Universal Poisson statistics of a passive tracer diffusing in dilute active suspensions

TL;DR

This paper derives a universal Poisson statistical model for a passive tracer in dilute active suspensions, revealing that the tracer's behavior can be exactly described by a spatial Poisson process with independent scattering events, valid across dimensions and interaction types.

Contribution

It provides a first-principles derivation of the Poisson statistics for tracers in active suspensions, connecting Markovian dynamics to non-Markovian processes through a variable transformation.

Findings

Tracer statistics are exactly represented as a spatial Poisson process.

The Poisson representation holds in any dimension and for arbitrary interactions.

The work analytically derives a non-Markovian process from Markovian dynamics.

Abstract

The statistics of a passive tracer immersed in a suspension of active self-propelled particles (swimmers) is derived from first principles by considering a perturbative expansion of the tracer interaction with the microscopic swimmer field. To first order in the swimmer density, the tracer statistics is exactly represented as a spatial Poisson process combined with independent swimmer-tracer scattering events, rigorously reducing the multi-particle dynamics to two-body interactions. The Poisson representation is valid in any dimensions and for arbitrary interaction forces and swimmer dynamics. It provides in particular an analytical derivation of the coloured Poisson process introduced in [K. Kanazawa et al.; Nature 579, 364 (2020)] highlighting that such a non-Markovian process can be obtained from Markovian dynamics by a variable transformation.