  The starting point of our analysis is a class of one-dimensional interacting
particle systems with two species. The particles are confined to an interval
and exert a nonlocal, repelling force on each other, resulting in a nontrivial
equilibrium configuration. This class of particle systems covers the setting of
pile-ups of dislocation walls, which is an idealised setup for studying the
microscopic origin of several dislocation density models in the literature.
Such density models are used to construct constitutive relations in plasticity
models.
  Our aim is to pass to the many-particle limit. The main challenge is the
combination of the nonlocal nature of the interactions, the singularity of the
interaction potential between particles of the same type, the non-convexity of
the the interaction potential between particles of the opposite type, and the
interplay between the length-scale of the domain with the length-scale $\ell_n$
of the decay of the interaction potential. Our main results are the
$\Gamma$-convergence of the energy of the particle positions, the evolutionary
convergence of the related gradient flows for $\ell_n$ sufficiently large, and
the non-convergence of the gradient flows for $\ell_n$ sufficiently small.
