Macroscopic fluctuation theory of interacting Brownian particles

We apply the macroscopic fluctuation theory (MFT) to study the large-scale dynamical properties of Brownian particles with arbitrary pairwise interaction. By combining it with standard results of equilibrium statistical mechanics for the collective diffusion coefficient, the MFT gives access to the exact large-scale dynamical properties of the system, both in- and out-of-equilibrium. In particular, we obtain exact results for dynamical correlations between the density and the current of particles. For one-dimensional systems, this allows us to obtain a precise description of these correlations for emblematic models, such as the Calogero and Riesz gases, and for systems with nearest-neighbor interactions such as the Rouse chain of hardcore particles or the recently introduced model of tethered particles. Tracer diffusion with the single-file constraint (but for arbitrary pairwise interaction) is also studied. For higher-dimensional systems, we quantitatively characterize these dynamical correlations by relying on standard methods such as the virial expansion.

💡 Research Summary

The paper presents a comprehensive application of macroscopic fluctuation theory (MFT) to interacting Brownian particles with arbitrary pairwise potentials. Starting from the microscopic overdamped Langevin equations for N particles, the authors introduce the microscopic density field and then perform a diffusive coarse‑graining (space scaled by Λ≫1, time by Λ²) to obtain a macroscopic density ρ(x,t) and current j(x,t). The central MFT ansatz writes the current as a sum of a deterministic Fickian term and a weak Gaussian noise term:
 j = –D(ρ)∇ρ – σ(ρ)Λ^{-d/2}η,
where D(ρ) is the collective diffusion coefficient, σ(ρ) the mobility, and η a space‑time white noise. The conservation law ∂ₜρ+∇·j=0 together with this constitutive relation yields a stochastic hydrodynamic description that is amenable to a path‑integral formulation. By introducing a conjugate field H, the probability of a trajectory is expressed as exp