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

TL;DR

This work analyzes the canonical ensemble within Tsallis statistics using the conventional (linear) expectation value and the equilibrium temperature $T_{\mathrm{eq}}$. It derives a probability representation where $p_i$ depends on energy through a $q$-power law, and expresses the energy, Tsallis entropy $S_q$, and heat capacity for a system of $N$ harmonic oscillators via Barnes zeta functions, with the distribution itself explicitly depending on $N$ and $q$. The results show that the energy $U$ and Rényi entropy $S_{Rq}$ as well as the heat capacity $C$ exhibit only weak dependence on $N$ and $q$, while $S_q$ retains $N$ and $q$ sensitivity; the probability distribution has power-law tails and remains sensitive to $N$ and $q$, highlighting a natural description of power-law phenomena. The methodology, grounded in $T_{\mathrm{eq}}$ rather than the Lagrange-multiplier temperature, provides a versatile framework that can describe a broad class of systems where nonextensive statistics and power-law behavior are relevant, with harmonic oscillators serving as a foundational testbed.

Abstract

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.

Figures (11)

• Figure 1: The scaled energies $u_{\mathrm{R}}$ and the scaled energies divided by $N$ ($u_{\mathrm{R}}/N$), as functions of $t_{\mathrm{eq}}$ at $q=0.98$ for $N=1, 5, 10, 15$, and $20$.
• Figure 2: The scaled energies $u_{\mathrm{R}}$ as functions of $t_{\mathrm{eq}}$ at $N=20$ for $q=0.96, 0.965, 0.97, 0.975$, and $0.98$.
• Figure 3: The Tsallis entropies $S_{q}$ and the Tsallis entropies divided by $N$ ($S_{q}/N$), as functions of $t_{\mathrm{eq}}$ at $q=0.98$ for $N=1, 5, 10, 15$, and $20$.
• Figure 4: The Tsallis entropies $S_{q}$ as functions of $t_{\mathrm{eq}}$ at $N=20$ for $q=0.96, 0.965, 0.97, 0.975$, and $0.98$.
• Figure 5: The Rényi entropies $S_{\mathrm{R}q}$ and the Rényi entropies divided by $N$ ($S_{\mathrm{R}q}/N$), as functions of $t_{\mathrm{eq}}$ at $q=0.98$ for $N=1, 5, 10, 15$, and $20$.