Current of interacting particles inside a channel of exponential cavities: Application of a modified Fick-Jacobs equation

Recently a nonlinear Fick-Jacobs equation has been proposed for the description of transport and diffusion of particles interacting through a hard-core potential in tubes or channels of varying cross section [Suárez et al., Phys. Rev. E 91, 012135 (2015)]. Here we focus on the analysis of the current and mobility when the channel is composed by a chain of asymmetric cavities and a force is applied in one or the opposite direction, for both interacting and noninteracting particles, and compare analytical and Monte Carlo simulation results. We consider a cavity with a shape given by exponential functions; the linear Fick-Jacobs equation for noninteracting particles can be exactly solved in this case. The results of the current difference (when a force is applied in opposite directions) are more accurate for the modified Fick-Jacobs equation for particles with hard-core interaction than for noninteracting ones.

Figure 1

Asymmetric cavity described by a combination of exponential functions; see Eq. (2). The parameters used are $A_1=4,a_1=0.035,\ell =50$, and $L=200$, where the distances are expressed in terms of the lattice spacing $a$ used in the MC simulations.

Figure 2

Concentration as a function of position when the force is applied to the right. $\delta =0.05$ and $c=0.4$. The dashed line was obtained from Eq. (1), and $▵$ corresponds to the noninteracting particles simulation, averaged over ${10}^6$ configurations. The continuous line is the corresponding theoretical solutions obtained from Eq. (7) for a system with interacting particles, and $▴$ corresponds to the interacting particles simulations, averaged over ${10}^8$ configurations. The time is ${10}^5$ MC time steps and thereafter (steady state). The length of the horizontal axis, $0\le x\le 200$, corresponds to the size of one cavity; see Fig. 1.

Figure 3

Current of particles when the force is applied to the left ($J_{-}<0$) or to the right ($J_{+}>0$). The continuous lines are the corresponding theoretical solutions obtained from Eq. (7). The dots correspond to numerical simulations: ■: $c=0.1$; $•:c=0.2$; $▴:c=0.4$. The dashed line is the solution to Eq. (1), and $\square$ are the corresponding simulation with noninteracting particles for $c=0.4$. In all cases we perform averages over $2\times {10}^6$ configurations. The time is ${10}^5$ MC time steps.

Figure 4

Difference in the current as a function of the force. The mean concentration is $c=0.1$. The dashed line was obtained from Eq. (1), and □ corresponds to the non-interacting particles simulation. The continuous line is the corresponding theoretical solution obtained from Eq. (7) and ■ corresponds to the hard-core interacting particles simulation. We perform averages over at least $2.5\times {10}^6$ configurations, and the time is ${10}^5$ MC time steps in both cases.

Figure 5

Difference in the current as a function of the force. The mean concentration is $c=0.4$. The dashed line was obtained from Eq. (1), and $▵$ corresponds to the noninteracting particles simulation. The continuous line is the corresponding theoretical solution obtained from Eq. (7), and $▴$ corresponds to the hard-core interacting particles simulation. We perform averages over $5\times {10}^6$ configurations, and the time is ${10}^5$ MC time steps in both cases.

Figure 6

Mobility as a function of the force when the force is applied to the right. The dashed line was obtained from Eq. (1), and □ corresponds to the noninteracting particles simulation. The continuous lines are the corresponding theoretical solutions obtained from Eq. (7) for each mean concentration: $c=0.1$ (■), $c=0.2(•)$, $c=0.3\left(⧫\right)$, and $c=0.4\left(▴\right)$. In all cases we perform averages over $2\times {10}^6$ configurations. The time is ${10}^5$ MC time steps.

Figure 7

Mobility as a function of the force when the force is applied to the left. The dashed line was obtained from Eq. (1), and □ corresponds to the noninteracting particles simulation. The continuous lines are the corresponding theoretical solutions obtained from Eq. (7) for each mean concentration: $c=0.1$ (■), $c=0.2(•)$, $c=0.3\left(⧫\right)$, and $c=0.4\left(▴\right)$. In all cases we perform averages over $2\times {10}^6$ configurations. The time is ${10}^5$ MC time steps.