  In this paper, we investigate a generalised model of $N$ particles undergoing
second-order non-local interactions on a lattice. Our results have applications
across many research areas, including the modelling of migration, information
dynamics and Muller's ratchet -- the irreversible accumulation of deleterious
mutations in an evolving population. Strikingly, numerical simulations of the
model are observed to deviate significantly from its mean-field approximation
even for large population sizes. We show that the disagreement between
deterministic and stochastic solutions stems from finite-size effects that
change the propagation speed and cause the position of the wave to fluctuate.
These effects are shown to decay anomalously as $(\log N)^{-2}$ and $(\log
N)^{-3}$, respectively -- much slower than the usual $N^{-1/2}$ factor. Our
results suggest that the accumulation of deleterious mutations in a Muller's
ratchet and the loss of awareness in a population may occur much faster than
predicted by the corresponding deterministic models. The general applicability
of our model suggests that this unexpected scaling could be important in a wide
range of real-world applications.
