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We consider in this paper random batch interacting particle methods forsolving the Poisson-Nernst-Planck (PNP) equations, and thus thePoisson-Boltzmann (PB) equation as the equilibrium, in the external unboundeddomain. To justify the simulation in a truncated domain, an error estimate ofthe truncation is proved in the symmetric cases for the PB equation. Then, therandom batch interacting particle methods are introduced which are $O(N)$ pertime step. The particle methods can not only be considered as a numericalmethod for solving the PNP and PB equations, but also can be used as a directsimulation approach for the dynamics of the charged particles in solution. Theparticle methods are preferable due to their simplicity and adaptivity tocomplicated geometry, and may be interesting in describing the dynamics of thephysical process. Moreover, it is feasible to incorporate more physical effectsand interactions in the particle methods and to describe phenomena beyond thescope of the mean-field equations.

Some random batch particle methods for the Poisson-Nernst-Planck and Poisson-Boltzmann equations