Beyond Poisson-Boltzmann: Numerical sampling of charge density fluctuations

TL;DR

This paper introduces a novel numerical method for sampling charge density fluctuations in Coulomb systems by propagating a Langevin-like SPDE, simplifying computations and enabling local treatment of long-range interactions.

Contribution

It develops a local, stochastic PDE-based approach to sample charge fluctuations around the mean-field Poisson-Boltzmann solution, improving computational efficiency.

Findings

Successfully implemented a finite-volume SPDE approach

Demonstrated preliminary results on like-charge ions in counter-ion bath

Reduced numerical complexity of Coulomb fluctuation sampling

Abstract

We present a method aimed at sampling charge density fluctuations in Coulomb systems. The derivation follows from a functional integral representation of the partition function in terms of charge density fluctuations. Starting from the mean-field solution given by the Poisson-Boltzmann equation, an original approach is proposed to numerically sample fluctuations around it, through the propagation of a Langevin like stochastic partial differential equation (SPDE). The diffusion tensor of the SPDE can be chosen so as to avoid the numerical complexity linked to long-range Coulomb interactions, effectively rendering the theory completely local. A finite-volume implementation of the SPDE is described, and the approach is illustrated with preliminary results on the study of a system made of two like-charge ions immersed in a bath of counter-ions.