This paper characterizes the limits of a large system of interacting particles distributed on the real line.The interaction occurring among neighbors involves two kinds of independent actions with different rates.This...This paper characterizes the limits of a large system of interacting particles distributed on the real line.The interaction occurring among neighbors involves two kinds of independent actions with different rates.This system is a generalization of the voter process,of which each particle is of type A or a.Under suitable scaling,the local proportion functions of A particles converge to continuous functions which solve a class of stochastic partial differential equations driven by Fisher-Wright white noise.To obtain the convergence,the tightness of these functions is derived from the moment estimate method.展开更多
We consider in this paper random batch interacting particle methods forsolving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann(PB) equation as the equilibrium, in the external unbounded domai...We consider in this paper random batch interacting particle methods forsolving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann(PB) equation as the equilibrium, in the external unbounded domain. To justify thesimulation in a truncated domain, an error estimate of the truncation is proved inthe 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 cannot 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 chargedparticles in solution. The particle methods are preferable due to their simplicity andadaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effectsand interactions in the particle methods and to describe phenomena beyond the scopeof the mean-field equations.展开更多
文摘This paper characterizes the limits of a large system of interacting particles distributed on the real line.The interaction occurring among neighbors involves two kinds of independent actions with different rates.This system is a generalization of the voter process,of which each particle is of type A or a.Under suitable scaling,the local proportion functions of A particles converge to continuous functions which solve a class of stochastic partial differential equations driven by Fisher-Wright white noise.To obtain the convergence,the tightness of these functions is derived from the moment estimate method.
基金This work is partially supported by the National Key R&D Program of China,Project Number 2021YFA1002800The work of L.Li was partially sponsored by the Strategic Priority Research Program of Chinese Academy of Sciences,Grant No.XDA25010403,and NSFC 11901389,12031013The work of J.-G.Liu was supported by NSF DMS-2106988.
文摘We consider in this paper random batch interacting particle methods forsolving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann(PB) equation as the equilibrium, in the external unbounded domain. To justify thesimulation in a truncated domain, an error estimate of the truncation is proved inthe 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 cannot 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 chargedparticles in solution. The particle methods are preferable due to their simplicity andadaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effectsand interactions in the particle methods and to describe phenomena beyond the scopeof the mean-field equations.
