A NONEXTENSIVE ENTROPY APPROACH TOKAPPA-DISTRIBUTIONS

Letter to the Editor

MANFRED P. LEUBNERInstitute for Theoretical Physics, University of Innsbruck, Technikerstrasse 25, A-6020 Innsbruck,

Austria; E-mail: manfred.leubner@uibk.ac.at

(Received 18 April 2001; accepted 18 February 2002)

Abstract. Most astrophysical plasmas are observed to have velocity distribution functions exhibitingnon-Maxwellian suprathermal tails. The high energy particle populations are accurately representedby the family of kappa-distributions where the use of these distributions has been unjustly criticizedbecause of a perceived lack of theoretical justification. We show that distributions very close tokappa-distributions are a consequence of the generalized entropy favored by nonextensive statistics,which provides the missing link for power-law models of non-thermal features from fundamentalphysics. With regard to the physical basis supplied by the Tsallis nonextensive entropy formalismwe propose that this slightly modified functional form, qualitatively similar to the traditional kappa-distribution, be used in fitting particle spectra in the future.

1. Introduction

A variety of space observations indicate clearly the ubiquitous presence of supra-thermal particle populations in astrophysical plasma environments (Mendis andRosenberg, 1994). The family of kappa velocity space distributions, introducedfirst by Vasyliunas (1968), is recognized to be highly appropriate for modelingspecific electron and ion components of different plasma states.

Numerous magnetospheric interaction processes and instabilities were studiedsuccessfully within the concept of kappa-distributions (Xue et al., 1996; Chastonet al., 1997) ranging from plasma sheet ion and electron spectra (Christon et al.,1991) to a combined electron-proton-hydrogen atom aurora (Decker et al., 1995).The saturation of ring current particles towards a kappa-distribution was studiedby Lui and Rostoker (1995) and accurate fittings of observed electron flux spectrawere performed by a power law at high energies (Janhunen and Olsson, 1998). Fur-thermore, it was demonstrated by Leubner (2000a) that the Jovian banded whistlermode emission can be interpreted within a kappa-distribution approach and thatmirror instability thresholds are drastically reduced in suprathermal space plasmas(Leubner and Schupfer, 2000; Leubner and Schupfer, 2001). Energetic tail distri-butions play a key role in coronal plasma dynamics (Maksimovic et al., 1997) andhigh resolution plasma observations near 1 AU confirm that even the distribution

Astrophysics and Space Science 282: 573–579, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

574 M.P. LEUBNER

function of heavy solar wind ions are well fitted by a kappa-distribution (Collier etal., 1996).

The generation of velocity space distributions exhibiting pronounced energetictails was frequently interpreted as a consequence of several different accelerationmechanisms. Those include besides DC parallel electric fields or field-aligned po-tential drops in reconnection regions also wave-particle interaction due to kineticAlfvén wave turbulence (Hui and Seyler, 1992; Leubner, 2000b) and cyclotroninteractions (Gomberoff et al., 1995). Clear evidence of the importance of supra-thermal particle populations in space plasmas have motivated the developmentof a modified plasma dispersion function for kappa-distributions (Summers andThorne, 1991; Mace and Hellberg, 1995).

The use of the family of kappa-distributions to model the observed non-thermalfeatures of electron and ion structures was frequently criticized since a profoundderivation in view of fundamental physics was not available. A classical analysisaddressed to this problem was performed by Hasegawa et al. (1985) demonstratingthat kappa-distributions turn out as consequence of the presence of suprathermalradiation fields in plasmas and Collier (1993) considers the generation of kappa-like distributions using velocity space Lévy flights. Furthermore, a justification forthe formation of power-law distributions in space plasmas due to electron acceler-ation by whistler mode waves was proposed by Ma and Summers (1998; 1999a;1999b) and a kinetic theory was developed showing that kappa-like velocity spacedistributions are a particular thermodynamic equilibrium state (Treumann, 1999).Here we demonstrate that the family of kappa-distributions results as consequenceof the entropy generalization in nonextensive statistics, providing thus the missinglink for the use of kappa-distributions from fundamental physics.

Theory

A generalization of the Boltzmann-Gibbs–Shannon entropy formula for statisticalequilibruim was recognized to be required for systems subject to spatial or temporallong-range interactions making their behavior nonextensive. Any extensive form-alism fails whenever a physical system includes long-range forces or long-rangememory. In particular, this situation is usually found in astrophysical environmentsand plasma physics where, for example, the range of interactions is comparableto the size of the system considered. A generalized entropy is required to possessthe usual properties of positivity, equiprobability, concavity and irreversibility butsuitably extending the standard additivity to nonextensivity.

In view of the difficulties arising in this conjunction within the Boltzmann-Gibbs standard statistical mechanics motivated Tsallis (1988, 1995) to introduce athermo-statistical theory based on the generalized entropy of the form

Sq = kB

1 − �pq

i

q − 1(1)

A NONEXTENSIVE ENTROPY APPROACH TO KAPPA-DISTRIBUTIONS 575

where pi is the probability of the ith microstate, kB is the Boltzmann’s constant andq is a parameter quantifying the degree of nonextensivity and is commonly referredto as the entropic index. A crucial property of this entropy is the pseudoadditivitysuch that

Sq(A + B) = Sq(A) + Sq(B) + (1 − q)Sq(A)Sq(B) (2)

for given subsystems A and B in the sense of factorizability of the microstate prob-abilities. For q → 1 the standard Boltzmann-Gibbs-Shannon extensive entropy isrecovered as

Sq = −kB�pi ln pi. (3)

Applying the transformation

1

q − 1= −κ (4)

to Equation (1) and restricting to values −1 < q ≤ 1 yields the generalized entropyof the form

Sκ = κkB

(�p

1−1/κ

i − 1)

(5)

for 12 < κ ≤ ∞. With regard to Silva et al. (1998) one finds the corresponding one

dimensional equilibrium velocity space distribution in kappa notation as

f (v) = Aκ

[1 + 1

κ

v2

v2th

]−κ

(6)

where the normalization constant reads

Aκ = N

vth

1√κ

�[κ]�[κ − 1/2] . (7)

Here N denotes the particle density, vth = √2kBT /m is the thermal velocity

where T and m are the temperature and the mass, respectively, of the speciesconsidered. The case κ = ∞ corresponds to q = 1 wherefrom the Maxwellequilibrium distribution is recovered. Figure 1a illuminates schematically the non-thermal behavior of the distribution function (6) for some values of the spectralindex kappa.

Consistently, the one-dimensional equilibruim velocity space distribution in theq-nonextensive framework is written as

f (v) = Aq

[1 − (q − 1)

v2

v2th

]1/(q−1)

(8)

where

Aq = N

vth

√1 − q

�[1/(1 − q)]�[1/(1 − q) − 1/2] (9)

576 M.P. LEUBNER

Figure 1. Schematic plot of the family of kappa-distributions. The subplots (a) and (b) demonstratethe one-dimensional functional dependence of Equation (6) and (12), respectively. For both, κ = 2corresponds to the outermost curve followed by values κ = 3, 4, 6 and 10. The innermost curverepresents with κ = ∞ an isotropic Maxwellian.

for −1 < q ≤ 1. In analogy, the isotropic three dimensional velocity spacedistribution is found as

f (v) = Bκ

[1 + 1

κ

v2

v2th

]−κ

(10)

identical to the distribution function for a plasma in a suprathermal radiation fielddiscussed by Hasegawa (1985) where the normalization constant reads

Bκ = N

π3/2v3th

1

κ3/2

�[κ]�[κ − 3/2] (11)

for 32 < κ ≤ ∞. Equation (6) and (10) denote a reduced form of the standard

kappa-distribution used for astrophysical applications and written conventionallyas

f (v) = Aκ

[1 + 1

κ

v2

v2th

]−(κ+1)

(12)

with the normalization constant

Aκ = N

π3/2v3th

1

κ3/2

�[κ + 1]�[κ − 1/2] . (13)

A NONEXTENSIVE ENTROPY APPROACH TO KAPPA-DISTRIBUTIONS 577

For a variation of the spectral index kappa Figure 1 demonstrates the differ-ent behavior between the one dimensional case of the distribution function (12),generally used for space applications and modeling, as compared to Equation (6)resulting from the modified entropy approach. Since (6) and (12) differ only in theexponent no exact mapping is possible when substituting κ → κ +1. Nevertheless,highly similar structures are found for both with the appropriate values of kappa,hence favoring relation (6) for astrophysical velocity space distribution modelingdue to the physical background.

An effective thermal speed θ = vth

√(κ − 3/2)/κ is commonly defined from

the moments of the distribution function (12) where the spectral index kappa isconventionally limited to positive values κ > 3/2 and we note that negative valuesof kappa have never been considered for space plasma applications. Contrary, thegeneralization to negative values of kappa identical to q ≥ 1, discussed in theframework of nonextensive statistics as well, generates according to Equation (6)or (10) a thermal cutoff at the maximum allowed velocity

vmax =√

κ2kBT

m(14)

hence providing an additional physical interpretation of the spectral index kappaof non-thermal plasma structures. To our knowledge there is presently no observa-tional evidence for distributions corresponding to q ≥ 1 in space plasmas available.On the other hand it should be mentioned that any observed thermal cutoff invelocity distributions would not have been recognized or interpreted as being aconsequence of a possible nonextensivity of the system.

Finally we note that Collier (1995) has evaluated the Boltzmann entropy forkappa-distribution functions. Upon introducing the corresponding nonextensivevelocity space distribution the resulting kappa-dependent entropy is found to obeyqualitatively the same functional dependence. Since the suprathermal tails are morepronounced in the modified entropy case as compared to the standard kappa func-tion, see Figure 1, also the entropy enhancement for low values of the spectralindex, corresponding to a large fraction of suprathermal particle populations, is in-creased relatively to the traditional kappa-distribution case. Both approaches mergefor large values of kappa towards the Maxwellian limit.

Conclusions

Nonextensive statistics was successfully applied to a number of astrophysical andcosmological scenarios. Those include stellar polytropes (Plastino and Plastino,1993), the solar neutrino problem (Kaniadakis and Lavagno, 1996), peculiar velo-city distributions of galaxies (Lavagno et al., 1998) and generally systems withlong range interactions and fractal like space-times. Cosmological implications

578 M.P. LEUBNER

were discussed (Torres et al., 1997) and recently an analysis of plasma oscilla-tions in a collisionless thermal plasma was provided from q-statistics (Lima etal., 2000). On the other hand, kappa-distributions are highly favored in any kindof space plasma modeling (Mendis and Rosenberg, 1994, among others) where areasonable physical background was not apparent. A comprehensive discussion ofkappa distributions in view of experimentally favored non-thermal tail formationsis provided by Leubner and Schupfer (2000) where also typical values of the indexκ are quoted and referenced for different space plasma environments.

In the present analysis the missing link to fundamental physics is providedwithin the framework of an entropy modification consistent with nonextensivestatistics. The family of kappa distribution are obtained from the positive defin-ite part 1

2 ≤ κ ≤ ∞, corresponding to −1 ≤ q ≤ 1 of the general statisticalformalism where in analogy the spectral index kappa is a measure of the degreeof nonextensivity. Since the main theorems of the standard Maxwell-Boltzmannstatistics admit profound generalizations within nonextensive statistics (Plastino etal., 1994; Rajagopal, 1995, 1996; Chame and Mello, 1997; Lenzi et al., 1998), ajustification for the use of kappa-distributions in astrophysical plasma modeling isprovided from fundamental physics.

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