  We consider the collective behaviour of active particles that locally align
with their neighbours. Agent-based simulation models have previously shown that
in one dimension, these particles can form into a flock that maintains its
stability by stochastically alternating its direction. Until now, this
behaviour has been seen in models based on continuum field equations only by
appealing to long-range interactions that are not present in the simulation
model. Here, we derive a set of stochastic field equations with local
interactions that reproduces both qualitatively and quantitatively the
behaviour of the agent-based model, including the alternating flock phase. A
crucial component is a multiplicative noise term of the voter model type in the
dynamics of the local polarization whose magnitude is inversely proportional to
the local density. We show that there is an important subtlety in determining
the physically appropriate noise, in that it depends on a careful choice of the
field variables used to characterise the system. We further use the resulting
equations to show that a nonlinear alignment interaction of at least cubic
order is needed for flocking to arise.
