Effective diffusion of confined active Brownian swimmers

We theoretically find the effect of confinement and thermal fluctuations on the diffusivity of a spherical active swimmer moving inside a two-dimensional narrow cavity of general shape. The explicit formulas for the effective diffusion coefficient of a swimmer moving inside two particular cavities are presented. We also compare our analytical results with Brownian dynamics simulations and we obtain excellent agreement.

Figure 1

Schematics of the two studied cavities. (a) A zig-zag cavity of width $w\left(x\right)$, narrow cross-section $b,$ and period $L.$ (b) A semicircular cavity of width $w\left(x\right),$ radius $R,$ period $L,$ and narrow cross-section $2\alpha$. The cavities shown are the result of the performed Brownian simulations. Here, each color represents the followed random path of a given active particle.

Figure 2

Effective diffusion coefficient $D_E$ [Eq. (15)] for two zig-zag cavities with $\lambda =0.5$ and $\lambda =1$, normalized by $D_0=D_B+U^2/2D_{\Omega }$ (free active diffusion), plotted as a function of $L/b$ (solid blue line). Brownian simulation results are shown as red circles.

Figure 3

Brownian simulation results for the msd at short times. Panel (a) shows a transition of the msd with respect to time from a linear to a quadratic dependence and, finally, again to a linear dependence. Panel (b) shows the typical linear dependence of the msd with time for long times for a zig-zag cavity with $L/b=10$.

Figure 4

Effective diffusion coefficient $D_E$ for a semicircular cavity [Eq. (17)] normalized by $D_0=D_B+U^2/2D_{\Omega }$ (free active diffusion), plotted as a function of $\alpha /R$ (solid blue line). Brownian simulation results are shown as red circles. Inset: A typical comparison among the theoretical (blue dashed line) and numerical (red circles) msd of a cavity with $\alpha /R=0.4$.

Figure 5

Theoretical and numerical effective diffusion coefficient $D_E$ for a zig-zag and semicircular cavity for different swimming speeds. Panel (a) shows the results for the zig-zag cavity. Panel (b) shows the results for the semicircular cavity. In both figures, the solid lines are the theory while the red circles are the Brownian simulations results. Error bars (giving the standard deviation) in panel (a) may be smaller than the circles indicating the numerical value.