Figure 1

(a) Distribution of kinetic energies $K_s$ and $K_b$, for small and big balls, respectively, and distribution of kinetic energy $K_t$ and potential energy $U_t$ for the weighted-ball thermometer, at airspeed $670\mathrm{cm}/\mathrm{s}$ and area density 0.43. Inset: Scatter plot of instantaneous kinetic and potential energy for the thermometer. (b) Distribution of thermometer total mechanical energy. Inset: The thermometer, as seen from above, is a weighted hollow sphere whose position degrees of freedom are measured by the coordinates $\mathbf{r}$ of its geometric center and of a surface spot $\delta \mathbf{r}$ marked radially opposite the center of mass. The solid black curves in both (a) and (b) are the statistical mechanics predictions based on total average mechanical energy $T_t=⟨K_t+U_t⟩/2$.Reuse & Permissions

Figure 2

Mobility vs tilt frequency for $1^{\prime \prime }$ diameter test spheres at airspeed $760\mathrm{cm}/\mathrm{s}$ and area density 0.57. Symbol types distinguish test spheres and tilt amplitudes; solid blue square: Teflon, 18.45 g, 0.52°; hollow (solid) red diamond: acrylic, 10.12 g, 2.35° (0.52°); hollow (solid) green triangle: wood, 6.4 g, 0.52° (0.32°). Inset: Instantaneous velocity vs instantaneous force for the wood test sphere at $1.67\times {10}^{-3}\mathrm{Hz}$ and tilt amplitude 0.52°. Mobility is the slope of the best fit to $v\propto F$ (black line).Reuse & Permissions

Figure 3

Effective temperatures vs (a) area fraction at airspeed $670\mathrm{cm}/\mathrm{s}$ and (b) vs driving airspeed at area fraction 0.57. The granular temperature $T_g$ is based on the ball kinetic energies; the thermometer temperature $T_t$ is based on the mechanical energy of weighted-sphere oscillators with two different masses, 3.83 g for upward triangles and 6.82 g for downward triangles; the Einstein temperature $T_e$ is the ratio of diffusivity to mobility. The solid green curve is the weighted average of all of these measures. Measurement uncertainties for $T_g$ and $T_t$ are comparable to symbol size, while the uncertainties in $T_e$ are indicated by error bars.Reuse & Permissions

Figure 4

(a) Inverse mobility scaled by ${\mu }_o={\tau }_c/m$, (b) dimensionless extent of subdiffusive plateau in mean-squared displacement, and (c) average number of grains in kinetic heterogeneities, all vs inverse effective temperature scaled by ball weight times diameter. Open and closed symbols are for the polypropylene balls used in Figs. 1, 2, 3, along two trajectories as labeled. The $\times$ symbols are based on earlier data [18] for a bidisperse mixture of 0.635 and 0.873 cm diameter steel balls, and the $+$ symbols are based on earlier data [20] for a bidisperse mixture of 0.318 and 0.397 cm diameter steel balls; for both, jamming is approached by a change in the packing fraction at constant airspeed. In (a), the diamond, $\times$, and $+$ symbols represent $T_{\mathrm{eff}}/D$. In (a)–(c), the dashed lines are Arrhenius fits, $\propto \exp (E/T_{\mathrm{eff}})$, all with the same energy barrier indicated by the vertical gray line. The solid curves are power-law fits ${\mu }_o/\mu \propto {\tau }_r/{\tau }_c\propto (mg{D}_b/T_{\mathrm{eff}}{)}^{2.0\pm 0.5}$ and $n^{*}\propto (mg{D}_b/T_{\mathrm{eff}}{)}^{0.7\pm 0.2}$.Reuse & Permissions