The Entropic Transport Identity: Transport Coefficients as Entropy Curvature in BKM-Selfadjoint Quantum Markov Semigroups

We prove the Entropic Transport Identity (ETI): for any primitive, BKM-selfadjoint Lindbladian on a finite-dimensional matrix algebra with a simple spectral gap Δ, the Green–Kubo transport coefficient of an observable J in the slow-mode eigenspace satisfies
κ(J) = ‖J‖²_BKM / Δ
where ‖J‖²_BKM is the Bogoliubov–Kubo–Mori norm. Under the same hypotheses, the asymptotic relative-entropy production rate satisfies Σ̇ = 2Δ, yielding the entropy form κ(J) · Σ̇ = 2‖J‖²_BKM: transport coefficient times entropy production rate equals twice the BKM norm squared.
The identity is model-agnostic (holds for any primitive Lindbladian satisfying quantum detailed balance), parameter-free (no scattering time is introduced), and exact in finite dimension. The thermodynamic limit is controlled via Γ-convergence of Dirichlet forms and Mosco convergence of resolvents, establishing uniform spectral gap stability.
Physical content: transport is entropy curvature. The Green–Kubo coefficient of any observable measures the curvature of relative entropy in the direction of the slow mode. Electrical resistance, thermal conductivity, and viscosity are all instances of the same geometric quantity — the BKM spectral gap — viewed through different observables.
The Fr öhlich electron-phonon model is treated as an explicit example. Spectral gap stability for the linearized collision operator is proved via Dirichlet form coercivity (angular and radial mechanisms), closing the thermodynamic limit without open assumptions.