  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.
