Active particles in noninertial frames: How to self-propel on a carousel

Typically the motion of self-propelled active particles is described in a quiescent environment establishing an inertial frame of reference. Here we assume that friction, self-propulsion, and fluctuations occur relative to a noninertial frame and thereby the active Brownian motion model is generalized to noninertial frames. First, analytical solutions are presented for the overdamped case, both for linear swimmers and for circle swimmers. For a frame rotating with constant angular velocity (“carousel”), the resulting noise-free trajectories in the static laboratory frame are trochoids if these are circles in the rotating frame. For systems governed by inertia, such as vibrated granulates or active complex plasmas, centrifugal and Coriolis forces become relevant. For both linear and circling self-propulsion, these forces lead to out-spiraling trajectories which for long times approach a spira mirabilis. This implies that a self-propelled particle will typically leave a rotating carousel. A navigation strategy is proposed to avoid this expulsion, by adjusting the self-propulsion direction at will. For a particle, initially quiescent in the rotating frame, it is shown that this strategy only works if the initial distance to the rotation center is smaller than a critical radius $R_c$ which scales with the self-propulsion velocity. Possible experiments to verify the theoretical predictions are discussed.

Figure 1

Five examples of circular motion in the rotating frame (left column) and the mapping back to the static laboratory frame (right column) via the rotation matrix indicated by the double arrow. The static origin of the rotation is marked as a bullet point. (a) Degenerate case of a linear swimmer ${\omega }_s=0$. (b) Special case ${\omega }_s/\omega =-1$ leading to circular trajectories in both frames. (c) Rational ratio ${\omega }_s/\omega >-1$ leading to epitrochoids. (d) Incommensurate frequencies when ${\omega }_s/\omega$ is irrational leading to covering of a ringlike area in the laboratory frame. (e) Rational ratio ${\omega }_s/\omega <-1$ leading to hypotrochoids.

Figure 2

Same as Fig. 1, but now for the mean trajectory $\overline{{\vec{r}}^{\prime }\left(t\right)}$ in the rotating frame (left column) and for the mean trajectory $\overline{\vec{r}\left(t\right)}$ in the laboratory frame (right column). (a) For a linear swimmer where the mean trajectory in the rotating frame is a linear segment which is correspondingly distorted due to the rotation. (b) For a circle swimmer where the mean trajectory in the rotating frame is a logarithmic spiral which is again transformed in the laboratory system.

Figure 3

Double logarithmic plot of the mean-square displacement as a function of time $t$ (a) for linear swimmers and (b) for circle swimmers. Length and time units are the persistence length ${\ell }_p=v_0/D_r$ and persistence time $t_p=1/D_r$. Energy units are in $k_{B}T$. In these units, the further parameters are (a) $M=0$, $\gamma =1$, ${\gamma }_R=0.1$, $\omega =1$ and (b) $M=2$, $\gamma =1$, ${\gamma }_R=0.1$, $\omega =1$. Reference slopes of 1 and 2 are also indicated.

Figure 4

Static balance of the centrifugal force, the self-propulsion force, and the friction force in a frame corotating with angular velocity $\omega +{\omega }_s$ where all forces are time independent.

Figure 5

Typical trajectories from the analytical solution of Eq. (25) in the laboratory frame. The unstable particular solution with radius $b$ is indicated as a black circle. The trajectory approaches a logarithmic spiral. The length unit is $v_0/\omega$ and the parameters are $\gamma /m\omega =2$, $\gamma =0.1$, $m=0.5$ and (a) ${\omega }_0/\omega =2$ and (b) ${\omega }_0/\omega =5$.

Figure 6

Same as Fig. 1 but now for a frame accelerated along the $x$ axis with a constant linear acceleration $a$. (a) The trajectory of a linear swimmer transforms into a parabola. (b) The swimming path of a circle swimmer in the noninertial frame is a “stretched trochoid” in the laboratory frame.

Figure 7

Same as Fig. 2 but now for a frame accelerated along the $x$ axis with a constant linear acceleration $a$. The mean displacements are shown for the overdamped case for (a) linear swimmers and (b) circle swimmers. The straight linear segment transforms into a stretched exponential if the initial orientation is perpendicular to the acceleration $\vec{a}$. The spira mirabilis of a circle swimmer transforms into a stretched spiral.