https://doi.org/10.1140/epjb/s10051-022-00309-w
Regular Article - Statistical and Nonlinear Physics
Thermodynamics of the independent harmonic oscillators with different frequencies in the Tsallis statistics in the high physical temperature approximation
Department of Domestic Science, Koriyama Women’s University, 963-8503, Koriyama, Fukushima, Japan
Received:
1
October
2021
Accepted:
2
March
2022
Published online:
17
March
2022
We study the thermodynamic quantities in the system of the N independent harmonic oscillators with different frequencies in the Tsallis statistics of the entropic parameter q (
) with escort average. The norm equations are derived, and the physical quantities are calculated with the physical temperature. It is found that the number of oscillators is restricted below 
. The energy, the Rényi entropy 
, and the Tsallis entropy 
are obtained by solving the norm equations approximately at high physical temperature and/or for small deviation 
. The energy is q-independent at high physical temperature when the physical temperature is adopted, and the energy is proportional to the number of oscillators and physical temperature at high physical temperature. The form of the Rényi entropy is similar to that of the von-Neumann entropy, and the Tsallis entropy is given through the Rényi entropy. The physical temperature dependence of the Tsallis entropy is different from that of the Rényi entropy. The Tsallis entropy is bounded from the above, while the Rényi entropy increases with the physical temperature. The ratio of the Tsallis entropy to the Rényi entropy is small at high physical temperature. The relation between the physical temperature 
and the temperature T (the inverse of the Lagrange multiplier) is obtained, and the quantity as a function of T and q can be obtained through . We calculate the free energy 
which is defined with and and the free energy 
which is defined with T and . The relation between 
and and the relation between 
and are shown.
© The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2022
