Boltzmann Equations for Spin and Charge Relaxations in Superconductors

In a superconductor coupled with a ferromagnetic metal, spin and charge imbalances can be induced by injecting spin-polarized electron current from the ferromagnetic metal. We theoretically study a nonequilibrium distribution of quasiparticles in the presence of spin and charge imbalances. We show that four distribution functions are needed to characterize such a nonequilibrium situation, and derive a set of linearized Boltzmann equations for them by extending the argument by Schmid and Sch\"{o}n based on the quasiclassical Green's function method. Using the Boltzmann equations, we analyze the spin imbalance in a thin superconducting wire weakly coupled with a ferromagnetic electrode. The spin imbalance induces a shift $\delta\mu$ ($- \delta \mu$) of the chemical potential for up-spin (down-spin) quasiparticles. We discuss how $\delta \mu$ is relaxed by spin-orbit impurity scattering.