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We prove a compactness and integral-representation theorem for sequences offamilies of lattice energies describing atomistic interactions defined onlattices with vanishing lattice spacing. The densities of these energies maydepend on interactions between all points of the corresponding latticecontained in a reference set. We give conditions that ensure that the limit isan integral defined on a Sobolev space. A homogenization theorem is alsoproved. The result is applied to multibody interactions corresponding todiscrete Jacobian determinants and to linearizations of Lennard-Jones energieswith mixtures of convex and concave quadratic pair-potentials.

An integral-representation result for continuum limits of discrete energies with multi-body interactions