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

TL;DR

This paper extends active Langevin dynamics to include time-dependent inertia, providing analytical solutions and introducing the 'Langevin rocket' model to optimize self-propelled particle reach, applicable to various physical and biological systems.

Contribution

It generalizes the active Langevin model with time-dependent parameters and introduces the Langevin rocket concept for optimizing propulsion strategies.

Findings

Analytical solutions for correlation functions are derived.

Superdiffusive behavior emerges with slow parameter variations.

The Langevin rocket model predicts maximum reach based on mass ejection strategies.

Abstract

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, 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 the mean-square displacement and the orientational and velocity autocorrelation functions. If the parameters exhibit a slow power-law in time, we obtain anomalous superdiffusion with a non-trivial 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.