Self-Propelled Particles with Fluctuating Speed and Direction of Motion in Two Dimensions

We study general aspects of active motion with fluctuations in the speed and the direction of motion in two dimensions. We consider the case in which fluctuations in the speed are not correlated to fluctuations in the direction of motion, and assume that both processes can be described by independent characteristic time scales. We show the occurrence of a complex transient that can exhibit a series of alternating regimes of motion, for two different angular dynamics which correspond to persistent and directed random walks. We also show additive corrections to the diffusion coefficient. The characteristic time scales are also exposed in the velocity autocorrelation, which is a sum of exponential forms.

Figure 1

Scaled mean squared position as a function of scaled time for persistent random walks. The lines correspond to Eq. (9) for rescaled variables. The thin solid red line corresponds to $\mu =0$, and $\mu =100$ for the other curves, with $\gamma ={10}^{-3}$ (dashed green line), $\gamma ={10}^3$ (dotted blue line), and $\gamma ={10}^6$ (thick solid black line).

Figure 2

Scaled mean squared position, $\xi =x\alpha /⟨v⟩$, as a function of the scaled time, $\tau =\alpha t$ for directed random walks. The lines correspond to Eq. (11) for the rescaled variables, with $c={10}^{-4}$. The thin solid red line corresponds to $\mu =0$. The other curves correspond to $\mu =100$, with $\gamma ={10}^{-3}$ (dashed green line), $\gamma ={10}^3$ (dotted blue line), and $\gamma ={10}^6$ (thick solid black line).

Figure 3

(a) Rescaled mean squared displacement and (b) velocity autocorrelation as a function of scaled time for persistent random walks. The lines in (b) correspond to Eq. (12) for rescaled variables. In both panels, the solid red line corresponds to $\mu =0$. For the other curves $\mu =1.31$, with $\gamma ={10}^{-1}$ (dashed green line) and $\gamma =10$ (dotted blue line). Dots correspond to numerical simulations as described in the text.