We present a comprehensive analytical study of the classical harmonic oscillator within the framework of Tsallis nonextensive statistics. Exact expressions for the thermodynamic properties — including internal energy U q , free energy F q , heat capacity C L,q , and entropy S q —are derived. The derived properties all reduce to their standard Boltzmann-Gibbs forms in the limit q → 1. A central result is that the entropic index q fundamentally alters the temperature scaling laws: the internal energy scales as U q ∝ T 2−q and the heat capacity follows C L,q ∝ T 1−q . Our results indicate that as q → 2, the system attains a temperature-independent entropy state, reflecting the presence of strong nonextensive constraints.” Our results demonstrate that the harmonic oscillator, a cornerstone model of physics, serves as an ideal prototype for understanding how nonextensivity modifies core thermodynamic behavior. Moreover, these results demonstrate that nonextensive statistics provide an essential generalization for modeling systems where standard thermodynamics fails, such as those with long-range interactions or nanoscale confinement.
Sandouqa et al. (Fri,) studied this question.