Stochastic Dynamic Density Functional Theory for Biased Random Organization

The critical point of Biased Random Organization (BRO), a non-equilibrium absorbing-state dynamical model, has recently been suggested as an alternative definition for the Random Close Packing of spheres. Although BRO has been extensively studied in numerical simulations, an analytical model of the process remains elusive. Dean's method, first introduced to study interacting Brownian particles in a thermal bath, is a powerful approach to systematically coarse-grain stochastic systems. Here, by mapping BRO to a stochastic differential equation describing the particle dynamics in the continuum limit, and extending Dean's method to systems exhibiting anisotropic, multiplicative Gaussian noise, we derive the mean field evolution of the coarse-grained density. Our analytical results, combined with numerical simulations, shed light on the density evolution and steady-state distribution of a system of particles evolving according to BRO. Although motivated by BRO, our results are generically applicable to any dynamical system having fluctuations due to pairwise interactions, which can be modeled as anisotropic, multiplicative Gaussian noise, such as mini batch gradient descent on particle systems.