Generalized Boltzmann-Gibbs Distribution and the Electronic Partition Function Paradox

TL;DR

The paper generalizes the Boltzmann-Gibbs framework by introducing a spectrum-dependent nonadditive entropy $S_{q,s}$ and a scale factor $k_s$ to produce finite, smooth thermodynamics for systems with unbounded spectra. Through constrained maximizing, it derives a $q$-exponential family $p_i \propto \exp_{2-q}(-\beta E_i)$ with $\beta$ set by energy constraints, recovering BG as $q\to1$. It applies the theory to the continuous-limit spectrum, the harmonic oscillator, the one-dimensional box, and the hydrogen atom, obtaining closed-form temperature-energy relations and a two-tier hydrogen model that circumvents the Electronic Partition Function Paradox; in particular, $q=0.5$ yields $c_V=k$ for hydrogen. The work highlights the critical role of $k_s$ in linking microscopic disorder to macroscopic temperature and discusses the need to constrain $q$ empirically for physical applicability.

Abstract

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 $k_{s}=k^{q}(e_{\max}/(W^σ-1))^{1-q}$, $σ>0$, 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 $β$ 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 $q\rightarrow1$.

Figures (7)

• Figure 1: Graphic depiction of $S_{q,s}(p_{*}(U))$, $U\in[0,1]$, $k=1$, for $q=0.2, 0.3, 0.8$, and the limiting Boltzmann-Gibbs (BG) case (i.e., $q\rightarrow 1$).
• Figure 2: Graphic depiction of the temperature and the internal energy for a system with an energy spectrum that uniformly approaches a continuous, unbounded limit with a common degeneracy, using Boltzmann-Gibbs entropy and $S_{q,s}$ with $q=0.6,0.8, 0.95$, for $k=1$.
• Figure 3: Graphic depiction of $\rho(e,T)$ in (\ref{['eqBGDistributionClassical2']}) and (\ref{['eqBGGeneralized']}) with $q=0.6,0.8, 0.95$, for $k=T=1$.
• Figure 4: Graphic depiction of the temperature and the internal energy for the harmonic oscillator using Boltzmann-Gibbs entropy and $S_{q,s}$ with $q=0.6,0.8, 0.95$, for $k=\hbar\omega=1$.
• Figure 5: Graphic depiction of the temperature and the internal energy for the one-dimensional box using Boltzmann-Gibbs entropy and $S_{q,s}$ with $q=0.6,0.8, 0.95$, for $k=\hbar^{2}\pi^{2}/2mL^{2}=1$.

Theorems & Definitions (3)

• Example 1
• Lemma 1
• proof : Proof of Lemma \ref{['lemmaEletronicPartition']}