The paper discusses the inhomogeneous kinetic equation for mixed neutrinos, focusing on the issue of missing energy conservation during fast flavor conversion. The authors identify a fundamental矛盾: the self-induced exponential growth of small inhomogeneities in flavor space strongly violates energy conservation. They derive the missing gradient terms in the kinetic equation, which account for the exchange of refractive energy with neutrino kinetic energy through gradients of flavor coherence. The usual equations remain sufficient to describe flavor evolution but do not conserve energy, which is crucial for understanding the final state of fast flavor conversions. The paper provides a detailed derivation of the new terms in the kinetic equation and shows that they are necessary to maintain energy conservation. The authors also demonstrate that the new equations conserve entropy, which is another important physical quantity. The findings have implications for the understanding of fast flavor conversions in dense neutrino environments, such as those occurring in stellar core collapse and binary neutron star mergers.The paper discusses the inhomogeneous kinetic equation for mixed neutrinos, focusing on the issue of missing energy conservation during fast flavor conversion. The authors identify a fundamental矛盾: the self-induced exponential growth of small inhomogeneities in flavor space strongly violates energy conservation. They derive the missing gradient terms in the kinetic equation, which account for the exchange of refractive energy with neutrino kinetic energy through gradients of flavor coherence. The usual equations remain sufficient to describe flavor evolution but do not conserve energy, which is crucial for understanding the final state of fast flavor conversions. The paper provides a detailed derivation of the new terms in the kinetic equation and shows that they are necessary to maintain energy conservation. The authors also demonstrate that the new equations conserve entropy, which is another important physical quantity. The findings have implications for the understanding of fast flavor conversions in dense neutrino environments, such as those occurring in stellar core collapse and binary neutron star mergers.