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

TL;DR

This paper clarifies the physical meaning of the Kac prescription used to make long-range interacting systems extensive by analyzing scaling arguments for finite and infinite systems.

Contribution

It provides a clear physical interpretation of the Kac prescription, addressing its justification and implications in long-range systems.

Findings

Kac prescription rescales interactions to achieve extensivity

Physical interpretation differs between finite and infinite systems

Clarifies common confusions about energy scaling in long-range systems

Abstract

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.