Annals of Physics 342 (2014) 78–82

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Annals of Physics

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Implication of Tsallis entropy in theThomas–Fermi model for self-gravitatingfermionsKamel Ourabah, Mouloud Tribeche ∗

Plasma Physics Group, Theoretical Physics Laboratory, Faculty of Physics, University of Bab-Ezzouar,USTHB, B.P. 32, El Alia, Algiers 16111, Algeria

h i g h l i g h t s

• Thomas–Fermi approach for self-gravitating fermions.• A generalized Thomas–Fermi equation is derived.• Nonextensivity preserves a scaling property of this equation.• Nonextensive approach to Jeans’ instability of self-gravitating fermions.• It is found that nonextensivity makes the Fermionic system unstable at shorter scales.

a r t i c l e i n f o

Article history:Received 23 July 2013Accepted 27 November 2013Available online 4 December 2013

Keywords:Self-gravitating fermionsTsallis statistical mechanicsThomas–Fermi model

a b s t r a c t

The Thomas–Fermi approach for self-gravitating fermions isrevisited within the theoretical framework of the q-statistics.Starting from the q-deformation of the Fermi–Dirac distributionfunction, a generalized Thomas–Fermi equation is derived. It isshown that the Tsallis entropy preserves a scaling property of thisequation. The q-statistical approach to Jeans’ instability in a systemof self-gravitating fermions is also addressed. The dependence ofthe Jeans’ wavenumber (or the Jeans length) on the parameter q istraced. It is found that the q-statistics makes the Fermionic systemunstable at scales shorter than the standard Jeans length.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

The Thomas–Fermi (TF) model [1,2] was originally introduced in order to describe the electrondistribution and the Coulomb potential around the nucleus. It was used to describe in a semi-classical

∗ Corresponding author. Tel.: +213 21 24 73 44; fax: +213 21 24 73 44.E-mail address:mouloudtribeche@yahoo.fr (M. Tribeche).

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K. Ourabah, M. Tribeche / Annals of Physics 342 (2014) 78–82 79

way, large atoms, solids as well as astrophysical objects. The model consists simply to consider theelectrons as forming a gas of fermions, obeying the Fermi–Dirac distribution and occupying thephase space uniformly with two fermions per h3, under the effect of the Pauli principle. The Poissonequation for the electrostatic potential gives the Thomas–Fermi equation, describing self-consistentlythe variation of the potential.

Surprisingly, a similar model was introduced to describe systems of fermions interacting viagravitational forces [3]. Such systems are ubiquitous in a variety of astrophysical situations, such asneutrons in a neutron star or neutrinos inmodels of darkmatter. The Thomas–Fermi approach for suchsystems is based on the same hypothesis, except that the Poisson equation for the electrostatic fieldis now replaced by the Newton–Poisson equation (Newtonian limit of Einstein’s equation) describingthe gravitational field.

Though 85 years old, the TF model remains actual and is still subject to developments andnew mathematical refinements [4–7]. Among them, a generalization in the recently introducedframework of Tsallis statistical mechanics [8,5]. The latter is a generalization of the conventionalBoltzmann–Gibbs–Shannon (BGS) statistical mechanics, based on a one parameter generalization inthe concept of the entropy in a way that it becomes non-additive (see [9] and references therein foran actual view of the theory). The deformed distributions arising from the Tsallis entropy have beenrecently observed in a variety of systems. Among these experimental evidences, the distributioncharacterizing themotion of cold atoms in dissipative optical lattices [10] or the velocity distributionsin driven dissipative dusty plasma [11]. The Tsallis statistical mechanics seems to be the frameworkdescribing systems where the BGS statistics shows its limits: systems with long-range interactions,long time memory, systems evolving in a fractal space–time, or systems out of equilibrium. Systemsof self-gravitating fermions may present such characteristics.

This paper is dedicated to a generalization of the TF model of self-gravitating fermions in theframework of Tsallis statistical mechanics. From the mathematical point of view, this procedure isidentical to that already proposed byMartinenko and Shivamoggi in the atomic context [5]. However,onemay expect that the effects of the Tsallis q-statisticshave amore appreciable significance in the caseof gravitational interactions. In fact, the Coulombian interactions have an effective short-range effectdue to the Debye shielding, whereas the gravitational interactions are unshielded and have a long-range nature. Moreover, the thermodynamics of such systems shows some peculiar features, differentfromusual systems, such as a negative specific heat and an absence of global entropymaxima [12]. Theintroduction of the q-statistics for self-gravitating systems appears to be useful in many ways [13,14]and the fact that the gravity is a long-range force suggests that the self-gravitating system is one ofthe most preferable testing grounds for the framework of Tsallis q-statistics [12]. After a brief reviewof the TF model ideas and the Tsallis statistics, we will derive and discuss in the first section, theThomas–Fermi equation describing a system of fermions interacting via Newtonian gravity. In thesecond section, we will explore the effect of Tsallis entropyon the Jeans instability of self-gravitatingFermionic matter. Concluding remarks are presented in the last section.

2. Generalized Thomas–Fermi equation

The Tsallis statistical mechanics is based on a one parameter generalization of the Shannon entropy.The generalized q-entropy reads [8]

Sq = kB

1 −ipqi

q − 1(1)

where pi is the probability of the i-th microstate and q a real parameter measuring the degree of non-additivity of the entropy and then the correlations in the system. kB stands for the Boltzmann constantandwewill from now take it equal to unity. In the limit q → 1, the generalized entropy reduces to theconventional one and its additivity is recovered. The Tsallis q-entropy leads to deformed distributions.For fermions, the latter reads

f (q)(E) =1

1 + [1 +(q−1)

T [E − µ]]q/(q−1)with 1 +

(q − 1)T

[E − µ] > 0 (2)

80 K. Ourabah, M. Tribeche / Annals of Physics 342 (2014) 78–82

where µ stands for the chemical potential. It can be obtained by a maximization of the q-entropy (1)subject to constraints of conservation of the number of particles and total energy [15,5]. Of course, inthe limit q → 1, it reduces to the conventional Fermi–Dirac distribution function. Note also that in thezero-temperature limit, it reduces to theHeaviside step function corresponding to an ideal degenerateFermi gas, regardless the value of the entropic index q. The generalized Fermionic distribution (2) canbe used to generalize the Thomas–Fermi model to the Tsallis statistical mechanics regime. The densityis then obtained in terms of the Fermi–Dirac distribution. It reads

n (r) =gfh3

f (q)(E)d3p (3)

where gf stands for the spin degeneracy factor (gf = 2 for Majorana fermions and 4 for Diracfermions). The gravitational potential φ (r) satisfies the Newton–Poisson equation

▽2 φ = 4πGm2n (r) . (4)

The density (3) can be expressed as

n (r) =gf

4π2h3 (2mT )3/2 Θ(q)1/2

µ − φ ((r))

T

(5)

where

Θ(η)(q)n =

∞

0

yndy1 + [1 + (q − 1)β[y − η]]q/(q−1)

× H[1 + (q − 1)(y − η)] (6)

are the q-analogs of the Fermi–Dirac integrals (i.e., for q → 1, they reduce to the ordinary Fermi–Diracintegrals). These integrals have been evaluated by Martinenko and Shivamoggi in the atomic context,in the weakly nondegenerate case (i.e., |η| ≫ 1). They can be expanded as

Θ(q)n (η) ∼

1n + 1

ηn+1I(q)0 + ηnI(q)1 +n2ηn−1I(q)2 + · · · (7)

where

I(q)n = q

∞

−∞

zn[1 + (q − 1)z]1/(q−1)dz1 + [1 + (q − 1)z]q/(q−1)

2 × H[1 + (q − 1)(y − η)]. (8)

The integrals (8) can be estimated numerically (see [16,5] for a table of integrals for different valuesof the entropic index q). The integral I(q)0 is easily calculated and found to be equal to 1. Introducingthe normalized reduced potential and spatial variable

Φ =r

mGM⊙

(µ − φ)

x = r/R0 (9)

whereM⊙ is the solar mass, and

R0 =

3π

4√2m4gf G3/2M1/2

⊙

2/3

(10)

Eq. (4) becomes

1xd2Φdx2

= −32β−3/2Θ

(q)1/2

β

Φ

x

(11)

where β = T0/T and T0 = mGM⊙/R0. In the limit q → 1, the integral Θ(q)1/2 reduces to the ordinary

Fermi–Dirac integral and Eq. (11) leads to the usual Thomas–Fermi equation. In the zero-temperaturelimit, it reduces to the Lane–Emden differential equation [17], regardless the value of the entropic

K. Ourabah, M. Tribeche / Annals of Physics 342 (2014) 78–82 81

index q. It may be worth noting the interesting scaling property of Eq. (11): if Φ (x) is a solution of theequation at a defined temperature T and a cavity radius R, then Φ (x) = A3Φ (Ax) is also a solutionbut at the temperatureT = A4T and cavity radiusR = R/A. The ordinary Thomas–Fermi equationhas this property [18] and it is interesting to note that the effects of Tsallis entropy do not break thissymmetry.

3. Jeans instability for fermionic systems

The effect of q-statistics on Jeans’ instability has been studied recently for a classical idealgas [19,13], using the q-generalization of the Maxwellian distribution function [20]. To the best ofour knowledge, this problem has never been addressed for Fermionic systems. Let us consider thelinearized, collisionless Boltzmann equation [21]

∂ f1∂t

+ v∂ f1∂x

−∂φ0

∂x∂ f1∂v

−∂φ1

∂x∂ f0∂v

= 0 (12)

and the Newton–Poisson equation written in terms of the distribution f1

▽2 φ1 = 4πG

f1d3v. (13)

Following Jeans and Ref. [21], we assume the unperturbed initial state to be uniform and time-independent, i.e., φ0 = 0, and consider solutions of the form

f1 (x, v, t) = fa (v) exp [i (kv − ωt)] (14)φ1 (x, t) = φa (v) exp [i (kx − ωt)] . (15)

We then obtain the following dispersion relation

1 +4πGmk2

k ∂ f0∂v

kv − ωd3v = 0 (16)

where the distribution function for a gravitating system consisting of fermions, following q-statistics,can be written as

f0 =gfm3

h3

1

1 +

1 +

(q−1)T

p22m − µ

q/(q−1) . (17)

Assuming k along the z-axis, we have

k2 =8π2Gm4gf

h3

+∞

−∞

vzdvz

vz −ωk

1

1 +

1 + (q − 1)

mv2z2T − η

q/(q−1) (18)

where η = µ/T . The Jeans wavenumber is given byk(q)J

2= k2 (ω = 0) =

8√2π2Gm4gfh3

√m

Θ(q)−1/2 (η) . (19)

As is evident from Eq. (19), the Jeans wavenumber depends sensitively on the entropic index q. In thelimit q → 1, it reduces exactly to the Jeans number for a system of self-gravitating semi-degeneratefermions [21]. In the same way, we can define the Jeans wavelength as λ

(q)J = 2π/k(q)

J . For densityfluctuations having k ≺ k(q)

J (λ ≻ k(q)J ), the system will be gravitationally unstable.

In Fig. 1 the variation of α(q) = k(q)J /k(1)

J with respect to η in the weakly nondegenerate limit(i.e., making use of the expansion (7)) is shown. Fig. 1 shows that as the q-statistical characterof thefermions increases, the Jeans wavenumber increases (the Jeans wavelength is reduced). This meansthat q-statisticsmakes the Fermionic system unstable at scales shorter than the standard Jeans length.This is in agreement with the results in the case of a classical ideal gas [19]. As one may expect, in thezero-temperature limit (|η| → ∞), the q-statistics has little effect on the Jeans instability.

82 K. Ourabah, M. Tribeche / Annals of Physics 342 (2014) 78–82

Fig. 1. Plot of α(q) = k(q)J /k(1)

J with respect to η for different values of the entropic index q = 1.01 (solid line), 1.20 (dashedline), and 1.50 (dotted line).

4. Conclusion

In this paper, the Thomas–Fermi model for self-gravitating fermions is revisited within thetheoretical framework of the Tsallis statistical mechanics. The generalized Thomas–Fermi equationdescribing a system of fermions interacting via Newtonian-gravity is derived. It is shown that theTsallis q-statistics preserves a scaling property of the equation. The Jeans instability for a systemof self-gravitating fermions is also examined, in the framework of q-statistics. It is found that theJeans wavenumber is q-dependent. In the limit q → 1, the latter reduces to the Jeans wavenumbercorresponding to a system of semi-degenerate self-gravitating fermions. Due to q-statistics, theinstability condition is weakened and the system becomes unstable for wavelengths (wavenumbers)smaller (larger) than their standard counterpart.

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