Neutrino oscillations in inhomogeneous matter

An equation is derived for the evolution of the density matrix for neutrinos propagating, and mixing, in inhomogeneous matter. A measure of the decrease in the efficiency of the Mikheyev-Smirnov-Wolfenstein (MSW) transformation of ${\nu }_e$ to ${\nu }_x$ is defined, and is proved to increase steadily as the neutrino traverses a lumpy medium. It is estimated that, at the least, 1% density variations in the solar core on a scale of 1000 km would be required in order to disrupt the MSW predictions for the solar-neutrino problem. In the case of the supernova core, 0.01% variations on the scale of the c.m. system would be sufficient to alter drastically the results of scenarios which have been proposed.