  We reveal a dramatic departure of electron thermodiffusion in solids relative
to the commonly accepted picture of the ideal free-electron gas model. In
particular, we show that the interaction with the lattice and impurities,
combined with a strong material dependence of the electron dispersion relation,
leads to counterintuitive diffusion behavior, which we identify by comparing a
single-layer two-dimensional electron gas (2DEG) and graphene. When subject to
a temperature gradient $\nabla T$, thermodiffusion of massless Dirac electrons
in graphene exhibits an anomalous behavior with electrons moving along $\nabla
T$ and accumulating in hot regions, in contrast to normal electron diffusion in
a 2DEG with parabolic dispersion, where net motion against $\nabla T$ is
observed, accompanied by electron depletion in hot regions. These findings have
fundamentally importance for the understanding of the spatial electron dynamics
in emerging material, establishing close relations with other branches of
physics dealing with electron systems under nonuniform temperature conditions.
