Effect of finite-size fluctuations in the small-N regime

Characterize the impact of finite-size fluctuations on the emergent swarm density when the number of agents N is small for the stochastic swarming model defined by the measurement-based movement and collision-ejection reactions (Eqs.(1)–(2)) and its mean-field counterpart (Eq.(MFpatternFormation)).

Background

The paper quantifies pattern formation accuracy and convergence in the large-N limit via mean-field and small-fluctuation analyses, and identifies finite-N crossovers where errors saturate. However, for small numbers of agents, fluctuations can be significant and may lead to deviations from mean-field predictions.

The authors explicitly flag the need to understand the role of finite-size fluctuations in the small-N regime, including how these stochastic effects alter steady-state error and convergence behavior relative to the target density.