Spontaneous Ratchet Effect in a Granular Gas

The spontaneous clustering of a vibrofluidized granular gas is employed to generate directed transport in two different compartmentalized systems: a granular fountain in which the transport takes the form of convection rolls, and a granular ratchet with a spontaneous particle current perpendicular to the direction of energy input. In both instances, transport is not due to any system-intrinsic anisotropy, but arises as a spontaneous collective symmetry breaking effect of many interacting granular particles. The experimental and numerical results are quantitatively accounted for within a flux model.

Figure 1

(a) Schematic cross section of a granular fountain exhibiting spontaneous symmetry breaking into a “cold” and a “hot” compartment and a concomitant spontaneous circular particle flow. (b) By folding out the geometry of several adjacent fountains, and adding cyclical boundary conditions, a granular ratchet is obtained with almost unchanged populations of the respective compartments and the particle currents between them (here shown for $K=4$ compartments).

Figure 2

Bifurcation diagram for the granular fountain. Depicted are the steady state fractions of particles $n_k$, $k\in \{1,2\}$, in the two compartments from Fig. 1a for different values of the driving frequency $f$ in units of $B\propto f^{-2}$ from Eq. (1). Experimental results are indicated by the black error bars; the other symbols represent molecular dynamics simulations using uniform (asterisks) and clustered (stars) initial conditions. The gray lines represent the theoretical prediction from the flux model (1, 2, 3) for $\lambda =0.018$: solid lines correspond to stable configurations; dashed lines to unstable ones.

Figure 3

Bifurcation diagram of the $K=4$ granular ratchet. Symbols: results from molecular dynamics simulations. Gray lines: prediction from the flux model (1, 2, 4) for $\lambda =0.018$. Solid lines correspond to stable configurations; dashed lines to unstable ones. For further details regarding parameters and methods, see main text. For small $B$ the uniform distribution, $n_1=n_2=n_3=n_4=1/4$, is the unique steady state. At $B=1$ a pair of symmetry broken stable states takes over, namely, $n_2=n_3\ne n_4=n_1$, indicated by (blue) asterisks. They exhibit no net particle flow and are called fluxless clustered states. For slightly larger $B$ another pair of stable states emerges, both of the form $n_1=n_3\ne n_2=n_4$ [see Fig. 1b], indicated by (red) stars. They do carry a net flow (cf. Fig. 4) and are called ratchet states. For $B>6.8$ only the fluxless stable states survive.

Figure 4

Running time average (over 5 s) of the net particle flux $\Phi (t)$ for the same molecular dynamics simulations as in Fig. 3 with $B=3.0$, measured through the rightmost slit (a), and through the rightmost hole (b) in Fig. 1b. The upper (blue) curves in each plot correspond to the clustered state with densely populated leftmost and rightmost compartments in Fig. 1b. The lower (red) curves belong to the ratchet state with densely populated compartments 1 and 3 in Fig. 1b. For both states the straight line corresponds to the long-time asymptote of the particle current.

Figure 5

Fraction $p_K$ of ratchet solutions versus number $K(\ge 4)$ of compartments in a granular ratchet system with a constant initial particle number $N/K$ per compartment. Symbols represent numerical solutions of the (deterministic) flux model (1, 2, 4) for $B=3$ and $\lambda =0.018$, averaged over initial distributions with a superimposed randomization, mimicking fluctuations due to the finite particle number within each compartment. The straight line is a linear best fit.