  This paper is concerned with equilibrium configurations of one-dimensional
particle system with non-convex nearest-neighbour and next-to-nearest-neighbour
interactions and its passage to the continuum. The goal is to derive
compactness results for a $\Gamma$-development of the energy with the novelty
that external forces are allowed. In particular, the forces may depend on
Lagrangian or Eulerian coordinates. Our result is based on a new technique for
deriving compactness results which are required for calculating the first-order
$\Gamma$-limit: instead of comparing a configuration of $n$ atoms to a global
minimizer of the $\Gamma$-limit, we compare the configuration to a minimizer in
some subclass of functions which in some sense are "close to" the
configuration. This new technique is required due to the additional presence of
forces with non-convex potentials. The paper is complemented with the study of
the minimizers of the $\Gamma$-limit.
