  We predict a thermodynamic magnon recoil effect for domain wall motions in
the presence of temperature gradients. All current thermodynamic theories
assert that a magnetic domain wall must move toward the hotter side, based on
equilibrium thermodynamic arguments. Microscopic calculations on the other hand
show that a domain wall can move either along or against the direction of heat
currents, depending on how strong the heat currents are reflected by the domain
wall. We have resolved the inconsistency between these two approaches by
augmenting the theory in the presence of thermal gradients by incorporating in
the free energy of domain walls by a heat current term present in
nonequilibrium steady states. The condition to observe a domain wall
propagation toward the colder regime is derived analytically and can be tested
by future experiments.
