Coherent-state field-theoretic simulations of fluctuation-dominated reaction–diffusion systems

Stochastic reaction–diffusion systems can exhibit fluctuation-dominated kinetics that is not captured by well-mixed rate equations, particularly at and below the upper critical dimension. Here, we develop and benchmark a coherent-state field-theoretic simulation framework derived from the Doi–Peliti representation of microscopic reaction–diffusion dynamics. Using single-species annihilating random walks, A + A → ∅, as a test model, we compare two complementary numerical treatments: direct complex Langevin (CL) sampling of the Doi–Peliti coherent-state action in fictitious time and a semi-classical (Semi-C) representation, leading to a stochastic Langevin dynamics in real time. In three dimensions, both approaches recover mean-field (MF) density decay within their respective stability windows, while the Semi-C method remains stable over substantially longer real times. In two dimensions, the Semi-C calculations exhibit finite-time deviations from MF kinetics and reaction-rate-dependent crossover behavior consistent with marginal fluctuation effects. In one dimension, the Semi-C treatment captures the expected early to intermediate crossover toward slower-than-MF decay but develops a late-time plateau when the quadratic annihilation contribution becomes statistically unresolved with the available trajectory ensemble. In both one and two dimensions, the direct CL approach exhibits poor stability and is thus impractical. Semi-C simulations were used to access equal-time connected structure factors and investigate non-uniform initial conditions, revealing fluctuation-induced spatial anticorrelations and finite-time memory of initial geometry. Overall, these results establish that Semi-C outperforms CL in fluctuation-dominated regimes of reaction–diffusion models formulated using the Doi–Peliti coherent-state framework. This work also provides a foundation for extensions to multispecies reactions, reversible association, aggregation, and chemically active soft matter.