Stochastic drift in discrete waves of non-locally interacting-particles

TL;DR

This paper examines how finite-size effects cause stochastic drift in wave propagation of non-locally interacting particles, revealing slower decay of fluctuations than traditional models predict, with implications for various biological and physical systems.

Contribution

It demonstrates that finite-size effects lead to anomalously slow decay of stochastic fluctuations in non-local particle systems, challenging mean-field approximations.

Findings

Finite-size effects cause wave speed fluctuations.

Decay of stochastic effects scales as $( ext{log } N)^{-2}$ and $( ext{log } N)^{-3}$.

Implications for faster mutation accumulation in Muller's ratchet.

Abstract

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.