Transport of interacting particles in a chain of cavities: Description through a modified Fick-Jacobs equation

We study the transport process of interacting Brownian particles in a tube of varying cross section. To describe this process we introduce a modified Fick-Jacobs equation, considering particles that interact through a hard-core potential. We were able to solve the equation with numerical methods for the case of symmetric and asymmetric cavities. We focused in the concentration of particles along the direction of the tube. We also preformed Monte Carlo simulations to evaluate the accuracy of the results, obtaining good agreement between theory and simulations.

Figure 1

(a) Symmetric and (b) asymmetric cavities used for testing the theory. The distances are expressed in terms of the lattice spacing $a$. See the text for further details.

Figure 2

Site concentration in the stationary state against position $x$ in a symmetric lattice for (a) $\overline{c}=0.5$ and (b) $\overline{c}=0.8$. For each mean concentration, the values of the force are $f=0.01$ (red squares), $0.1$ (black circles), and $0.5$ (blue triangles). Parameters of the cavity in (19) are $\alpha =10.5$, $\gamma =4$, and $L_x=100$. Full lines were obtained with 15 terms of Eq. (16). Dots were obtained with numerical simulations. The average is taken over ${10}^6$ samples between $t={10}^4$ and $t={10}^8$ Monte Carlo time steps.

Figure 3

Stationary state of site concentration along the direction of the channel in an asymmetric cavity for (a) $\overline{c}=0.2,$ (b) $\overline{c}=0.5,$ and (c) $\overline{c}=0.8$. In each case, the following values of the force were used: $f=0.1$ (blue triangles), $0.2$ (black squares), and $0.5$ (red circles). Solid lines were obtained with 12 terms of Eq. (16). Dots were obtained with simulations (with ${10}^5$ samples between $t={10}^4$ and $t={10}^7$ Monte Carlo time steps).

Figure 4

Numerical simulation of average density or site occupation frequency in an asymmetric cavity in the stationary state. Values of the mean concentration are (a) $\overline{c}=0.2$, (b) $\overline{c}=0.5$, and (c) $\overline{c}=0.8$. In all cases the force is $f=0.5$ (to the right). The number of samples is ${10}^5$.

Figure 5

Site concentration along the direction of the channel for interacting (blue circles) and noninteracting (red squares) particles in an asymmetric cavity. Solid lines were obtained with 12 terms of Eq. (16). Dots were obtained with numerical simulations. The parameters are $\overline{c}=0.5$ and $f=0.5$. Inset: Same information but for $\overline{c}=0.1$. In all cases, numerical simulations were averaged over ${10}^6$ samples between $t={10}^4$ and $t={10}^8$ Monte Carlo time steps.