Generalized Boltzmann-Gibbs Distribution and the Electronic Partition Function Paradox

This paper generalizes the entropy maximization problem leading to the Boltzmann-Gibbs distribution through the nonadditive entropy $S_{q,s}(p)=k_{s}\sum^{W}_{i\geq1}p_{i}\ln_{q}1/p_{i}$, $q\in(0,1)$, which is a rescaled version of $S_{q}$ \cite{Tsallis1988} by a factor $k_{s}=k^{q}(e_{\max}/(W^{\sigma}-1))^{1-q}$, $\sigma>0$, varying according to the underlying energy spectrum and satisfying $k_{s}\rightarrow k$ (Boltzmann constant) as $q\rightarrow 1$. The maximization problem based on $S_{q,s}$ is used to derive analytical generalizations of the Boltzmann-Gibbs distribution for the case of an energy spectrum that uniformly approaches a continuous, unbounded limit with a common degeneracy, the harmonic oscillator, and the one-dimensional box. Furthermore, I demonstrate that this generalized problem yields a two-tier model with finite structural parameters $\beta$ for the hydrogen atom in free space and, therefore, can be used to circumvent the Electronic Partition Function Paradox and obtain a family of well-defined thermodynamic behaviors indexed by $q\in(0,1)$. In particular, for $q=0.5$, the specific heat of the free hydrogen atom becomes the Boltzmann constant. Finally, it is shown that all the limiting processes involved in these four cases lead naturally to the same definition of the scale factor $k_{s}$ that characterizes $S_{q,s}$ (``s'' stands, in this context, for ``spectrum'') in order to grant finite, smooth, macroscopically observable temperature values that are related to the entropic functional by $\partial S_{q,s}/\partial U=1/\vert T\vert^{q}_{\pm}$, which recovers $\partial S_{BG}/\partial U=1/T$ \cite{Clausius1865} as $q\rightarrow1$.