The textbook thermophoretic force which acts on a body in a fluid is proportional to the local temperature gradient; the same could be assumed to hold for a diffusive nanometer sized object physisorbed on a 2D layer such as graphene.
By means of a Non-Equilibrium Molecular Dynamics (NEMD) study of a test system - a gold nanocluster adsorbed on free-standing graphene clamped between two different temperatures - we find a phoretic force which for relatively large submicron lengths L is parallel to, but roughly
independent of, the gradient magnitude. This signals a nonconventional thermophoresis that is ballistic in character. Analysis shows that the phoretic force is dominated by flexural phonons, whose flow is indeed known to be ballistic and distance-independent up to the relatively long
scattering lengths that precede the standard diffusive regime.
Interestingly, ordinary harmonic phonons only carry pseudomomentum and could not exert a force. Yet, the monolayer supports a specific anharmonic coupling between corrugation and 2D density which endow the flexural phonons with some real momentum, part of which is transmitted
to the adsorbate through scattering.