2026/06/05 by Abdelmalek Boumali · 1 voice
Physics and Astronomy · #physics.gen-ph
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 Bq(ε) = expq(-⟨β⟩ε), independent of f(β). We compute the partition function, entropy S, and specific heat Cv for both 1D oscillators using excitation energies εn = En - E0 to remove the rest-energy shift and enforce third-law behaviour Cv → 0 as T = 1/⟨β⟩ → 0. Appropriate degeneracies (gn = 1 for KGO; g0 = 1, gn = 2 for n ≥ 1 for DO) are applied. Hyperstatistics successfully (i) reproduces the high-temperature Boltzmann limit Cv → 2kB, (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 ≪ 1 and ⟨β⟩ 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.