Density Fluctuations in Stochastic Kinematic Flows

TL;DR

This paper systematically examines finite-size stochastic fluctuations in particle models of kinematic flows, revealing phenomena like traffic quasi-jams, pattern formation, sub-diffusion, and wave fluctuations.

Contribution

It provides a comprehensive analysis of stochastic effects in finite particle systems with density-dependent velocity and diffusion, highlighting new emergent behaviors.

Findings

Identification of quasi-jams in traffic models

Observation of stochastic pattern formation

Detection of anomalous sub-diffusion and wave fluctuations

Abstract

At the macroscopic scale, many important models of collective motion fall into the class of kinematic flows for which both velocity and diffusion terms depend only on particle density. When total particle numbers are fixed and finite, simulations of corresponding microscopic dynamics exhibit stochastic effects which can induce a variety of interesting behaviours not present in the large system limit. In this article we undertake a systematic examination of finite-size fluctuations in a general class of particle models whose statistics correspond to those of stochastic kinematic flows. Doing so, we are able to characterise phenomena including: quasi-jams in models of traffic flow; stochastic pattern formation amongst spatially-coupled oscillators; anomalous bulk sub-diffusion in porous media; and travelling wave fluctuations in a model of bacterial swarming.