Active motion on curved surfaces

A theoretical analysis of active motion on curved surfaces is presented in terms of a generalization of the telegrapher equation. Such a generalized equation is explicitly derived as the polar approximation of the hierarchy of equations obtained from the corresponding Fokker-Planck equation of active particles diffusing on curved surfaces. The general solution to the generalized telegrapher equation is given for a pulse with vanishing current as initial data. Expressions for the probability density and the mean squared geodesic displacement are given in the limit of weak curvature. As an explicit example of the formulated theory, the case of active motion on the sphere is presented, where oscillations observed in the mean squared geodesic displacement are explained.

Figure 1

Time dependence of the dimensionless mean squared geodesic displacement $\langle s^2\left(r\right)\rangle /R^2=\langle {\theta }^2\left(t\right)\rangle$ for different values of the ratio of the sphere radius $R$ to the persistent length $L$. In the left panel, $R/L=0.25$, 0.5, 1.0, 2.0, and 5.0, the effects of persistence are marked by the $t^2$ dependence and by the oscillations around the asymptotic value $〈{\theta }^2\left(t\right)〉=({\pi }^2-4)/2$, that characterizes the uniform distribution on the whole sphere (dotted lines). In the right panel $R/L=10$, 20, and 100, the effects of persistence are diminished and $\langle s^2\left(t\right)\rangle$ starts exhibiting the standard linear dependence for times larger than ${\gamma }^{-1}$. The thick-gray lines mark the linear $t$ and quadratic $t^2$ time dependence.

Figure 2

Shown is one of the two possible integration contours $\mathrm{\Gamma }$ to evaluate the integral in Eq. (C2). These contours are symmetric respect to the imaginary axis and they enclose the poles ${\alpha }_{+}$ and ${\alpha }_{-}$, respectively.