Brownian self-driven particles on the surface of a sphere

We present the dynamics of overdamped Brownian self-propelled particles moving on the surface of a sphere. The effect of self-propulsion on the diffusion of these particles is elucidated by determining their angular (azimuthal and polar) mean-square displacement. Short- and long-times analytical expressions for their angular mean-square displacement are offered. Finally, the particles' steady marginal angular probability density functions are also elucidated.

Figure 1

Schematics of the studied problem.

Figure 2

Comparison between theory and Brownian dynamics simulations for passive particles diffusing on a sphere. (a, b) Angular mean-square displacements versus time. The theoretical short times [Eqs. (10) and (11)] and long times [Eqs. (16) and (17)] results for different initial conditions are presented in dashed lines, while the numerical results are in solid lines. (c) Time evolution of the marginal polar PDF. The analytical results for $p(\theta ;t)$ are presented in dashed and solid lines, whereas the numerical results are represented with bars. (d, e) Marginal steady PDFs. The theoretical expressions given by Eq. (13) and (14) are presented in solid (red) lines, while the numerical results appear with bars. (f) The theoretical value $\langle cos\theta \rangle =exp\left(-2D_{T}t/r^2\right)$ (solid red line) is compared with its numerical value (blue circles).

Figure 3

Comparison between theory and Brownian dynamics simulations for active particles diffusing on a sphere. (a, b) Time evolution of the marginal angular PDFs for $(P{e}_R,\overline{\kappa })$ set to $\left(1.95,0.0021\right)$. Here, Eqs. (22) and (23) were numerically solved using the statistics generated from Eqs. (18, 19, 20, 21). (c, d) Steady marginal angular PDFs. From these figures one infers that for ABPs $p\left(\theta \right)=sin\left(\theta \right)/2$ and $p\left(\varphi \right)=1/2\pi$ (black dashed lines). In the same figures, we superpose in red solid lines, the steady marginal PDFs for the passive case. Note that for the polar and azimuthal angles, passive and active marginal steady PDFs coincide. (e, f) Curvature effect on the angular MSD. Here, the dimensionless number $\overline{\kappa }$ was taken as $\overline{\kappa }=\left\{0.0001,0.0002,0.0005,0.0015\right\}$. The theoretical results given by Eqs. (24) and (25) are presented in dotted-dashed black lines.

Figure 4

(a, b) Comparison between theoretical results and Brownian dynamics simulations for different swimming velocities namely, $P{e}_R=\left\{0,0.41,0.81\right\}$, and for the same curvature parameter $\overline{\kappa }=0.0015.$ Here, the dashed lines represent the theoretical long time MSD results given by Eqs. (24) and (25). The short times results [Eqs. (10) and (17)] are represented in dotted-dashed lines. Note that for comparison purposes, we have included in solid red lines, the results for passive particles. (c) Passive Brownian particles ($P{e}_{{}_R}=0$) on a sphere with $\overline{\kappa }=0.0015$ and for $\tilde{t}=\{5\left(\text{red}\right),25\left(\text{yellow}\right),50\left(\text{green}\right)\}$. (d) Active Brownian particles ($P{e}_{{}_R}=0.81$) with $\overline{\kappa }=0.0015$ and for $\tilde{t}=\{0.74\left(\text{red}\right),1.29\left(\text{yellow}\right),1.79\left(\text{green}\right)\}$. In both cases, ${\theta }_0=0.4$ and ${\varphi }_0=\pi$.