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We consider a large class of interacting particle systems in 1D described byan energy whose interaction potential is singular and non-local. This classcovers Riesz gases (in particular, log gases) and applications to plasticityand approximation theory of functions. While it is well established that theminimisers of such interaction energies converge to a certain particle densityprofile as the number of particles tends to infinity, any bound on the rate ofthis convergence is only known in special cases by means of quantitativeestimates. The main result of this paper extends these quantitative estimatesto a large class of interaction energies by a different proof. The proof relieson one-dimensional features such as the convexity of the interaction potentialand the ordering of the particles. The main novelty of the proof is thetreatment of the singularity of the interaction potential by means of acarefully chosen renormalisation.

Quantitative estimate of the continuum approximations of interacting particle systems in one dimension