Thermodynamics of the independent harmonic oscillators with different frequencies in the Tsallis statistics

Abstract: 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 $q$ ($1<q<2$) with escort average. The self-consistent equation is derived, and the physical quantities are calculated with the physical temperature. It is found that the number of oscillators is restricted below $1/(q-1)$. The energy, the R\'enyi entropy, and the Tsallis entropy are obtained by solving the self-consistent equation approximately at high physical temperature and/or for small deviation $q-1$. 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 R\'enyi entropy is similar to that of von-Neumann entropy, and the Tsallis entropy is given through the R\'enyi entropy. The physical temperature dependence of the Tsallis entropy is different from that of R\'enyi entropy. The Tsallis entropy is bounded from the above, while the R\'enyi entropy increases with the physical temperature. The ratio of the Tsallis entropy to the R\'enyi entropy is small at high physical temperature.