  We derive the thermomagnonic torque associated with smooth magnetic textures
subjected to a temperature gradient, in the framework of the stochastic
Landau-Lifshitz-Gilbert equation. Our approach captures on equal footing two
distinct contributions: (1) A local entropic torque that is caused by a
temperature dependence of the effective exchange field, the existence of which
had been previously suggested based on numerics and (2) the well-known
spin-transfer torque induced by thermally-induced magnon flow. The dissipative
components of two torques have the same structure, following a common
phenomenology, but opposite signs, with the twice larger entropic torque
leading to a domain-wall motion toward the hotter region. We compare the
efficiency of the torque-driven domain-wall motion with the recently proposed
Brownian thermophoresis.
