Hyperstatistical thermodynamics of the one-dimensional Klein-Gordon and Dirac oscillators: a closed-form q-generalized Boltzmann factor and a quantitative comparison with Beck's superstatistics

Abstract

We revisit the thermodynamics of the one-dimensional Klein-Gordon (KGO) and Dirac (DO) oscillators within two frameworks of generalized statistics: Beck's asymptotic superstatistics and the recently introduced hyperstatistics. In hyperstatistics, a $γ$-distribution of domain Boltzmann factors yields, after Laplace transformation and averaging over a normalisable density $f(β)$, the closed-form q-generalized Boltzmann factor $B_q(\varepsilon) = \exp_q(-\langleβ\rangle\varepsilon)$, independent of $f(β)$. We compute the partition function, entropy $S$, and specific heat $C_v$ for both 1D oscillators using excitation energies $\varepsilon_n = E_n - E_0$ to remove the rest-energy shift and enforce third-law behaviour $C_v \to 0$ as $T = 1/\langleβ\rangle \to 0$. Appropriate degeneracies ($g_n = 1$ for KGO; $g_0 = 1$, $g_n = 2$ for $n \geq 1$ for DO) are applied. Hyperstatistics successfully (i) reproduces the high-temperature Boltzmann limit $C_v \to 2k_B$, (ii) is structurally independent of $f(β)$, (iii) avoids the unphysical negative regions of the Beck polynomial bracket, and (iv) systematically distinguishes KGO from DO by capturing the enhanced entropy and sharper specific-heat structure caused by spin-induced degeneracy. The frameworks agree quantitatively for $q - 1 \ll 1$ and $\langleβ\rangle E \lesssim 2$, but diverge at high temperatures where Beck's polynomial expansion loses validity and the exact hyperstatistical q-exponential remains positive, monotonic, and analytic. Ultimately, hyperstatistics provides a numerically stable and analytically tractable alternative to asymptotic superstatistics for relativistic oscillators, naturally extensible to higher dimensions and external magnetic fields.