Stochastic transport of interacting particles in periodically driven ratchets

An open system of overdamped, interacting Brownian particles diffusing on a periodic substrate potential $\mathcal{U}(x+l)=\mathcal{U}\left(x\right)$ is studied in terms of an infinite set of coupled partial differential equations describing the time evolution of the relevant many-particle distribution functions. In the mean-field approximation, this hierarchy of equations can be replaced by a nonlinear integro-differential Fokker-Planck equation. This is applicable when the distance $a$ between particles is much less than the interaction length $\lambda$, i.e., a particle interacts with many others, resulting in averaging out fluctuations. The equation obtained in the mean-field approximation is applied to an ensemble of locally $(a⪡\lambda ⪡l)$ interacting (either repelling or attracting) particles placed in an asymmetric one-dimensional substrate potential, either with an oscillating temperature (temperature rachet) or driven by an ac force (rocked ratchet). In both cases we focus on the high-frequency limit. For the temperature ratchet, we find that the net current is typically suppressed (or can even be inverted) with increasing density of the repelling particles. In contrast, the net current through a rocked ratchet can be enhanced by increasing the density of the repelling particles. In the case of attracting particles, our perturbation technique is valid up to a critical value of the particle density, above which a finite fraction of the particles starts condensing in a liquidlike state near the substrate minima. The dependence of the net transport current on the particle density and the interparticle potential is analyzed in detail for different values of the ratchet parameters.

Figure 1

(Color online) Deviation of the statistical correlation function $G$ from its uncorrelated (mean-field) value $G=1$, versus distance $x-x^{\prime }$ between particles, normalized by the interaction length $\lambda$. The interaction was taken as $W\left(x\right)=g(\lambda -\mid x\mid )∕{\lambda }^2$, while the substrate potential $\mathcal{U}(x∕l)$ is shown in the inset. The downward arrow in the inset marks the $x=0$ position where $G(x-x^{\prime };x=0)$ was numerically computed. $G$ substantially differs from 1 (its mean-field value) over distances $\mid x-x^{\prime }\mid \sim a$, when the average distance $a=1∕n$ between particles is much smaller than $\lambda$. In the opposite limit $a>\lambda$ (shown by the dashed line), we obtain $G<1$, on scales $\mid x-x^{\prime }\mid <\lambda$. This result depends on neither temperature nor the substrate potential $\mathcal{U}$ in the range of parameters studied. Thus, we have numerically verified Eq. (11), which is used for the derivation of the mean-field approximation.

Figure 2

Graphical solution of the transcendental equation (21) for a subcritical density of attracting particles. With increasing density of the particles, both values ${\varphi }_0\left(x_{\min }\right)$ and ${\varphi }_0\left(x_{\max }\right)$ increase. Horizontal dotted lines represent $Z\left({\varphi }_0\right)=Z\mathbf{\left(}{\varphi }_0\left(x_{\min }\right)\mathbf{\right)}=C{e}^{-{\mathcal{U}}^{\min }∕k_{B}T}$ (upper) and $Z\left({\varphi }_0\right)=Z\mathbf{\left(}{\varphi }_0\left(x_{\max }\right)\mathbf{\right)}=C{e}^{-{\mathcal{U}}^{\max }∕k_{B}T}$ (lower), respectively. As soon as $Z\mathbf{\left(}{\varphi }_0\left(x_{\min }\right)\mathbf{\right)}$ reaches the maximum value $\max \left[Z\left({\varphi }_0\right)\right]=k_{B}T∕e\mid g\mid$, the transcendental equation admits no solution and a phase transition occurs. Note that the decreasing $Z\left({\varphi }_0\right)$ branch is unstable. As shown in the inset, one can always find the solution to Eq. (21) for repelling particles since $Z\left({\varphi }_0\right)$ is a monotonic increasing function of ${\varphi }_0$. We used $\mid g\mid ∕k_{B}T=\pm 0.3$ to plot the function $Z\left({\varphi }_0\right)$.

Figure 3

(a) The spatial dependence of the effective temperature $T^{\mathrm{eff}}$ and (b) the effective potential ${\mathcal{U}}^{\mathrm{eff}}-{\mathcal{U}}^{\mathrm{eff}}\left(0\right)$ for repelling particles and for different values of their density $n$. Both the effective temperature $T^{\mathrm{eff}}$ and the effective potential energy ${\mathcal{U}}^{\mathrm{eff}}$ are shown in arbitrary units. The bare substrate potential is chosen as $\mathcal{U}\left(x\right)={\mathcal{U}}^{\mathrm{ramp}}\left(x\right)\equiv \sin (2\pi x∕l)+(1∕2)\sin (4\pi x∕l)+(1∕3)\sin (6\pi x∕l)$. Mutual repulsion of particles out of the potential wells causes the flattening the “effective” potential with increasing $n$. The positions of the maxima of ${\mathcal{U}}^{\mathrm{eff}}\left(x\right)$ coincide with the minima of $T^{\mathrm{eff}}\left(x\right)$, and vice versa. This indicates that $T^{\mathrm{eff}}\left(x\right)$ and ${\mathcal{U}}^{\mathrm{eff}}\left(x\right)$ have opposite asymmetry.

Figure 4

Same as in Fig. 3, but for attracting particles. The “effective” potential wells deepen due to particle clustering around its minima. The effective temperature $T^{\mathrm{eff}}$ has a maximum where ${\mathcal{U}}^{\mathrm{eff}}$ has a maximum, and vice versa. The condensation of the attracting particles in the potential minima $x=x_{\min }$ occurs as $T^{\mathrm{eff}}\left(x_{\min }\right)$ drops to zero.

Figure 5

Net velocity $V_{\mathrm{dc}}$ versus density $n$ of repelling particles for temperature (a) and rocked ratchets (b) with the substrate potential $\mathcal{U}\left(x\right)={\mathcal{U}}^{\text{ramp}}\left(x\right)$ defined in Fig. 3. The dash-dotted lines correspond to the case of noninteracting particles, while the dotted lines show the high-density limits. The inset in (a) shows the case of two current inversions with increasing particle density for the temperature ratchet with $\mathcal{U}\left(x\right)=\sin (2\pi x∕l)+0.2\sin (4\pi x∕l-0.45)-0.06\sin (6\pi x∕l-0.45)$. 

Figure 6

Net average velocity $V_{\mathrm{dc}}$ versus density $n$ of attracting particles. Notation is as in Fig. 5 with the same substrate potential $\mathcal{U}\left(x\right)={\mathcal{U}}^{\mathrm{ramp}}\left(x\right)$ used in Fig. 3: (a) temperature ratchet; (b) rocked ratchet. Here, the vertical dotted lines mark the critical particle density $n_{\mathrm{crit}}$ where the condensation takes place.

Figure 7

Schematics of the $n$ dependence of $V_{\mathrm{dc}}$ for a rocked ratchet. (a) The original potential (solid line) and flatter effective potential (dashed line) due to repelling interaction among particles. (b) The small amplitude of the ac force, which tilts the effective potential from ${\mathcal{U}}^{\mathrm{eff}}-Ax$ (upper panel) to ${\mathcal{U}}^{\mathrm{eff}}+Ax$ (lower panel), could not produce a net motion in the bare potential (solid line) at low temperatures because of the potential barriers. However, the suppressed barriers (dashed line) for a high density of repelling particles can be overcome resulting in directed particle motion. (c) At large amplitudes, the ac particle motion in the bare potential gets rectified as the tilt is strong enough. Indeed, a particle (solid circle) moves easily only when the potential is tilted to ${\mathcal{U}}^{\mathrm{eff}}-Ax$ [upper panel in (c)]; the suppression of the barriers also activates a substantial particle flow in the opposite direction, thus reducing the ratchet rectification power. For the attracting particles, the effective potential deepens with increasing $n$. The dependence of $V_{\mathrm{dc}}$ on $n$ for attracting particles is discussed in the text.