The use of the Pearson differential equation to test energetic distributions in space physics as Kappa distributions; implication for Tsallis nonextensive entropy: II

Shizgal, Bernie D.

doi:10.1007/s10509-021-04033-2

Cited by 6 publications

(2 citation statements)

References 36 publications

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“…For p = 0, this is a hard sphere cross section for which the eigenvalues of the Fokker-Planck operator are known [49]. [50][51][52] and given by…”

Section: Model Systems That Yield Kappa Distributions and Distributio...mentioning

confidence: 99%

“…The increase of the high speed portion of the distribution without bound is evident and decreases with a decrease in E/n from curve a to curve d. In figure 7(B), we show the distributions for E/n = 0.001Td and different values of p with distributions increasing without bound especially for the larger p values. These nonequilibrium distributions result from the dynamical information in the Fokker-Planck equation and they cannot be derived from entropy functionals [33,52]. In figure 8, we show the behaviour of f x ln ss ( ( ))for E/n equal to (A) 0.001 Td and (B) 0.002 Td for decreasing values of p. The distribution function exhibit long 'tails' perhaps analogous to a Kappa distribution and there is no entropic principle for their calculation.…”

Section: Model Systems That Yield Kappa Distributions and Distributio...mentioning

confidence: 99%

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Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

Yin¹,

Shizgal

2023

Phys. Scr.

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The current paper considers the thermalization of an ensemble of electrons under the inﬂuence of an external electric ﬁeld and dilutely dispersed in one of the inert gas moderators, Argon, Krypton or Xenon for which the electron momentum transfer cross sections have deep Ramsauer-Townsend minima. As a consequence, the steady state electron distribution functions are bimodal over a small range of external electric ﬁeld strengths. The current work is directed towards the time evolution of the electron distribution function determined from the numerical solution of the Fokker-Planck equation. The kinetic theory of electrons dilutely dispersed in a heat bath of atoms at temperature Tb has a very long history. The solution of the Fokker-Planck equation can be expressed as a sum of exponentials of the form e−λnt where λn are the eigenvalues of the Fokker-Planck operator. Alternatively, a ﬁnite diﬀerence algorithm is used to solve the time dependent Fokker-Planck equation to give the time dependent electron energy distribution function. We demonstrate the evolution of the initial Maxwellian into a nonequilibrium bimodal distribution which cannot be rationalized with either the Gibbs-Boltzmann entropy or the Tsallis nonextensive entropy. Instead, the time dependent approach of an initial Maxwellian to the bimodal distribution is described in terms of the Kullback-Leibler entropy. We also demonstrate the inapplicability of the Boltzmann entropy nor the Tsallis entropy for a model system with a power law momentum transfer cross section of the form, σ(x) = σ0/xp, where x =with p = 2 is also employed to demonstrate a steady-state Kappa distribution which featuresqmev2/2kBTb is the reduced speed. This model prominently in space physics and other ﬁelds. For p > 2, we show distribution functions that increase without bound analogous to runaway electrons. The steady nonequilibrium distributions are interpreted as solutions of a Pearson ordinary diﬀerential equation.

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“…For p = 0, this is a hard sphere cross section for which the eigenvalues of the Fokker-Planck operator are known [49]. [50][51][52] and given by…”

Section: Model Systems That Yield Kappa Distributions and Distributio...mentioning

confidence: 99%

Section: Model Systems That Yield Kappa Distributions and Distributio...mentioning

confidence: 99%

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

Yin¹,

Shizgal

2023

Phys. Scr.

Self Cite

View full text Add to dashboard Cite

show abstract

Electric-field-dependent bimodal distribution functions for electrons in argon, xenon, and krypton owing to the Ramsauer-Townsend minima in the electron-atom momentum-transfer cross sections

Shizgal

2022

Phys. Rev. A

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Nonequilibrium distributions from the Fokker-Planck equation: Kappa distributions and Tsallis entropy

Oylukan,

Shizgal

2023

Phys. Rev. E

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The use of the Pearson differential equation to test energetic distributions in space physics as Kappa distributions; implication for Tsallis nonextensive entropy: II

Cited by 6 publications

References 36 publications

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

Electric-field-dependent bimodal distribution functions for electrons in argon, xenon, and krypton owing to the Ramsauer-Townsend minima in the electron-atom momentum-transfer cross sections

Nonequilibrium distributions from the Fokker-Planck equation: Kappa distributions and Tsallis entropy

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