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

2020/04/12 by Lei Li, Jian‐Guo Liu, Jian-Guo Liu +4 · 2 citations
Chemistry · Computer Science · Engineering · Mathematics · Physics and Astronomy · #37M05 #65N75 #78A30 #Applied mathematics #Boltzmann equation #Classical mechanics #Domain (mathematical analysis) #Electromagnetic Scattering and Analysis #Electrostatics and Colloid Interactions #FOS: Mathematics #Ion #Mathematical analysis #Mathematics #Nanopore and Nanochannel Transport Studies #Nernst equation #Numerical Analysis (math.NA) #Particle (ecology) #Physics #Poisson distribution #Poisson's equation #Poisson–Boltzmann equation #Quantum mechanics #Statistical physics #Statistics #Truncation (statistics) #cs.NA #math.NA #msc:37M05 #msc:65N75 #msc:78A30

paper · pdf · doi:10.48550/arxiv.2004.05614

published in arXiv (Cornell University) (Cornell University) · 36 pages, 16 figures

openalex publication_date 2020/04/12 · openalex created_date 2021/08/16 · arxiv created 2022/03/24 · arxiv updated 2022/03/25 · openalex updated_date 2026/08/06

Abstract

We consider in this paper random batch interacting particle methods for solving the Poisson-Nernst-Planck (PNP) equations, 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 the PB equation. Then, the random batch interacting particle methods are introduced which are 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 their simplicity and adaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effects and interactions in the particle methods and to describe phenomena beyond the scope of the mean-field equations.

Citations

• Die Berechnung optischer und elektrostatischer Gitterpotentiale
• Approximations for stochastic differential equations with reflecting convex boundaries
• Particle mesh Ewald: An N⋅log(N) method for Ewald sums in large systems
• Asymptotic analysis of the Poisson–Boltzmann equation describing electrokinetics in porous media
• A reduced basis method for the nonlinear Poisson-Boltzmann equation
• Molecular Mean-Field Theory of Ionic Solutions: A Poisson-Nernst-Planck-Bikerman Model

Cited by

• Propagation of chaos and approximation error of random batch particle system in the mean field regime

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