The dynamics of a magnetic active Brownian particle undergoing three-dimensional Brownian motion, both translation and rotation, under the influence of a uniform magnetic field is investigated. The particle self-propels at a constant speed along its magnetic dipole moment, which reorients due to the interplay between Brownian and magnetic torques, quantified by the Langevin parameter $\alpha$. In this work, the time-dependent active diffusivity and the crossover time (${\tau }^{\mathrm{cross}}$)—from ballistic to diffusive regimes—are calculated through the time-dependent correlation function of the fluctuations of the propulsion direction. The results reveal that, for any value of $\alpha$, the particle undergoes a directional (or ballistic) propulsive motion at very short times ($t\ll {\tau }^{\mathrm{cross}}$). In this regime, the correlation function decreases linearly with time, and the active diffusivity increases with it. It the opposite time limit ($t\gg {\tau }^{\mathrm{cross}}$), the particle moves in a purely diffusive regime with a correlation function that decays asymptotically to zero and an active diffusivity that reaches a constant value equal to the long-time active diffusivity of the particle. As expected in the absence of a magnetic field ($\alpha =0$), the crossover time is equal to the characteristic time scale for rotational diffusion, ${\tau }_{\mathrm{rot}}$. In the presence of a magnetic field ($\alpha >0$), the correlation function, the active diffusivity, and the crossover time decrease with increasing $\alpha$. The magnetic field regulates the regimes of propulsion of the particle. Here, the field reduces the period of time at which the active particle undergoes a directional motion. Consequently, the active particle rapidly reaches a diffusive regime at ${\tau }^{\mathrm{cross}}\ll {\tau }_{\mathrm{rot}}$. In the limit of weak fields ($\alpha \ll 1$), the crossover time decreases quadratically with $\alpha$, while in the limit of strong fields ($\alpha \gg 1$) it decays asymptotically as ${\alpha }^{-1}$. The results are in excellent agreement with those obtained by Brownian dynamics simulations.

• Received 31 August 2017

DOI:https://doi.org/10.1103/PhysRevE.96.052607