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

Abstract: Onsager reciprocity $L_{ij}=L_{ji}$ is a cornerstone of near-equilibrium thermodynamics derived from microscopic time-reversal symmetry. We develop a geometric framework in which entropy-weighted reparameterization of thermodynamic response functions leads to an effective asymmetry in cross-couplings without violating the microscopic Onsager theorem. Motivated by the parallel structure of heat capacities $C_p$ and $C_v$, we introduce entropy-weighted response variables $λ_p$, $λ_v$, $λ_s$, and $λ_t$. Their ratios $Γ_c=λ_p/λ_v=C_v/C_p$ and $Γ_m=λ_s/λ_t=κ_T/κ_S$ form thermodynamic invariants whose product equals unity in equilibrium. Within a differential-form representation of thermodynamic state space, equilibrium corresponds to exactness of the accessibility form $ω=λ_p\,dp+λ_v\,dv$ with $dω=0$, while non-equilibrium processes generate curvature $Ω=dω$, producing an effective asymmetry in the transformed coupling matrix. A microscopic theorem shows that entropy-weighted statistical ensembles with time-reversal asymmetry $χ(Γ)=W(ΘΓ)/W(Γ)\neq1$ generate an antisymmetric contribution to the Green--Kubo transport matrix. Atomic-scale analysis using the Transforma model reveals cross-derivative asymmetries across the $3d$ transition series, peaking at configuration anomalies in Cr and Cu. Temperature-dependent Raman spectroscopy of monolayer graphene exhibits statistically significant hysteresis loops (up to $30σ$), providing experimental evidence for thermodynamic curvature. These results unify microscopic irreversibility, atomic structure anomalies, and macroscopic hysteresis within a geometric interpretation of entropy-weighted thermodynamic coupling.