Granular Brownian motors: Role of gas anisotropy and inelasticity

We investigate the motion of a two-dimensional wedge-shaped object (a granular Brownian motor), which is restricted to move along the $x$ axis and cannot rotate as gas particles collide with it. We show that its steady-state drift, resulting from inelastic gas-motor collisions, is dramatically affected by anisotropy in the velocity distribution of the gas. We identify the dimensionless parameter providing the dependence of this drift on shape, masses, inelasticity, and anisotropy: The anisotropy leads to dramatically enhanced drift of the motor, which should easily be visible in experimental realizations.

Figure 1

A particle (black circle) colliding with the Brownian motor (triangular wedge with wedge angle $2{\theta }_0$). The angles of the edges $i\in \{0,1,2\}$ are measured counterclockwise from the positive $x$ axis to the outside of the motor, yielding ${\theta }_0$, ${\theta }_1=\pi -{\theta }_0$, and ${\theta }_2=3\pi /2$, respectively.

Figure 2

Ensemble drift $\langle V\rangle$ (main panel) and temperature $\mathcal{T}$ (inset) against $t$ for motors with mass ratio $\mathcal{M}=10$ and ${\theta }_0=\pi /4$. Blue lines (i) denote $r=1$ and $\alpha =1$, elastic collisions with an isotropic gas. Green lines (ii) denote $r=0.3$ and $\alpha =1$, strongly inelastic collisions with an isotropic gas. The motor relaxes to the values predicted by [2] (black horizontal lines). Red lines (iii) denote $r=1.0$ and $\alpha =1.02$, elastic collisions with a slightly anisotropic gas.

Figure 3

Data for all combinations of ${\theta }_0=\pi /4$, $r\in \{0.3,0.5,0.8\}$, and $\alpha \in \{1.02,1.007,1.002,1.0007,1.0002\}$ and ${\theta }_0=\pi /10$, $r=0.5$, and $\alpha \in \{1.02,1.007,1.002,1.0007,1.0002\}$. (a) Master plot for the motor drift where the inset illustrates the effect of varying $\alpha$. Curves for inelastic collisions with an isotropic gas with $r=0.3$ and $\alpha =1$ [straight blue line (i)] and elastic collisions with an anisotropic gas with $r=1$ and $\alpha =1.02$ [straight red line (ii)] are included for reference. All other curves show the drift for decreasing $\alpha \in \{1.02,1.007,1.002,1.0007,1.0002\}$ from top to bottom. (b) (top) Motor temperature $\mathcal{T}$ for ${\theta }_0=\pi /4$, $r\in \{0.3,0.5,0.8\}$, and $\alpha \in \{1.02,1.007,1.002,1.0007,1.0002\}$. (bottom) Difference between motor temperature and the asymptotic theory. For comparison, ${\theta }_0=\pi /10$, $r=0.5$, and $\alpha \in \{1.02,1.007,1.002,1.0007,1.0002\}$ are also shown ($☆$).