Statistics for an object actively driven by spontaneous symmetry breaking into reversible directions

TL;DR

This paper models the stochastic dynamics of objects driven by spontaneous symmetry breaking, revealing velocity distributions and spatial patterns, with implications for active matter systems like bacteria and droplets.

Contribution

It introduces a formal mapping of symmetry-breaking driven motion to a quantum harmonic oscillator with a repulsive delta potential, providing new analytical insights.

Findings

Velocity distribution is double-peaked or ditched.

Spatial statistics show outward propagating maxima.

Diffusion coefficients are derived from the quantum analogy.

Abstract

Propulsion of otherwise passive objects is achieved by mechanisms of active driving. We concentrate on cases in which the direction of active drive is subject to spontaneous symmetry breaking. In our case, this direction will be maintained, until a large enough impulse by an additional stochastic force reverses it. Examples may be provided by self-propelled droplets, gliding bacteria stochastically reversing their propulsion direction, or nonpolar vibrated hoppers. The magnitude of active forcing is regarded as constant, and we include the effect of inertial contributions. Interestingly, this situation can formally be mapped to stochastic motion under (dry, solid) Coulomb friction, however, with a negative friction parameter. Diffusion coefficients are calculated by formal mapping to the situation of a quantum-mechanical harmonic oscillator exposed to an additional repulsive delta-potential. Results comprise a ditched or double-peaked velocity distribution and spatial statistics showing outward propagating maxima when starting from initially concentrated arrangements.