Glass transition of hard spheres in high dimensions

TL;DR

This paper analyzes the liquid-glass transition of hard spheres in high dimensions using mode-coupling theory, revealing non-Gaussian behavior, differences in nonergodicity parameters, and a critical packing fraction that scales with dimension.

Contribution

It provides the first detailed analytical and numerical study of the glass transition of hard spheres in the limit of infinite dimensions, highlighting new scaling behaviors and critical parameters.

Findings

Non-Gaussian k-dependence of nonergodicity parameters up to d=800

Difference between self and collective nonergodicity parameters at k~d^{1/2}

Critical packing fraction scales as d^2 2^{-d} with a quadratic pre-factor

Abstract

We have investigated analytically and numerically the liquid-glass transition of hard spheres for dimensions $d\to \infty $ in the framework of mode-coupling theory. The numerical results for the critical collective and self nonergodicity parameters $f_{c}(k;d) $ and $f_{c}^{(s)}(k;d) $ exhibit non-Gaussian $k$ -dependence even up to $d=800$. $f_{c}^{(s)}(k;d) $ and $f_{c}(k;d) $ differ for $k\sim d^{1/2}$, but become identical on a scale $k\sim d$, which is proven analytically. The critical packing fraction $\phi_{c}(d) \sim d^{2}2^{-d}$ is above the corresponding Kauzmann packing fraction $\phi_{K}(d)$ derived by a small cage expansion. Its quadratic pre-exponential factor is different from the linear one found earlier. The numerical values for the exponent parameter and therefore the critical exponents $a$ and $b$ depend on $d$, even for the largest values of $d$.