Diffusion of Asymmetric Swimmers

Particles moving along curved trajectories will diffuse if the curvature fluctuates sufficiently in either magnitude or orientation. We consider particles moving at a constant speed with either a fixed or a Gaussian distributed magnitude of curvature. At small speeds the diffusivity is independent of the speed. At larger particle speeds, the diffusivity depends on the speed through a novel exponent. We apply our results to intracellular transport of vesicles. In sharp contrast to thermal diffusion, the effective diffusivity increases with vesicle size and so may provide an effective means of intracellular transport.

Figure 1

(a) Particle trajectory with GC dynamics in $d=2$ with $\tilde{v}=0.1$. The particle does not complete a circular loop before $\mathbf{K}$ changes significantly. (b) With $\tilde{v}=100$. The particle can complete many circular loop before $\mathbf{K}$ changes; however, straight segments are seen when $|\mathbf{K}|$ is small. The result is a characteristic “knotty wool” appearance. In both cases $K_0=1$.

Figure 2

Dimensionless diffusivities $\tilde{D}\equiv D{K}_0^2\tau$ for rotating (open circles, ${\tilde{D}}_{R2}$) and Gaussian (filled circles, ${\tilde{D}}_{G2}$) curvature dynamics in $d=2$, plotted against dimensionless particle speed $\tilde{v}\equiv v{K}_0\tau$. Also shown with solid lines are the small $\tilde{v}$ asymptote $\tilde{D}=1$ and the large $\tilde{v}$ best-fit asymptote $\tilde{D}\sim {\tilde{v}}^{{\lambda }_{2d}}$. The inset shows the effective exponents, with the solid line indicating the best-fit ${\lambda }_{2d}=0.98\pm 0.02$.

Figure 3

Dimensionless diffusivities $\tilde{D}\equiv D{K}_0^2\tau$ for rotating (open circles, ${\tilde{D}}_{R3}$) and Gaussian (filled circles, ${\tilde{D}}_{G3}$) curvature dynamics in $d=3$, plotted against dimensionless particle speed $\tilde{v}\equiv v{K}_0\tau$. Solid lines show the exact result ${\tilde{D}}_{R3}=1/3$, as well as the large $\tilde{v}$ power-law asymptote ${\tilde{D}}_{G3}\sim {\tilde{v}}^{{\lambda }_{3d}}$. The inset shows effective exponents between sequential points, with a solid line indicating the best-fit ${\lambda }_{3d}=0.71\pm 0.01$.