  We explore entanglement generation between multiple optically levitated
nanospheres interacting with a common optical cavity via the Coherent
Scattering optomechanical interaction. We derive the many-particle Hamiltonian
governing the unitary evolution of the system and show that it gives rise to
quantum correlations among the various partitions of the setup, following a
non-Markovian dynamics of entanglement birth, death and revivals. We also
consider the effects of coupling the system to external environments and show
that under reasonable experimental conditions entanglement between the
mechanical modes can survive even at room temperature. Its dependence upon the
number of nanoparticles, their initial temperature and coupling strength is
studied. A numerical toolbox to simulate the closed and open dynamics of
Gaussian optomechanical states and their informational measures is developed.
