Solution of the dynamics of liquids in the large-dimensional limit

TL;DR

This paper derives exact analytical expressions for the dynamics of spherical particles in high-dimensional liquids, revealing a glass transition at high density and suggesting improvements to sphere packing bounds.

Contribution

It provides a novel, elementary derivation of liquid dynamics in large dimensions, connecting dynamic and thermodynamic properties and improving bounds on sphere packings.

Findings

Exact expressions for time correlation functions in high dimensions

Identification of a glass transition at high density

Higher glass transition density than previous bounds

Abstract

We obtain analytic expressions for the time correlation functions of a liquid of spherical particles, exact in the limit of high dimensions $d$. The derivation is long but straightforward: a dynamic virial expansion for which only the first two terms survive, followed by a change to generalized spherical coordinates in the dynamic variables leading to saddle-point evaluation of integrals for large $d$. The problem is thus mapped onto a one-dimensional diffusion in a perturbed harmonic potential with colored noise. At high density, an ergodicity-breaking glass transition is found. In this regime, our results agree with thermodynamics, consistently with the general Random First Order Transition scenario. The glass transition density is higher than the best known lower bound for hard sphere packings in large $d$. Because our calculation is, if not rigorous, elementary, an improvement in the bound for sphere packings in large dimensions is at hand.