a Half-Logistic distribution when

λ = 2 , \alpha = 1 and \kappa = 0 .

• The ordinary Logistic distribution is a particular case of a κ-Logistic distribution, when

\kappa = 0 .

Applications

The κ-Logistic distribution has been applied in several areas, such as:

• In quantum statistics, the survival function of the κ-Logistic distribution represents the most general expression of the Fermi-Dirac function, reducing to the Fermi-Dirac distribution in the limit

\kappa → 0 .^{[2] [3] [4]}

See also

• Giorgio Kaniadakis
• Kaniadakis statistics
• Kaniadakis distribution
• Kaniadakis κ-Exponential distribution
• Kaniadakis κ-Gaussian distribution
• Kaniadakis κ-Gamma distribution
• Kaniadakis κ-Weibull distribution
• Kaniadakis κ-Erlang distribution

External links

• Kaniadakis Statistics on arXiv.org

Notes and References

1. Kaniadakis . G. . 2021-01-01 . New power-law tailed distributions emerging in κ-statistics (a) . Europhysics Letters . 133 . 1 . 10002 . 10.1209/0295-5075/133/10002 . 0295-5075. 2203.01743 . 2021EL....13310002K . 234144356 .
2. Santos . A.P. . Silva . R. . Alcaniz . J.S. . Anselmo . D.H.A.L. . 2011 . Kaniadakis statistics and the quantum H-theorem . Physics Letters A . en . 375 . 3 . 352–355 . 10.1016/j.physleta.2010.11.045. 2011PhLA..375..352S . free .
3. Kaniadakis . G. . 2001 . H-theorem and generalized entropies within the framework of nonlinear kinetics . Physics Letters A . en . 288 . 5–6 . 283–291 . 10.1016/S0375-9601(01)00543-6. cond-mat/0109192 . 2001PhLA..288..283K . 119445915 .
4. Lourek . Imene . Tribeche . Mouloud . 2017 . Thermodynamic properties of the blackbody radiation: A Kaniadakis approach . Physics Letters A . en . 381 . 5 . 452–456 . 10.1016/j.physleta.2016.12.019. 2017PhLA..381..452L .