Enhancing Physical Realism in Mean-Field Continuum Models through Maximum Capacity Constraints on Concentration Fields in Active Matter

Mean-field continuum models are commonly used to simulate systems of multiple agents in suspension, such as bacteria or engineered micro-machines. Ensuring realistic physical constraints in computational models of concentration fields is essential for accurately representing such systems. We present a novel method to enforce a maximum capacity constraint on a computed concentration field within a specified domain, restricting the concentration values to a physical range. Our approach involves formulating a partial differential equation (PDE) for a concentration correction field that preserves the overall mass while ensuring the concentration remains within the physical range of capacity, and minimizes loss of energy. This projection step is incorporated into a mean-field continuum model constructed from the first three orientational moments of a probability density function that solves the Smoluchowski equation. This method enhances the physical realism of the simulation results, making them more applicable to real-world scenarios. We demonstrate the efficacy of our approach through various applications, highlighting its potential to improve the fidelity of simulations in fields ranging from biology to materials science.