Connecting a Discrete Ionic Simulation to a Continuum

An important problem in simulating ions in solution is the connection of the finite simulation region to the surrounding continuum bath. In this paper we consider this connection for a simulation of uncharged independent Brownian particles and discuss the relevance of the results to a simulation of charged particles (ions). We consider a simulation region surrounded by a buffer embedded in a continuum bath. We analyze the time course of the exchange process of particles between the simulation region and the continuum, including re-entrances of particles that left the simulation. We partition the particle population into (i) those that have not yet visited the simulation and (ii) those that have. While the arrival process into the simulation of population (i) is Poissonian with known rate, that of population (ii) is more complex. We identify the ordered set of re-entrance times of population (ii) as a superposition of an infinite number of delayed terminating renewal processes, where the renewal periods may be infinite with positive probability. The ordered entrance times of populations (i) and (ii) form the pooled process of injection times of particles into the simulation. We show that while the pooled process is stationary, it is not Poissonian but rather has infinite memory. Yet, under some conditions on the sizes of the simulation and buffer regions, it can be approximated by a Poisson process. This seems to be the first result on the time course of a discrete simulation of a test volume embedded in a continuum.