Granular Brownian ratchet model

We show by numerical simulations that a nonrotationally symmetric body, whose orientation is fixed and whose center of mass can slide along a rectilinear guide, under the effect of inelastic collisions with a surrounding gas of particles, displays directed motion. We present a theory which explains how the lack of time reversal induced by the inelasticity of collisions can be exploited to generate a steady average drift. In the limit of a heavy ratchet, we derive an effective Langevin equation whose parameters depend on the microscopic properties of the system and obtain a fairly good quantitative agreement between the theoretical predictions and simulations concerning effective friction, diffusivity, and average velocity.

Figure 1

Sketch of the 2D model. The triangle is constrained to move only in the $\widehat{x}$ (left-right) direction, while its orientation is fixed, i.e., it cannot rotate. Gas particles collide against it and occasionally receive energy from an external bath, in the form of uncorrelated isotropic random kicks.

Figure 2

(Color online) Top frame: Averaged trajectory of the tracer in MD (full symbols) and DSMC (empty symbols) with $M=10$. Bottom frame: Rescaled mean squared displacement $d_2\left(t\right)$ (see text) for the same choices of parameters. The power law $\sim x$ and $\sim x^2$ are drawn for reader’s convenience (dotted straight lines).

Figure 3

Rescaled average velocities and energies of the tracer, both from MD (full symbols) and DSMC (empty symbols) simulations. Averages have been obtained with ${10}^3$ MD dynamics and ${10}^4$ DSMC dynamics. Gas particles have always $m=1$. In MD, $T_g=m⟨v_g^2⟩∕2$ changes when changing the parameters ($\alpha$ and $M$), in DSMC it is always $T_g=1$. The dashed line represents the theorical predictions obtained from Langevin theory.

Figure 4

Rescaled self-correlation functions of the tracer velocity with different choices of parameters, elastic and inelastic, from MD and DSMC, against rescaled time $\gamma t$. The bold dashed line is the theoretical prediction coming from Eq. (11).