Visnyk of the Lviv University. Series Physics
56 (2019) ñ. 65-75
DOI: https://doi.org/10.30970/vph.56.2019.65 ## Superadditive model of the ideal Fermi-gas near absolute zeroB. Sobko, A. Rovenchak |

Nonadditive statistics can be used to describe various systems, for instance, in fractional structures and systems with long-range interactions. In this paper, we focus on finding the low-temperature limits of thermodynamic functions of a system obeying a nonadditive modification of the Fermi-statistics.
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For integrity, we briefly show how the Tsallis entropy is introduced by generalizing classical Boltzmann--Gibbs entropy. In the case of a system, which consists of two subsystems, unlike the additive entropy in the classical sense, the Tsallis entropy satisfies the relation S_{q} (A + B) = S_{q} (A) + S_{q} (B) + (1-q) S_{q} (A) S_{q} (B), hence, is a non-additive quantity. For q>1 we have, therefore, S_{q} (A + B) < S_{q} (A) + S_{q} (B), so in this case the Tsallis entropy is called \textit{subadditive}, and for q<1 the respective inequality becomes S_{q} (A + B)> S_{q} (A) + S_{q} (B) , which corresponds to a \textit{superadditive} entropy. It is this case that we will consider in this article.
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In the low temperature limit we have obtained the expansion of energy per particle as a series over temperature T as follows: E/N=E_{0}/N+\alpha(q,s)T+\mathcal{O}(T^{2}). The analytical expression for the coefficient \alpha(q, s), which is a factor at T in the energy expansion, is obtained. Here, s is the power in the energy density g(\eps)\sim \eps^{s-1}. Note that \alpha(q, s) does not depend on system parameters. Obviously, this factor is nothing but the specific heat of the system in the limit of T\to0.
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The temperature dependences of the specific heat are calculated for different parameters of nonadditivity q and the power s in the density of states. For all values in the range of q_{\rm min}V to a certain value \alpha(q,s) different from zero. Such a violation of the third law of thermodynamics is known for nonadditive systems. As q approaches unity, \alpha(q, s) tends to zero linearly in (1-q).
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The described properties of superadditive modification of an ideal Fermi gas are not only of academic interest, but also provide grounds for using this model in effective description of some physical problems, in particular in the theory of magnetic systems and cosmology.

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