Varying Time Scales in the Travel Distance of Active Brownian Particles
Active Brownian motion refers to the dynamics of a self-propelled particle that changes its direction of movement randomly. The equations of motion for an active Brownian particle in two-dimensional space can be written as a coupled set of two stochastic differential equations describing the translational and angular motion, respectively [1,2].
In case of an active particle one finds that the (ensemble-averaged) dynamics of the travel distance takes place on varying time scales. The short-term travel distance is dominated by overdamped Brownian diffusion yielding a growth proportional to the square root of time (as for passive Brownian particles). However, in an intermediate ballistic regime the travel distance begins to scale linearly. This scaling is caused by the finite duration for which a specific orientation of a particle typically persists. In the long-term dynamics, the randomization of the orientation is again associated with a diffusive motion and therefore proportional to the square root of t [3,4].
The plots in this animation illustrate this change in scales over time in the form of travel distance over time (below) as well as the motion in the two-dimensional space for some representative realizations (right). Note that also the displayed time scale changes over time in order to capture all three regimes of dynamics.
The equations of motion are solved by means of the Euler-Maruyama method. The parameters are chosen as v0=10, D1=1, and D2=0.05. The ensemble average is approximated by a set of 1000 trajectories.
[1] G. Volpe, S. Gigan, and G. Volpe, "Simulation of the active Brownian motion of a microswimmer", Am. J. Phys. 82(7), 659 (2014).
[2] A. Callegari and G. Volpe, "Numerical simulations of active brownian particles" in Flowing Matter, 211 (2019).
[3] J. R. Howse et al. "Self-motile colloidal particles: from directed propulsion to random walk", PRL 99(4), 048102 (2007).
[4] C. C. Maass et al., "Swimming droplets", Annu. Rev. Condens. Matter Phys. 7, 171 (2016).