  We develop a theoretical formalism for the study of light-induced motion of
atoms trapped in a two-dimensional (2D) array, considering the effect of
multiple scattering of light between the atoms. We find that the atomic motion
can be described by a collective diffusion equation, wherein laser-induced
dipole-dipole forces couple the motion of different atoms. This coupling leads
to the formation of collective mechanical modes of the array atoms, whose
spatial structure and stability depend on the parameters of the illuminating
laser and the geometry of the 2D array. We demonstrate the application of our
formalism for the analysis of light-induced heating of the 2D array. The
presented approach should be useful for treating the optomechanical properties
of recently proposed quantum optical platforms made of atomic arrays.
