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We consider a class of particle systems which appear in various applicationssuch as approximation theory, plasticity, potential theory and space-fillingdesigns. The positions of the particles on the real line are described as aglobal minimum of an interaction energy, which consists of a nonlocal,repulsive interaction part and a confining part. Motivated by the applications,we cover non-standard scenarios in which the confining potential weakens as thenumber of particles increases. This results in a large area over which theparticles spread out. Our aim is to approximate the particle interaction energyby a corresponding continuum interacting energy. Our main results are bounds onthe corresponding energy difference and on the difference between the relatedpotential values. We demonstrate that these bounds are useful to problems inapproximation theory and plasticity. The proof of these bounds relies onconvexity assumptions on the interaction and confining potentials. It combinesrecent advances in the literature with a new upper bound on the minimizer ofthe continuum interaction energy.

Convergence rates for energies of interacting particles whose distribution spreads out as their number increases