Top: Time dependence of the mean-squared displacements (MSD) at constant temperature, T = 10−8, and various volume fractions increasing from top to bottom across the jamming transition. Open squares indicate the microscopic time scale τ0, filled squares indicate the long timescale t ⋆, which marks roughly the convergence of the MSD to its long-time limit. Both time scales decrease with φ, but their ratio is maximum near φ J . Bottom: Volume fraction dependence of the Debye-Waller (DW) factor (long-time limit of the MSD) for different temperatures. The T = 0 jamming singularity at φ J ≈ 0.6466 (vertical line) strongly influences the DW factor when T becomes very small, T < 10−7, and φ is very close to the critical density, 0.64 < φ J < 0.655.
Top: Frequency dependence of the density of states at constant temperature, T = 10−8, and densities increasing from top to bottom. Parameters are as in Fig. 1 . Open squares indicate the microscopic frequency ω0 = π/τ0, while filled squares indicate the low-frequency crossover ω⋆ = π/t ⋆. Bottom: Volume fraction dependence of t ⋆ = π/ω⋆ for different temperatures. The vertical line indicates φ J .
Volume fraction dependence of the microscopic time scale τ0, Eq. (8) , at various temperatures decreasing from bottom to top using symbols as in Fig. 1 . The vertical line indicates φ J . These data were used to draw the open squares in Figs. 1 and 2 . Inset: τ0 is replotted as a function of the scaling variable . Upper and lower dashed lines correspond to T → 0 limits for hard and soft spheres, respectively, Eqs. (9) and (10) .
Time dependence of the four-point susceptibility, Eq. (19) , for T = 10−8, and various volume fractions across φ J . The filled squares indicate the timescale t ⋆. Note that χ4(t) saturates at long times to a system size dependent plateau.
Diverging susceptibility and length scale near the jamming transition. Top: Volume fraction dependence of the dynamic susceptibility χ4(t = t ⋆), Eq. (22) , for various temperatures showing the emergence of “vibrational heterogeneity” at low enough T close to φ J (shown with the vertical line). Bottom: Four-point correlator S 4(q, t = t ⋆) at T = 10−8 and various volume fractions exhibits a nonmonotonic density dependence (note the nonmonotonic evolution of the φ labels).
Test of the scaling behavior of S 4(q) using the assumption that . Note that this data collapse involves no free parameter since S 4 and result from independent measurements. Dashed lines are fits to the Ornstein-Zernike functional form, Eq. (28) .
Limit of validity of harmonic approximation above jamming. Top: The density of state d(ω) at φ = 0.655 (φ − φ J ≈ 0.0084) measured for various temperatures through Eq. (5) . The T = 0 DOS is obtained from diagonalization of the dynamical matrix. Bottom: Convergence of the first moment of the density of state to its T = 0 limit below a crossover temperature scale T ⋆ = 10−3(φ − φ J )2 which is independent of system size. The vertical line indicates T/T ⋆ = 1.
The structure of the three scaling regimes where power law divergences and “anomalous” vibrational motion is observed. Regimes I and II are described by T = 0 harmonic theories, while dynamics in Regime III is fully anharmonic. These regimes are separated by the crossover temperature T ⋆ ∼ 10−3(φ − φ J )2 discussed in Sec. VII . Note the small range of parameters where the effect of the T = 0 jamming critical point are felt and Eqs. (34) and (37) hold.
Limit of validity of (effective) harmonic approximation below jamming. Top: Temperature-scaled density of state at φ = 0.630 and various temperatures converges to T = 0 limit below T ⋆ ≈ 10−7. Bottom: Convergence of the height of the first peak of the DOS to its T = 0 value below the crossover temperature scale T ⋆(φ) = 10−3(φ J − φ)2. The vertical line indicates T/T ⋆ = 1.
The crossover temperature scale for the onset of anharmonicity T ⋆ (bottom data with filled circles) compared to the temperature scale T pt (top data with open symbols different symbols for different packings) quantifying the relevance of the high order terms in the perturbative expansion of the energy. The lines indicate T ⋆∝E gs and . Clearly, these two temperature scales differ.
The radial distribution function g(r) at finite T (black/thick lines), at T = 0 (blue/thin line), and its particle-resolved version g i (r) (red/dashed lines) for a randomly chosen particle i at φ = 0.66 and various temperatures. While contacts between i and its neighbors are well-defined at low T, they cannot be resolved when T becomes larger than T ⋆ which is ≈4 × 10−7 for this particular density.
An enlarged view of the (temperature, volume fraction) phase diagram reporting the critical regimes shown in Fig. 9 , and the approximate location of the experimental studies aimed at studying anomalous vibrational dynamics in colloidal systems: Gosh et al. 10 use PMMA hard spheres, while Chen et al. 12 and Caswell et al. 16 study PNIPAM microgel particles. Previous studies all lie too far away from the jamming transition to detect the dynamic criticality associated to the transition.
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