Description using equilibrium temperature in the canonical ensemble within the framework of the Tsallis statistics employing the conventional expectation value

We studied the thermodynamic quantities and the probability distribution, expressing the probability distribution as a function of the energy, in the canonical ensemble within the framework of the Tsallis statistics, which is characterized by the entropic parameter $q$, employing the conventional expectation value (the linear average). We treated the power-law-like distribution. The equilibrium temperature, which is often called the physical temperature, was employed, and the probability distribution described with the equilibrium temperature was derived. The Tsallis statistics represented by the equilibrium temperature was applied to $N$ harmonic oscillators, where $N$ is the number of the oscillators. The expressions of the energy, the Tsallis entropy, and the heat capacity were obtained. The expressions of these quantities and the expression of the probability distribution were obtained when the differences between adjacent energy levels are the same. These quantities and the distributions were numerically calculated. The $q$ dependences of the energy, the Rényi entropy, and the heat capacity are weak. In contrast, the Tsallis entropy depends on $q$. The probability distribution as a function of the energy depends on $N$ and $q$. The results provide a basis for describing power-law-like phenomena in the Tsallis statistics. The present formulation is expected to apply to various phenomena, because the harmonic oscillator plays a fundamental role in describing classical and quantum systems.

💡 Research Summary

The paper investigates the canonical ensemble of the Tsallis statistics when the conventional (linear) expectation value is employed, focusing on the so‑called Tsallis‑1 formulation. A central theme is the use of the equilibrium temperature (T_{\mathrm{eq}}) – often called the physical temperature – which is defined through the condition of thermal equilibrium between subsystems. By relating the Lagrange multiplier (\beta) to the derivative of the Tsallis entropy (S_q) with respect to the internal energy (U), the authors obtain the simple relation
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