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 behaviors 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 characterize phenomena including quasi-jams in models of traffic flow; stochastic pattern formation among spatially coupled oscillators; anomalous bulk subdiffusion in porous media; and travelling wave fluctuations in a model of bacterial swarming.


  1. kinematic flows
  2. interacting particle systems
  3. traffic modelling
  4. organism swarming
  5. anomalous sub diffusion

MSC codes

  1. 60H15
  2. 60H30
  3. 58J65
  4. 92C35
  5. 76Z05

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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1000 - 1024
ISSN (online): 1095-712X


Submitted: 3 May 2022
Accepted: 20 December 2022
Published online: 23 May 2023


  1. kinematic flows
  2. interacting particle systems
  3. traffic modelling
  4. organism swarming
  5. anomalous sub diffusion

MSC codes

  1. 60H15
  2. 60H30
  3. 58J65
  4. 92C35
  5. 76Z05



Department of Mathematical Sciences, The University of Bath, Bath BA2 7AY, UK.
Department of Mathematical Sciences, The University of Bath, Bath BA2 7AY, UK.
Department of Mathematical Sciences, The University of Bath, Bath BA2 7AY, UK.

Funding Information

Funding: This work was funded by EPSRC.

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