  We consider a large class of interacting particle systems in 1D described by
an energy whose interaction potential is singular and non-local. This class
covers Riesz gases (in particular, log gases) and applications to plasticity
and approximation theory of functions. While it is well established that the
minimisers of such interaction energies converge to a certain particle density
profile as the number of particles tends to infinity, any bound on the rate of
this convergence is only known in special cases by means of quantitative
estimates. The main result of this paper extends these quantitative estimates
to a large class of interaction energies by a different proof. The proof relies
on one-dimensional features such as the convexity of the interaction potential
and the ordering of the particles. The main novelty of the proof is the
treatment of the singularity of the interaction potential by means of a
carefully chosen renormalisation.
