I. INTRODUCTION Spintronics aims at using the information carried by the electrons’ spin in solids. For this purpose, establishing reliable methods to create, transfer and detect the spin current is an urgent task. Compared to the charge transports, spin transports have one serious fundamental difficulty. That is the non-conservation of the spin in solids. This limits the range of the spin transmission to be less than the spin diffusion length, which is typically µ m scale in metals. The non-conservation of the spins is expressed by a source term in the continuity equation for the spin Here s and j$_{s}$ are the spin density and spin current, respectively, α = x, y, z is the spin direction and T is the spin relaxation torque resulting in the non-conservation of the spin. In the most cases in metals, the dominant origin of T is the spin-orbit interaction. Although the relaxation torque term is essential in spin transports, it has so far been treated only on the phenomenological ground. The continuity equation is equivalent to the Boltzmann equation, which is useful in discussing the spin transports. The Boltzmann equation for the distribution function of each spin channel was discussed by Son et al. [1] and later by Valet and Fert [2] in the context of the giant magnetoresistance in multilayer systems. In their analysis, they approximated the spin relaxation torque as proportional to the inverse of a spin relaxation time τ$_{sf}$ and to some unknown function representing a driving force for the spin accumulation. The driving force was written in terms of what they called the spin chemical potential µ$_{s}$ . The relaxation torque was approximated as T z = µ$_{s}$/τ$_{sf}$ . They argued that µ$_{s}$ satisfies the diffusion equation, ∇ $^{2}$µ$_{s}$ = − ℓ − 2 sf µ$_{s}$ , with the diffusion length ℓ$_{sf}$ ∝ √ τ$_{sf}$ . Microscopic calculation for µ$_{s}$ has not been done so far. The diffusion equation for the spin has been widely used to discuss recent spin transports in metallic junctions [3]. The decay of spin transport has been confirmed in non-local spin injection experiments [4–6], which indicate that the spin diffusion decays with a decay length of 350-500nm in Cu and 100nm in Au at room temperature. Although the spin diffusion equation appears to be so far successful, the phenomenological treatment of the spin relaxation term and the spin chemical potential must be improved to consider the spin transport seriously. 2 ˙ s α + ∇ · j α s = T $^{α}$. (1)