The dimensional evolution of structure and dynamics in hard sphere liquids

Abstract

The formulation of the mean-field, infinite-dimensional solution of hard sphere glasses is a significant milestone for theoretical physics. How relevant this description might be for understanding low-dimensional glass-forming liquids, however, remains unclear. These liquids indeed exhibit a complex interplay between structure and dynamics, and the importance of this interplay might only slowly diminish as dimension d increases. A careful numerical assessment of the matter has long been hindered by the exponential increase of computational costs with d. By revisiting a once common simulation technique involving the use of periodic boundary conditions modeled on D d lattices, we here partly sidestep this difficulty, thus allowing the study of hard sphere liquids up to d = 13. Parallel efforts by Mangeat and Zamponi [Phys. Rev. E 93, 012609 (2016)] have expanded the mean-field description of glasses to finite d by leveraging standard liquid-state theory, and thus help bridge the gap from the other direction. The relatively smooth evolution of both structure and dynamics across the d gap allows us to relate the two approaches, and to identify some of the missing features that a finite-d theory of glasses might hope to include to achieve near quantitative agreement.

Gradient descent dynamics and the jamming transition in infinite dimensions

“…[14][15][16][17][18][19]. Because finite-dimensional MB systems converge quite slowly to the asymptotic d → ∞ limit [44][45][46][47], here we focus on the simpler Random Lorentz Gas (RLG) [35,36], which is a single particle tracer with d degrees of freedom embedded in a sea of random obstacles. In the limit d → ∞, the MB problem can be mapped onto the RLG [35,36], via a simple rescaling described in appendix C. In short, a given value of density ϕ in the RLG corresponds to twice that value in the MB problem.…”

Comment on "Explicit Analytical Solution for Random Close Packing in $d = 2$ and $d = 3$"

“…(Another caveat is that these liquid state descriptions do not capture singular features of g(r, ϕ) at RCP [10]. )Because PY fails to describe essential contact features of g(r, ϕ) in d > 3 liquids [11], the dimensional generalization here focuses on the CS form. Its one parameter, A(d), is known numerically for d = 2-13, and asymptotically scales as A(d) ∼ 3 (d+1)/2 2/(dπ) [11].…”

“…)Because PY fails to describe essential contact features of g(r, ϕ) in d > 3 liquids [11], the dimensional generalization here focuses on the CS form. Its one parameter, A(d), is known numerically for d = 2-13, and asymptotically scales as A(d) ∼ 3 (d+1)/2 2/(dπ) [11]. Equating the invariant in both the RCP and crystalline phases yields…”