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Flavor-dependent neutrino transport is described by a well-known kineticequation for occupation-number matrices in flavor space. However, as anoverlooked theoretical problem, we show that in the inhomogeneous case,neutrino-neutrino refractive energy is not conserved. We derive the missinggradient terms in the fast flavor limit (vanishing neutrino masses), and provethat the missing refractive energy is traded with the huge reservoir ofneutrino kinetic energy through gradients of the weak interaction potential.Even small changes of the kinetic energy accommodate the refractive energygained or lost. Flavor evolution alone is negligibly affected by the new terms.

Inhomogeneous Kinetic Equation for Mixed Neutrinos: Tracing the Missing Energy