Transport Determines Spectral Distance in Mesoscopic Thermodynamics

Abstract

We derive a distance between thermodynamic states from a positive Hermitian transport tensor within a mesoscopic canonical model. An exact first-order factorization of the evolution generator gives a commutator norm and the metric $G=2g^{-1}$, where $g$ denotes the symmetric transport tensor. Gradients of the antisymmetric tensor can change the evolution, but they leave this distance unchanged. The construction allows coordinate-dependent coefficients. For constant Onsager coefficients, a Clifford representation gives the same metric. In a two-channel thermoelectric system, the ratio of normalized principal distances is $\sqrt{1+ZT}+\sqrt{ZT}$. Transport coupling thus fixes the anisotropy of this metric. The construction requires neither an independent metric nor a physical spin variable.