Spin waves (or magnons) interact with magnetic domain walls (DWs) in a
complicated way that a DW can propagate either along or against magnon flow.
However, thermally activated magnons always drive a DW to the hotter region of
a nanowire of magnetic insulators under a temperature gradient. We
theoretically illustrate why it is surely so by showing that DW entropy is
always larger than that of a domain as long as material parameters do not
depend on spin textures. Equivalently, the total free energy of the wire can be
lowered when the DW moves to the hotter region. The larger DW entropy is
related to the increase of magnon density of states at low energy originated
from the gapless magnon bound states
