Equipartition of Rotational and Translational Energy in a Dense Granular Gas

Experiments quantifying the rotational and translational motion of particles in a dense, driven, 2D granular gas floating on an air table reveal that kinetic energy is divided equally between the two translational and one rotational degrees of freedom. This equipartition persists when the particle properties, confining pressure, packing density, or spatial ordering are changed. While the translational velocity distributions are the same for both large and small particles, the angular velocity distributions scale with the particle radius. The probability distributions of all particle velocities have approximately exponential tails. Additionally, we find that the system can be described with a granular Boyle’s law with a van der Waals-like equation of state. These results demonstrate ways in which conventional statistical mechanics can unexpectedly apply to nonequilibrium systems.

Figure 1

(a) Schematic of experiment, top view, with bumpers at sides. The piston provides CP boundary conditions, or can be fixed in place to create CV. (b),(c) Time series of average kinetic energy per particle at two values of $\varphi$ (CV boundary conditions).

Figure 2

Translational velocity distributions (a) $\mathcal{P}(v_y)$, with $⟨|v_y|⟩=19.9\mathrm{mm}/\mathrm{s}$, (b) $\mathcal{P}(v_x)$, with $⟨|v_x|⟩=19.4\mathrm{mm}/\mathrm{s}$ and (c) $\mathcal{P}(v_y)$ for particles which are at least $3r_L$ from a boundary, with $⟨|v_y|⟩=19.3\mathrm{mm}/\mathrm{s}$. (d) Rescaled rotational velocity distribution $\mathcal{P}(r\omega )$, with $⟨r|\omega |⟩=20.9\mathrm{mm}/\mathrm{s}$. Colors indicate conditions of run (large, bare, amorphous, unless otherwise specified).

Figure 3

(a) Time-averaged ratio of the translational kinetic energy to the total kinetic energy $⟨E_{\mathrm{tr}}/E⟩$ averaged over 100 s. Dashed line is $2/3$, corresponding to equipartition. (b) Average kinetic energy per particle as a function of $\varphi$. Error bars are typical standard deviation. Each data point indicates the average for 25 s of data (CP) or 100 s (CV). Color of symbols indicates $P$ (see Fig. 4 for key).

Figure 4

Data from CP experiments, corresponding to Eq. (3). Symbol color indicates $P$; symbol shapes as shown in Fig. 3.