  The spin Nernst effect is a phenomenon in which the spin current flows
perpendicular to a temperature gradient. Similar to the spin Hall effect, this
phenomenon also causes spin accumulation at the boundaries. Here, we study the
spin response to the gradient of the temperature gradient with the use of
Green's functions. Our formalism predicts physically observable spin
accumulation without the ambiguity regarding the definition of the spin current
or the magnetization correction. We prove the generalized Mott relation between
the electric and thermal responses assuming the presence of time-reversal
symmetry and the absence of inelastic scattering. We also find that thermal
spin accumulation vanishes for the three-dimensional Luttinger model but is
nonzero for the two-dimensional Rashba model with $\delta$-function nonmagnetic
disorder in the first Born approximation.
