Long-range systems, (non)extensivity, and the rescaling of energies

Systems with long-range interactions have seen a surge of interest in the past decades. In the wake of this surge, the use of a system size dependent rescaling, sometimes termed "Kac prescription," of the long-range pair potential has seen widespread use. This ad hoc modification of the Hamiltonian makes the energy extensive, but its physical justification and implications are a frequent source of confusion and misinterpretation. After all, in real physical $N$-body systems, the pair interaction strength does not scale with the number $N$ of constituents. This article presents, at an introductory level, scaling arguments that provide a clear physical interpretation of the "Kac prescription" for finite systems as well as in the thermodynamic limit.