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

Beyond Poisson-Boltzmann: Numerical sampling of charge density fluctuations