Time-dependent inertia of self-propelled particles: The Langevin rocket

Many self-propelled objects are large enough to exhibit inertial effects but still suffer from environmental fluctuations. The corresponding basic equations of motion are governed by active Langevin dynamics, which involve inertia, friction, and stochastic noise for both the translational and orientational degrees of freedom coupled via the self-propulsion along the particle orientation. In this paper, we generalize the active Langevin model to time-dependent parameters and explicitly discuss the effect of time-dependent inertia for achiral and chiral particles. Realizations of this situation are manifold, ranging from minirockets (which are self-propelled by burning their own mass), to dust particles in plasma (which lose mass by evaporating material), to walkers with expiring activity. Here we present analytical solutions for several dynamical correlation functions, such as mean-square displacement and orientational and velocity autocorrelation functions. If the parameters exhibit a slow power law in time, we obtain anomalous superdiffusion with a nontrivial dynamical exponent. Finally, we constitute the “Langevin rocket” model by including orientational fluctuations in the traditional Tsiolkovsky rocket equation. We calculate the mean reach of the Langevin rocket and discuss different mass ejection strategies to maximize it. Our results can be tested in experiments on macroscopic robotic or living particles or in self-propelled mesoscopic objects moving in media of low viscosity, such as complex plasma.

Figure 1

Self-propelled inertial particle with center position $\mathrm{R}\left(t\right)$ at time $t$ moving with its center in the two-dimensional $xy$ plane. The particle position is indicated as $\mathrm{R}\left(t\right)$ (black arrow). Moreover, the particle possesses an orientational degree of freedom that is characterized by a unit vector $\widehat{\mathbf{n}}=(cos\varphi ,sin\varphi )$ with $\varphi$ denoting the angle relative to the $x$-axis. The particle self-propels along its orientation with the velocity $v_0\widehat{\mathbf{n}}$ (red arrow). It may also experience a torque $M$ along the $z$-axis leading to rotational motion as indicated by the blue arrow. The translational motion is further influenced by a translational friction $\xi$ and the noise strength $D$ (as indicated by the light red horizontal double arrow) while the rotational motion is influenced by a rotational friction ${\xi }_r$ and the orientational noise strength $D_r$ (as indicated by the light blue curved double arrow).

Figure 2

Schematic illustration of the different special setups for an active inertial particle. The particle is shown as a dark-gray sphere and its inertia is characterized by the particle mass $m$ and its moment of inertia $J$. (a) Time-independent inertia with constant $m$ and $J$ as a reference situation (gray background). (b) Directed mass ejection: Per unit time, the mass $\dot{m}\left(t\right)$ is ejected centrally with a velocity $-u\widehat{\mathbf{n}}\left(t\right)$ along the particle orientation $\widehat{\mathbf{n}}\left(t\right)$ which leads to a change $-u\widehat{\mathbf{n}}\dot{m}\left(t\right)$ in the translational momentum of the particle and a time-dependent particle mass $m\left(t\right)$ but a constant moment of inertia $J$ (red background). (c) Isotropic mass evaporation. Here the translational and the angular momentum of the particle are both conserved, but the particle mass $m\left(t\right)$ and moment of inertia $J\left(t\right)$ are time-dependent (green background). (d) Isotropic change in the particle shape. Here again the linear and the angular momentum of the particle are both conserved, the particle mass $m$ is constant, but the moment of inertia $J\left(t\right)$ is time-dependent (blue background).

Figure 3

Mean displacement $\langle \mathrm{\Delta }\mathrm{R}\left(t\right)\rangle$ in the $xy$-plane for a chiral particle with initial orientation along the $x$-axis for different moment of inertia, $J=0.1{\xi }_r/D_r$ (orange), $J=1{\xi }_r/D_r$ (red), and $J=10{\xi }_r/D_r$ (purple). Lengths are given in units of $l_p=v_0/D_r$. The parameters are $\omega =4D_r, m=0.1\xi /D_r$. The starting point at $t=0$ is denoted by a black dot. The spira mirabilis of the overdamped limit is plotted on the left (black) for comparison.

Figure 4

Mean-square displacement as a function of time on a double-logarithmic plot. (a) For an achiral particle with fixed mass $m$ and varied moment of inertia $J$. (b) For a chiral particle with fixed mass $m$ and varied moment of inertia $J$. The fixed parameters are $m=0.1\xi /D_r, D=0$ and the moment of inertia is $J=0.1{\xi }_r/D_r$ (orange), $J=1{\xi }_r/D_r$ (red), or $J=10{\xi }_r/D_r$ (purple). (c) For an achiral particle with varied $m$ and fixed $J$. (d) For a chiral particle with varied $m$ and fixed $J$. Here, the fixed parameters are $J=0.1{\xi }_r/D_r, D=0$ and the mass is $m=0.1\xi /D_r$ (orange), $m=1\xi /D_r$ (red), or $m=10\xi /D_r$ (purple).

Figure 5

Long-time diffusion coefficient $D_L$ as a function of the moment of inertia $J$ for different circling frequencies $\omega =10D_r, \omega =1D_r, \omega =0.1D_r$, and $\omega =0$. The translational diffusion coefficient was set to zero, $D=0$. In the inset, the global maximum point $J_{\text{max}}$ for a given circling frequency $\omega$ is depicted. The corresponding maximal value $D_L\left(J_{\text{max}}\right)$ is shown as a red dot in the main figure.

Figure 6

Maximal reach $\mathrm{\Delta }{R}_{\infty }$ as a function of the propellant mass fraction $\zeta$ for several burn times $\mathrm{\Delta }t={10}^{-2}/{\gamma }_0, \mathrm{\Delta }t=1/{\gamma }_0$, and $\mathrm{\Delta }t={10}^2/{\gamma }_0$. The inset depicts the optimal propellant mass fraction ${\zeta }_{\text{max}}$ for a given burn time $\mathrm{\Delta }t$. The corresponding maximal value $R_{\infty }\left({\zeta }_{\text{max}}\right)$ is shown as a red dot in the main figure.

Figure 7

(a) The optimal mean reach $max\left(\overline{\mathrm{\Delta }{R}_{\infty }}\right)$ maximized with respect to the propellant mass fraction $\zeta$ and the burn time $\mathrm{\Delta }t$ as a function of the rotational noise strength $D_r/{\gamma }_0$. (b) Optimal burn time $\mathrm{\Delta }{t}_{\text{max}}$ as a function of the rotational noise strength $D_r/{\gamma }_0$. (c) Optimal propellant mass fraction ${\zeta }_{\text{max}}$ as a function of the rotational noise strength $D_r/{\gamma }_0$. The transition from the complete extended mass ejection strategy to that of the fractional instantaneous mass ejection is marked by vertical black lines at $D_{r,\text{crit}}\approx 0.72{\gamma }_0$ in all three figures.

Figure 8

Comparison of the different special setups for an achiral active particle ($\omega =0$) with time-dependent inertia: (a) orientation autocorrelation function $C(0,t)$, (b) velocity autocorrelation function $Z(0,t)$, (c) delay function $d(0,t)$, (d) mean displacement along the initial orientation $\langle \mathrm{\Delta }\mathrm{R}(0,t)\rangle ·{\widehat{\mathbf{n}}}_0$, (e) mean-square displacement $\langle \mathrm{\Delta }{\mathrm{R}}^2(0,t)\rangle$, and (f) the corresponding scaling exponent $\alpha (0,t)$ for time-independent inertia (dashed black), directed mass ejection (red), isotropic mass evaporation (green), and an isotropic change in the particle shape (blue). Velocities are given in units of $v_0$, times in $1/D_r$, and lengths in $l_p=v_0/D_r$. The time dependencies of the mass $m\left(t\right)$ and the moment of inertia $J\left(t\right)$ for the different setups are summarized in Table 1. The remaining parameters are $D=0, {\gamma }_0=\xi /m_0=0.1D_r$, and ${\gamma }_{r,0}={\xi }_r/J_0=0.1D_r$.

Figure 9

Same as in Fig. 8 for a chiral particle with a spinning frequency of $\omega =0.1D_r$.