Partitioning of energy in highly polydisperse granular gases

A highly polydisperse granular gas is modeled by a continuous distribution of particle sizes, $a$, giving rise to a corresponding continuous temperature profile, $T\left(a\right)$, which we compute approximately, generalizing previous results for binary or multicomponent mixtures. If the system is driven, it evolves toward a stationary temperature profile, which is discussed for several driving mechanisms in dependence on the variance of the size distribution. For a uniform distribution of sizes, the stationary temperature profile is nonuniform with either hot small particles (constant force driving) or hot large particles (constant velocity or constant energy driving). Polydispersity always gives rise to non-Gaussian velocity distributions. Depending on the driving mechanism the tails can be either overpopulated or underpopulated as compared to the molecular gas. The deviations are mainly due to small particles. In the case of free cooling the decay rate depends continuously on particle size, while all partial temperatures decay according to Haff’s law. The analytical results are supported by event driven simulations for a large, but discrete number of species.

Figure 1

(a) The stationary temperatures [Eqs. (6a, 6b)] in a three component (open disks) and five component (filled disks) mixtures compared to those of a highly polydisperse mixture for energy controlled driving $H^{\text{ec}}={10}^{-3}$ at density $n=5\times {10}^{-4}$ and coefficient of restitution $ϵ=0.9$. (b) Inverse cooling time ${\omega }_0$ in a two-dimensional system for a uniform size distribution of width $R=3$, coefficient of restitution $ϵ=0.9$ and density $n=2\times {10}^{-4}$. The symbols denote simulation results for $X=30$ species each with ${10}^4$ particles, while the solid line is the solution of Eq. (12).

Figure 2

(Color online) The stationary temperature in a three-dimensional driven system for a size distribution with $R_2=3R_1$, a coefficient of restitution $ϵ=0.9$ at a density $n=2\times {10}^{-4}$. Force controlled driving $H^{\text{fc}}\left(a\right)=1.875\times {10}^{-3}/m\left(a\right)$ (solid, red), energy controlled driving $H^{\text{ec}}=1.875\times {10}^{-3}$ (long dashed, blue) and velocity controlled driving $H^{\text{vc}}\left(a\right)=1.875\times {10}^{-3}m\left(a\right)$ (short dashed, green). For the simulation data (symbols) a system of 20 different particle species with ${10}^4$ particles each was used.

Figure 3

(Color online) Stationary velocity distribution [Eq. (11)] in a three-dimensional driven system; parameters, and symbols as in Fig. 2; for comparison the velocity distribution of an elastically colliding, molecular gas (dashed-dotted, black), and a Gaussian fit to the central part of the distribution (thin dotted line) are also shown.

Figure 4

Stationary velocity distribution relative to the elastic gas and separately for the two halves of smaller (full line) and larger (dashed line) particles; parameters as in Fig. 2; (a) force controlled driving, (b) energy controlled driving.

Figure 5

(Color online) (a) The mean temperature $\overline{T}$ [Eq. (5)] and (b) the relative temperature variance $\Delta T^2=\overline{T^2}/{\overline{T}}^2-1$ as a function of the relative width of the size distribution $\Delta$ for the three driving mechanisms $H^{\text{fc}}\left(a\right)={10}^{-2}/m\left(a\right)$ (solid, red), $H^{\text{ec}}\left(a\right)={10}^{-2}$ (long dashed, blue) and $H^{\text{vc}}\left(a\right)={10}^{-2}m\left(a\right)$ (short dashed, green), coefficient of restitution $ϵ=0.9$ and density $n=5\times {10}^{-4}$. The mean temperatures in (a) are rescaled such that they agree for a relative width of $\Delta =1$.