  The textbook thermophoretic force which acts on a body in a fluid is
proportional to the local temperature gradient. The same is expected to hold
for the macroscopic drift behavior of a diffusive cluster or molecule
physisorbed on a solid surface. The question we explore here is whether that is
still valid on a 2D membrane such as graphene at short sheet length. By means
of a non-equilibrium molecular dynamics study of a test system -- a gold
nanocluster adsorbed on free-standing graphene clamped between two temperatures
$\Delta T$ apart -- we find a phoretic force which for submicron sheet lengths
is parallel to, but basically independent of, the local gradient magnitude.
This identifies a thermophoretic regime that is ballistic rather than
diffusive, persisting up to and beyond a hundred nanometer sheet length.
Analysis shows that the phoretic force is due to the flexural phonons, whose
flow is known to be ballistic and distance-independent up to relatively long
mean-free paths. Yet, ordinary harmonic phonons should only carry crystal
momentum and, while impinging on the cluster, should not be able to impress
real momentum. We show that graphene, and other membrane-like monolayers,
support a specific anharmonic connection between the flexural corrugation and
longitudinal phonons whose fast escape leaves behind a 2D-projected mass
density increase endowing the flexural phonons, as they move with their group
velocity, with real momentum, part of which is transmitted to the adsorbate
through scattering. The resulting distance-independent ballistic thermophoretic
force is not unlikely to possess practical applications.
