Interacting particles on a rocked ratchet: Rectification by condensation

The transport of interacting particles subject to an external low-frequency ac force on a ratchetlike asymmetric substrate is studied via a nonlinear Fokker-Planck equation as well as via numerical simulations. With increasing the particle density, the ratchet current can either increase or decrease depending on the temperature, the drive amplitude, and the nature of the interparticle interaction. At low temperatures, attracting particles can condense randomly at some potential minima, thus breaking the discrete translational symmetry of the substrate. Depending on the drive amplitude, condensation results either in a drop to zero or in the saturation of the net particle velocity at densities above the condensation density—the latter case producing a very efficient rectification mechanism.

Figure 1

(Color online) Repelling particles: net velocity $V_{\mathrm{DC}}$ versus dimensionless pair coupling $\tilde{g}=gn∕k_{B}T$ for different values of $\tilde{A}$ in the regime of high activation, $q=Q∕k_{B}T=10$, and large anisotropy $\gamma$. $V_{\mathrm{DC}}\left(\tilde{g}\right)$ exhibits a broad peak for $\tilde{A}<20$. The normalized driving amplitude $\tilde{A}=lA∕k_{B}T$ is a measure of the strength of the ac drive, compared to the disordering thermal energy. These results were obtained by solving the mean-field set of equations (14). Inset in (a): two cells of the sawtooth substrate potential [Eq. (11)].

Figure 2

(Color online) Attracting particles: (a) Net velocity $V_{\mathrm{DC}}$ versus effective interaction coupling $\tilde{g}=gn∕k_{B}T$ for different values of driving amplitudes $\tilde{A}=Al∕k_{B}T$ in the regime of low activation, $q=1$, and large anisotropy, $\gamma =40$. There exists no mean-field solution for $\tilde{g}<{\tilde{g}}_c$ (left end points). (b) Gas-liquid phase diagram in the density-drive plane. The phase boundary between gas and liquid phases was obtained based on the criterion of existence of a solution to the mean-field equations (14). At zero driving $F_{\mathrm{DC}}=0$, the transition is an equilibrium transition. The condensation transition is driven out of equilibrium by increasing the DC force $F_{\mathrm{DC}}$. Black dots in (b) mark the left-end-points of curves in (a). These results were obtained by solving the mean-field set of equations (14).

Figure 3

(Color online) Net particle speed $V_{\mathrm{DC}}$ versus drive amplitude $A$ for repelling, attracting, and noninteracting particles at $\gamma =1.2$, $\tilde{g}=\pm 0.05$, $q=20$ (a) and $q=2$ (b). Panel (c) shows the curves $V_{\mathrm{DC}}$ versus effective interaction strength $\tilde{g}$ for the parameter values corresponding to the dots (1) and (2) in (a). These results were obtained by solving the mean-field set of equations (14).

Figure 4

$V_{\mathrm{DC}}$ versus $n$ for repelling particles. These data (symbols) were obtained by numerically solving the Langevin Eq. (1) for the potential $U\left(x\right)$ in Eq. (11) and the truncated pair potential $W\left(x\right)$ in Eq. (27); while the analytical curves were obtained by solving the mean-field set of equations (14). Other simulation parameters: $\nu =0.01$, $A=0.5$, $Q=1$, $l_1=0.9$, $\lambda =0.1$, $g_{\mathrm{MD}}=0.02$. (a) $T=0.2$ (solid circles) and (b) $T=0.05$ (open circles). Periodic boundary conditions were imposed over two unit cells of the piecewise linear potential (11). The corresponding analytical predictions for $g=g_{\mathrm{MD}}=0.02$ (dashed lines) and $g=g_{\mathrm{MF}}=0.02∕1.5\approx 0.0133$ (solid lines) are reported for comparison (see text).

Figure 5

$V_{\mathrm{DC}}$ versus $n$ for attracting particles with $A=0.8$, $T=0.2$ (solid squares) and $A=1$, $T=0.4$ (open squares). Periodic boundary conditions were imposed over two unit cells of the piecewise linear potential (11). These results were obtained by numerically solving the Langevin Eq. (1) for the potential $U\left(x\right)$ in Eq. (11) and the truncated pair potential $W\left(x\right)$ in Eq. (27). Other simulation parameters: $\nu =0.01$, $Q=1$, $l_1=0.9$, $\lambda =0.1$, and $g=-0.02$.

Figure 6

(Color online) (a) Net average velocity $V_{\mathrm{DC}}$ versus particle density $n$ for repelling (open circles) and attracting (solid squares) particles. These results were obtained by numerically solving the Langevin Eq. (1) for the potential $U\left(x\right)$ in Eq. (11) and the truncated pair potential $W\left(x\right)$ in Eq. (27). Other simulation parameters are $\nu =0.01$, $A=6$, $Q=1$, $l_1=0.9$, $\lambda =0.1$, $g=-0.02$, and $T=0.2$. (b) Spatial distribution of attracting particles for $n=7$, corresponding to data marked by a black circle in (a). One snapshot (normalized $\text{distribution}\equiv F_1∕n$) of the particles was taken at each drive period with the external force pushing in the “hard” (solid line, the left axis) or the “easy” (dotted line, the right axis) direction, respectively. Particle condensation at both potential minima is apparent in the latter case. In both panels, periodic boundary conditions were imposed over two unit cells of the piecewise linear potential (11).

Figure 7

Net velocity $V_{\mathrm{DC}}$ versus particle density $n$ for attracting particles with $A=15$ and $T=0.2$. These results were obtained by numerically solving the Langevin Eq. (1) for the potential $U\left(x\right)$ in Eq. (11) and the truncated pair potential $W\left(x\right)$ in Eq. (27). Other simulation parameters are $\nu =0.01$, $Q=1$, $l_1=0.9$, $\lambda =0.1$, and $g=-0.02$. Periodic boundary conditions were imposed over two unit cells of the piecewise linear potential (11). Because of the strong driving, there is no condensation and $V_{\mathrm{DC}}$ increases monotonically.

Figure 8

(Color online) Net velocity $V_{\mathrm{DC}}$ versus the number of particles $N$ for (a) attracting and (b) repelling particles. These results were obtained by numerically solving the Langevin Eq. (1) for the potential $U\left(x\right)$ in Eq. (11) and the truncated pair potential $W\left(x\right)$ in Eq. (27). Periodic boundary conditions were imposed over two (solid squares) and four potential unit cells (open squares). Other simulation parameters are $\nu =0.01$, $Q=1$, $l_1=0.9$, $\lambda =0.1$, $g=\pm 0.02$, $T=0.3$, and $A=6$. The open small circles are the simulation data for a four-cell chain with half the number of particles, i.e., $V_{\mathrm{DC}}$ versus $N∕2$.

Figure 9

(Color online) Spontaneous symmetry breaking destroys the equivalence of two neighboring potential cells at the phase transition. The normalized particle distributions $F_1(x∕l)∕N$ are shown for four different total numbers $N=2n$ of particles inside the two cells, corresponding to the net velocity $V_{\mathrm{DC}}\left(N\right)$ shown at the top. The normalized equilibrium distributions $F_1(x∕l)∕N$ for the same particle numbers are also shown in the right column. These results were obtained by numerically solving the Langevin Eq. (1) for the potential $U\left(x\right)$ in Eq. (11) and the truncated pair potential $W\left(x\right)$ in Eq. (27). The parameters used here are $\nu =0.01$, $Q=1$, $l_1=0.9$, $\lambda =0.1$, $g=-0.02$, $T=0.2$, and $A=0.5$. Both cells are equally occupied at densities lower than the condensation point (e.g., point 1); this equivalence is broken at higher densities (point 2 and 3).

Figure 10

(Color online) The mechanism of spontaneous symmetry breaking of Fig. 9 has counterparts in the theory of equilibrium phase transitions [e.g., the paramagnetic-ferromagnetic transition shown in (a) and certain dynamical instabilities of the type sketched in (b), see text]. (c) A schematic diagram of a simple realization of a four-cell ratchet device. Two coaxial cylinders, the outer one having a sawtooth-shaped inner cross section, rotate with angular frequency ${\Omega }_R$; subject to the resulting centrifugal force, the elastic particles experience an effective ratchetlike potential. An ac driving force can be generated by periodically switching the rotation frequency between two values ${\Omega }_R\pm \Delta {\Omega }_R$.