We consider in this paper random batch interacting particle methods for solving the Poisson-Nernst-Planck (PNP) equation, and thus the Poisson-Boltzmann (PB) equation as the equilibrium, in the external unbounded domain. To justify the simulation in a truncated domain, an error estimate of the truncation is proved in the symmetric cases for PB equation. Then, the random batch interacting particle methods are introduced which are $\mathcal{O}(N)$ per time step. The particle methods can not only be considered as a numerical method for solving the PNP and PB equations, but also can be used as a direct simulation approach for the dynamics of the charged particles in solution. The particle methods are preferable due to its simplicity and adaptivity to complicated geometry of the surfaces, and may be interesting in describing the dynamics of the physical process.

翻译：在本文中,我们考虑随机批量互动粒子方法来解决Poisson-Nernst-Planck(PNP)方程式,从而将Poisson-Boltzmann(PB)方程式视为外部无约束域的平衡。为了证明在短空域进行模拟的合理性,PB方程式的对称性案例证明了对短程的错误估计。然后,引入随机批量互动粒子方法,即每时间步骤$\mathcal{O}(N)$。粒子方法不仅可以被视为解决PNP和PB方程式的数值方法,还可以用作溶液中带电荷粒子动态的直接模拟方法。粒子方法更可取,因为粒子的简单性和适应地表的复杂几何性,并且可能有趣地描述物理过程的动态。