Multiscale Violation of Onsager Reciprocity: Thermomechanical Proof, Atomic Evidence, and Graphene Predictions

Onsager reciprocity (L_ij = L_ji) is a cornerstone of near-equilibrium thermodynamics, resting on microscopic reversibility and proper choice of conjugate forces and fluxes. We show that an entropy-weighted reparameterization of thermodynamic response functions—motivated by the parallel structure of C_p and C_v—leads to an effective coupling matrix whose cross-coefficients are generically asymmetric. This asymmetry is not a violation of Onsager's theorem but rather a geometric consequence of working in a transformed variable space where the usual force-flux pairing is modified. We unify three scales of analysis: (i) a rigorous proof that the ratios λ_p/λ_v = C_v/C_p and λ_s/λ_t = κ_T/κ_S are thermodynamic invariants, with their product I = (λ_p/λ_v)(λ_s/λ_t) = 1 in equilibrium; (ii) atomic-scale computational evidence from the Transforma model showing that analogous cross-derivative asymmetries appear in electronic structure, peaking at configuration anomalies (Cr, Cu); (iii) experimental predictions for graphene, where temperature-controlled measurements of Γ*(T) = λ_p/λ_v and hysteresis loops can probe the geometric structure directly. Raman measurements on monolayer graphene confirm reproducible hysteresis loops with areas up to 750 cm⁻¹·K (>30σ significance), providing the first experimental validation of entropy-weighted geometric curvature. The common thread is a geometric interpretation: equilibrium corresponds to flat geometry (d(dλ) = 0), while non-equilibrium processes manifest as curvature Ω = dω ≠ 0. Onsager reciprocity emerges as the flat-space limit of this entropy-weighted geometric framework.