This paper presents an inhomogeneous kinetic equation for mixed neutrinos, addressing the missing energy in flavor evolution. The traditional kinetic equations for neutrino flavor evolution do not conserve energy, as they fail to account for the exponential growth of small inhomogeneities that violate energy conservation. The authors show that this energy is instead traded with the large reservoir of neutrino kinetic energy through gradients of neutrino flavor coherence. They derive the missing gradient terms in the kinetic equation, demonstrating that energy conservation is violated in inhomogeneous settings but is restored when these terms are included. The paper begins by introducing the flavor evolution of neutrinos in a dense environment, described by a kinetic equation for the flavor density matrices. The traditional equations of motion (EOMs) are shown to not conserve energy, as they do not account for the exponential growth of inhomogeneities. The authors then derive the missing terms in the EOMs, showing that energy conservation is restored when these terms are included. They demonstrate this with a two-beam example, where the interaction energy changes dramatically as inhomogeneities grow. The solution shows that the energy is traded with the kinetic energy of neutrinos, and that the traditional EOMs are not consistent with energy conservation in inhomogeneous settings. The paper also discusses the derivation of the inhomogeneous kinetic equation, showing that the missing gradient terms are necessary to restore energy conservation. The authors show that the traditional EOMs are non-conservative in inhomogeneous settings, and that the missing gradient terms are necessary to restore energy conservation. They also show that the energy is traded with the kinetic energy of neutrinos, and that the traditional EOMs are not consistent with energy conservation in inhomogeneous settings. The paper concludes by discussing the implications of the missing energy in flavor evolution, showing that the traditional EOMs are non-conservative in inhomogeneous settings, and that the missing gradient terms are necessary to restore energy conservation. The authors show that the energy is traded with the kinetic energy of neutrinos, and that the traditional EOMs are not consistent with energy conservation in inhomogeneous settings. The paper also discusses the implications of the missing energy in flavor evolution, showing that the traditional EOMs are non-conservative in inhomogeneous settings, and that the missing gradient terms are necessary to restore energy conservation.This paper presents an inhomogeneous kinetic equation for mixed neutrinos, addressing the missing energy in flavor evolution. The traditional kinetic equations for neutrino flavor evolution do not conserve energy, as they fail to account for the exponential growth of small inhomogeneities that violate energy conservation. The authors show that this energy is instead traded with the large reservoir of neutrino kinetic energy through gradients of neutrino flavor coherence. They derive the missing gradient terms in the kinetic equation, demonstrating that energy conservation is violated in inhomogeneous settings but is restored when these terms are included. The paper begins by introducing the flavor evolution of neutrinos in a dense environment, described by a kinetic equation for the flavor density matrices. The traditional equations of motion (EOMs) are shown to not conserve energy, as they do not account for the exponential growth of inhomogeneities. The authors then derive the missing terms in the EOMs, showing that energy conservation is restored when these terms are included. They demonstrate this with a two-beam example, where the interaction energy changes dramatically as inhomogeneities grow. The solution shows that the energy is traded with the kinetic energy of neutrinos, and that the traditional EOMs are not consistent with energy conservation in inhomogeneous settings. The paper also discusses the derivation of the inhomogeneous kinetic equation, showing that the missing gradient terms are necessary to restore energy conservation. The authors show that the traditional EOMs are non-conservative in inhomogeneous settings, and that the missing gradient terms are necessary to restore energy conservation. They also show that the energy is traded with the kinetic energy of neutrinos, and that the traditional EOMs are not consistent with energy conservation in inhomogeneous settings. The paper concludes by discussing the implications of the missing energy in flavor evolution, showing that the traditional EOMs are non-conservative in inhomogeneous settings, and that the missing gradient terms are necessary to restore energy conservation. The authors show that the energy is traded with the kinetic energy of neutrinos, and that the traditional EOMs are not consistent with energy conservation in inhomogeneous settings. The paper also discusses the implications of the missing energy in flavor evolution, showing that the traditional EOMs are non-conservative in inhomogeneous settings, and that the missing gradient terms are necessary to restore energy conservation.