Exact theory of dense amorphous hard spheres in high dimension. I. The free energy

Abstract

We consider the theory of the glass transition and jamming of hard spheres in the large space dimension limit. Previous investigations were based on the assumption that the probability distribution within a "cage" is Gaussian, which is not fully consistent with numerical results. Here we perform a replica calculation without making any assumption on the cage shape. We show that thermodynamic functions turn out to be exact within the Gaussian ansatz -- provided one allows for arbitrary replica symmetry breaking --- and indeed agree well with numerical results. The actual structure function (the so-called non-ergodic parameter) is not Gaussian, an apparent paradox which we discuss. In this paper we focus on the free energy, future papers will present the results for the structure functions and a detailed comparison with numerical results.

14 pages

References in corpus (3)

• Jamming versus Glass Transitions
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Dimensional Study of the Caging Order Parameter at the Glass Transition

Cited by in corpus (31)

• Fractal free energy landscapes in structural glasses
• Jamming Criticality Revealed by Removing Localized Buckling Excitations
• Hopping and the Stokes-Einstein relation breakdown in simple glass formers
• Following the evolution of glassy states under external perturbations: compression and shear-strain
• Dynamic criticality at the jamming transition
• Mean-field avalanches in jammed spheres
• The Gardner transition in finite dimensions
• Long-Range Anomalous Decay of the Correlation in Jammed Packings
• Active hard-spheres in infinitely many dimensions
• The jamming transition in high dimension: an analytical study of the TAP equations and the effective thermodynamic potential
• Dimensional study of the dynamical arrest in a random Lorentz gas
• Random-field Ising model criticality in a glass-forming liquid
• Finite size effects in the microscopic critical properties of jammed configurations: A comprehensive study of the effects of different types of disorder
• Systematic expansion in the order parameter for replica theory of the dynamical glass transition
• Higher order corrections to the effective potential close to the jamming transition in the perceptron model
• Breakdown of Nonlinear Elasticity in Stress-Controlled Thermal Amorphous Solids
• Calorimetric glass transition in a mean field theory approach
• Inherent Structure Landscape of Hard Spheres Confined to Narrow Cylindrical Channels
• Transverse forces and glassy liquids in infinite dimensions
• The Super-Potts glass: a new disordered model for glass-forming liquids
• Vibrational density of states of jammed packing: mean-field theory
• Collective dynamics in a glass-former with Mari-Kurchan interactions
• Universal scaling for disordered viscoelastic matter near the onset of rigidity
• Theory of simple glasses
• Treating random sequential addition via the replica method
• Ergodicity breaking of an inorganic glass in aging near $T_g$ probed by elasticity relaxation
• A proof of an identity for the critical exponents of jamming
• Theory of the $\beta$-Relaxation Beyond Mode-Coupling Theory: A Microscopic Treatment
• An alternative, dynamic density functional-like theory for time-dependent density fluctuations in glass-forming fluids
• Universal Jamming Criticality and Self-Organizing Principles from Disorder to the Limit of Perfect Crystalline Order
• Probing the Gardner transition in an active quasi-thermal pressure-controlled granular system