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 Rényi 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 Rényi entropy is similar to that of 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 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.

arXiv comment8 pages

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• Thermodynamic relations and fluctuations in the Tsallis statistics
• A harmonic oscillator in nonadditive statistics and a novel transverse momentum spectrum in high-energy collisions
• Relation between the escort average in microcanonical ensemble and the escort average in canonical ensemble in the Tsallis statistics
• Ising model in the Rényi statistics: the finite size effects