Phase Transitions in Brownian Pumps

We study stochastic particle transport between two reservoirs along a channel, where the particles are pumped against a bias by a traveling wave potential. It is shown that phase transitions of period-averaged densities or currents occur inside the channel when exclusion interactions between the particles are taken into account. These transitions reflect those known for the asymmetric simple exclusion process. We argue that their occurrence is a generic feature of Brownian motors operating in open systems.

Figure 1

(a) Sketch of a channel with $N$ equidistant sites in contact with two particle reservoirs $L$ and $R$ to the left and right. (b) Illustration of the site energies ${ϵ}_i$ along the channel. The dashed blue line marks a linear increase of the (time-averaged) site energies from left to right by a bias $F>0$. A time-dependent traveling wave potential allows for pumping of particles against the bias. Its superposition with the bias potential yields the total potential marked by the solid red line.

Figure 2

Simulated current $\overline{J}$ as a function of $F$ for equilibrated-bath couplings with reservoir densities ${\rho }_L=0.1$ and ${\rho }_R=0.28$, and parameters $\lambda =4$, $\tau =2$, and $A=2$ of the traveling wave potential. The inset shows the density profiles for two values $F=-0.1$ and 0 before ($F<F_{\star }\cong 0.09$, ${\overline{\rho }}_B\cong 0.096$) and two values $F=0.2$ and 0.3 after ($F>F_{\star }$, ${\overline{\rho }}_B\cong 0.34$) the phase transition (symbols in the main plot and inset were assigned correspondingly). For $F=0.3$, the oscillations of the period-averaged density ${\overline{\rho }}_i$ were zoomed out.

Figure 3

(a) Period-averaged current $\overline{J}$ as a function of period-averaged density $\overline{\rho }$ in a bulk system for three different $F$ values, and fixed parameters $\lambda =4$, $\tau =2$, and $A=3$ of the traveling wave potential. Data were obtained by KMC simulations of a closed channel. (b) Phase diagram derived from (a) by applying the minimum and maximum current principles. Phase transitions of first order are marked by thick lines, and transitions of second order by thin lines. The assignment of line types (and colors) to the $F$ values is the same as in the legend of (a). The five different phases are labeled by roman numbers with respective bulk densities given in brackets. The cross marks the point $({\rho }_L,{\rho }_R)=(0.1,0.28)$ used in the example for demonstrating the influence of a phase transition on $\overline{J}$ in Fig. 4; in (a), the densities 0.1 and 0.28 are marked by vertical dashed lines.

Figure 4

Current $\overline{J}$ as a function of $F$ for fixed reservoir densities ${\rho }_L=0.1$ and ${\rho }_R=0.28$, and the same parameters of the traveling wave potential as in Fig. 3. The open circles refer to closed channels. The filled colored symbols refer to the simulated $\overline{J}$ in open channels for bulk-adapted couplings at $F=0.2$, 0.4, 0.6, and 0.8. The respective density profiles are shown in the inset.