Effective single particle theory for active particles using local density fluctuations

We characterize the dynamic non-equilibrium steady state behavior of active particles using density fluctuations in the system. We analyze the effective local density around a particle in the steady state and numerically calculate its mean, variance and autocorrelation. Thus, using local density and its statistical properties as a temporally correlated stochastic variable, we develop an effective single-particle theoretical model and analytically derive an expression for the particle's diffusivity as a function of the global packing density in the system. Our theory accurately predicts the transport properties of an active particle, validated against numerical simulations. Unlike mean-field theory, which fails at high packing densities due to significant density fluctuations from dynamic cluster formation, our model remains effective across all densities. It also captures the well-known phase transition beyond a critical packing density. The key novelty of our model lies in the introduction of a stochastic local density field, which encapsulates the effect of steric interactions on an active particle and helps predict single-particle behavior in a collection, a feature often absent in standard active matter models. This approach could be useful in experimental setups where fluctuations in local density around a tagged particle are measurable.