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Polytropes are self-gravitating fluid spheres used in astrophysics as crude approximation of more realistic stellar models. They possess equations that have scale parameters linked to mass, energy and entropy. Since Boltzmann distribution yields unphysical results, the use of generalized entropies, such as Tsallis and Kaniadakis entropies, had been proposed. Here we discuss how these entropies are related in polytrope solutions.
Tsallis and Kaniadakis Entropic Measures in Stellar Polytropes
Amelia Carolina Sparavigna
Department of Applied Science and Technology, Politecnico di Torino, Torino, Italy
Abstract: Polytropes are self-gravitating fluid spheres used in astrophysics as crude approximation of more realistic stellar models. They possess equations that have scale parameters linked to mass, energy and entropy. Since Boltzmann distribution yields unphysical results, the use of generalized entropies, such as Tsallis and Kaniadakis entropies, had been proposed. Here we discuss how these entropies are related in polytrope solutions.
Keywords: Entropy, Generalized Entropies.
In astrophysics, a polytrope refers to a solution of the Lane–Emden equation. This is an equation which gives the pressure as a function of density . These solutions are modelling self-gravitating fluid spheres that are called “polytropes” too, which are objects used as crude approximation to more realistic stellar models . The solution of the Lane-Emden equation, a dimensionless form of Poisson's equation for the gravitational potential, depends on a parameter which is the polytropic index n. It is written as P=Kρ(n+1)/n, where P is pressure, ρ is density and K a constant. If stellar structure is approximated with a polytrope having a given index, then two scaling parameters are needed to express the structure in physical units . The two parameters that we can use are a constant which is related to entropy and the stellar mass. Since Boltzmann distribution yields unphysical results, the Boltzmann entropy had been substituted by a generalized entropy, the Tsallis entropy . Another generalized entropy, the Kaniadakis entropy, had been recently proposed too, in . Here we discuss how these two entropies are related in polytrope solutions, and that the result given in  can be easily obtained from .
2. The entropies
Well-known is the entropy proposed by Claude Shannon in 1948 . He defined the entropy H of a discrete random variable X, as the expected value of the information content: H(X)= −Σi pi logb pi . The probability of i-event is pi and b is the base of the used logarithm. However, several entropies exist which are generalizing Shannon entropy. Among them we have Tsallis and Kaniadakis entropies [7,8], which are defined, with a corresponding choice of measurement units equal to 1, as follow:
In (1) and (2) we have the entropic indies q and κ. For its generalized additivity, the Kaniadakis entropy requires another function, defined as follow:
A detailed discussion of the generalized additivity of Tsallis and κ-entropy is given in . Tsallis and Kaniadakis entropies are linked:
Eq.(3) is a simpler form of an expression given in [10,11]. However, besides this relation, because of the generalized additivity possessed by the Kaniadakis entropy, we need also another relation:
In (3) and (4), we have Kaniadakis functions expressed by Tsallis entropy. As shown in , we can also write T expressed by means of Kaniadakis functions:
Let us have: κ=1−q. From (5) we have immediately the relation between Tsallis and Kaniadakis functions:
3. With polytropes
The relation (6) between Tsallis and Kaniadakis entropies can be useful in several problems. Here we consider its use in polytropes. In the previous equations, we have pi denoting the probability distribution. In Ref.4, it is used letter f for probability. From now on, we will use this notation. In , the distribution from Tsallis entropy is:
After Eq.6, we can write Eq.7 in the following manner:
Of course, (7) and (8) are the same equation. As a consequence, Kaniadakis distributions are linked to Tsallis distribution by:
From , a relation exists between polytrope index and entropic Tsallis index:
As a special case, for q→1, we find the isothermal situation. To have Eq.6, as shown in , we need κ=1−q or κ=q−1. Then, from (10), considering that we have for the Kaniadakis index, −1<κ<1:
And in fact, (11) is the relation that we find in . Using then the relation between Tsallis and Kaniadakis entropies and distributions we can easily finds results concerning several applications. Polytropes are an example of such a possible approach.
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Amelia Carolina Sparavigna, ORCID 0000-0003-4502-8974
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Sparavigna, A. (2015). Tsallis and Kaniadakis Entropic Measures in Stellar Polytropes. PHILICA.COM Article number 541.
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Reference 12 is published. Here the details:
Amelia Carolina Sparavigna (2015). Relations Between Tsallis and Kaniadakis Entropic Measures and Rigorous Discussion of Conditional Kaniadakis Entropy, International Journal of Sciences 4(10):47-50 DOI: 10.18483/ijSci.866