Approximate scale invariance in particle systems: A large-dimensional justification

¹ Laboratoire de Physique Théorique, École Normale Supérieure, PSL Research University, CNRS, Université Pierre et Marie Curie, Sorbonne Universités - 24 rue Lhomond, 75005 Paris, France
² Laboratoire de Physique Statistique, École Normale Supérieure, PSL Research University, CNRS, Université Pierre et Marie Curie - Sorbonne Universités, Université Paris Diderot, Sorbonne Paris Cité - 24 rue Lhomond, 75005 Paris, France

^(a) thibaud.maimbourg@lpt.ens.fr
^(b) kurchan.jorge@lps.ens.fr

Received: 9 May 2016
Accepted: 21 June 2016

Abstract

Systems of particles interacting via inverse-power law potentials have an invariance with respect to changes in length and temperature, implying a correspondence in the dynamics and thermodynamics between different “isomorphic” sets of temperatures and densities. In a recent series of works, it has been argued that such correspondences hold to a surprisingly good approximation in a much more general class of potentials, an observation that summarizes many properties that have been observed in the past. In this paper we show that such relations are exact in high-dimensional liquids and glasses, a limit in which the conditions for these mappings to hold become transparent. The special role played by the exponential potential is also confirmed.