About the two spin-channel model for ferromagnetic excitations and spin-dependent heat transfer equations

The two spin-channel model is generalized to the case of transport of ferromagnetic excitations in electric conductors and insulators. The two channels are defined by reducing the ferromagnetic degrees of freedom to a bivaluated variable, i.e. to an effective spin one-half. The reduction is performed after defining the local magnetic configuration space by a sphere $Sigma_x$, and integrating the relevant physical quantities over the two hemispheres $Sigma_x^{uparrow}$ and $Sigma_x^{downarrow}$. The configuration space is then extended to the $x$ direction for non-uniform magnetization excitations. The transport equations for both magnetic moments and magnetic energy are deduced, including the relaxation from one channel to the other. The heat transport equations for ferromagnets is deduced.