Absence of Hyperuniformity in Amorphous Hard-Sphere Packings of Nonvanishing Complexity.

We relate the structure factor S(k→0) in a system of jammed hard spheres of number density ρ to its complexity per particle Σ(ρ) by the formula S(k→0)=-1/[ρ^{2}Σ^{″}(ρ)+2ρΣ^{'}(ρ)]. We have verified this formula for the case of jammed disks in a narrow channel, for which it is possible to find Σ(ρ) and S(k) analytically. Hyperuniformity, which is the vanishing of S(k→0), will therefore not occur if the complexity is nonzero. An example is given of a jammed state of hard disks in a narrow channel which is hyperuniform when generated by dynamical rules that produce a nonextensive complexity.

Gardner Transition in Physical Dimensions.

The Gardner transition is the transition that at mean-field level separates a stable glass phase from a marginally stable phase. This transition has similarities with the de Almeida-Thouless transition of spin glasses. We have studied a well-understood problem, that of disks moving in a narrow channel, which shows many features usually associated with the Gardner transition. We show that some of these features are artifacts that arise when a disk escapes its local cage during the quench to higher densities. There is evidence that the Gardner transition becomes an avoided transition, in that the correlation length becomes quite large, of order 15 particle diameters, even in our quasi-one-dimensional system.

Marginally jammed states of hard disks in a one-dimensional channel.

We have studied a class of marginally jammed states in a system of hard disks confined in a narrow channel-a quasi-one-dimensional system-whose exponents are not those predicted by theories valid in the infinite dimensional limit. The exponent γ which describes the distribution of small gaps takes the value 1 rather than the infinite dimensional value 0.41269⋯. Our work shows that there exist jammed states not found within the tiling approach of Ashwin and Bowles. The most dense of these marginal states is an unusual state of matter that is asymptotically crystalline.

Glasslike behavior of a hard-disk fluid confined to a narrow channel.

Disks moving in a narrow channel have many features in common with the glassy behavior of hard spheres in three dimensions. In this paper we study the caging behavior of the disks that sets in at characteristic packing fraction ϕ(d). Four-point overlap functions similar to those studied when investigating dynamical heterogeneities have been determined from event-driven molecular dynamics simulations and the time-dependent dynamical length scale has been extracted from them. The dynamical length scale increases with time and, on the equilibration time scale, it is proportional to the static length scale associated with the zigzag ordering in the system, which grows rapidly above ϕ(d). The structural features responsible for the onset of caging and the glassy behavior are easy to identify as they show up in the structure factor, which we have determined exactly from the transfer-matrix approach.

Understanding the ideal glass transition: lessons from an equilibrium study of hard disks in a channel.

We use an exact transfer-matrix approach to compute the equilibrium properties of a system of hard disks of diameter σ confined to a two-dimensional channel of width 1.95σ at constant longitudinal applied force. At this channel width, which is sufficient for next-nearest-neighbor disks to interact, the system is known to have a great many jammed states. Our calculations show that the longitudinal force (pressure) extrapolates to infinity at a well-defined packing fraction ϕ(K) that is less than the maximum possible ϕ(max), the latter corresponding to a buckled crystal. In this quasi-one-dimensional problem there is no question of there being any real divergence of the pressure at ϕ(K). We give arguments that this avoided phase transition is a structural feature, the remnant in our narrow channel system of the hexatic to crystal transition, but that it has the phenomenology of the (avoided) ideal glass transition. We identify a length scale ξ̃(3) as our equivalent of the penetration length for amorphous order: In the channel system, it reaches a maximum value of around 15σ at ϕ(K), which is larger than the penetration lengths that have been reported for three-dimensional systems. It is argued that the α-relaxation time would appear on extrapolation to diverge in a Vogel-Fulcher manner as the packing fraction approaches ϕ(K).

Static and dynamical properties of a hard-disk fluid confined to a narrow channel.

The thermodynamic properties of disks moving in a channel sufficiently narrow that they can collide only with their nearest neighbors can be solved exactly by determining the eigenvalues and eigenfunctions of an integral equation. Using it, we have determined the correlation length ξ of this system. We have developed an approximate solution which becomes exact in the high-density limit. It describes the system in terms of defects in the regular zigzag arrangement of disks found in the high-density limit. The correlation length is then effectively the spacing between the defects. The time scales for defect creation and annihilation are determined with the help of transition-state theory, as is the diffusion coefficient of the defects, and these results are found to be in good agreement with molecular dynamics simulations. On compressing the system with the Lubachevsky-Stillinger procedure, jammed states are obtained whose packing fractions ϕJ are a function of the compression rate γ. We find a quantitative explanation of this dependence by making use of the Kibble-Zurek hypothesis. We have also determined the point-to-set length scale ξPS for this system. At a packing fraction ϕ close to its largest value ϕmax, ξPS has a simple power law divergence, ξPS∼1/(1-ϕ/ϕmax), while ξ diverges much faster, ln(ξ)∼1/(1-ϕ/ϕmax).

Transition state theory and the dynamics of hard disks.

The dynamics of two- and five-disk systems confined in a square has been studied using molecular dynamics simulations and compared with the predictions of transition state theory. We determine the partition functions Z and Z(‡) of transition state theory using a procedure first used by Salsburg and Wood for the pressure. Our simulations show this procedure and transition state theory are in excellent agreement with the simulations. A generalization of the transition state theory to the case of a large number of disks N is made and shown to be in full agreement with simulations of disks moving in a narrow channel. The same procedure for hard spheres in three dimensions leads to the Vogel-Fulcher-Tammann formula for their alpha relaxation time.