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Fugacity versus chemical potential in nonadditive generalizations of the ideal Fermi-gas

We compare two approaches to the generalization of the ordinary Fermi-statistics based on the nonadditive Tsallis $q$-exponential used in the Gibbs factor instead of the conventional exponential function. Both numerical and analytical calculations are made for the chemical potential, fugacity, energy, and the specific heat of the ideal gas obeying such generalized types of statistics. In the approach based on the Gibbs factor containing the chemical potential, high temperature behavior of the specific heat significantly deviates from the expected classical limit, while at low temperatures it resembles that of the ordinary ideal Fermi-gas. On the contrary, when the fugacity enters as a multiplier at the Gibbs factor, the high-temperature limit reproduces the classical ideal gas correctly. At low temperatures, however, some interesting results are observed, corresponding to non-zero specific heat at the absolute zero temperature or a finite (non-zero) minimal temperature. These results, though exotic from the first glance, might be applicable in effective modeling of physical phenomena in various domains.