Connecting a Discrete Ionic Simulation to a Continuum
An important problem in simulating ions in solution is the connection ofthe finite simulation region to the surrounding continuum bath. In this paper we consider this connection for a simulation ofuncharged independent Brownian particles and discuss the relevance ofthe results to a simulation ofcharged particles (ions). We consider a simulation region surrounded by a buffer embedded in a continuum bath. We analyze the time course ofthe exchange process ofparticles between the simulation region and the continuum, including re-entrances ofparticles that lef t 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 ofpopulation (i) is Poissonian with known rate, that ofpopulation (ii) is more complex. We identif y the ordered set ofre-entrance times ofpopulation (ii) as a superposition ofan infinite number ofdelayed terminating renewal processes, where the renewal periods may be infinite with positive probability. The ordered entrance times ofpopulations (i) and (ii) f orm the pooled process ofinjection times ofparticles 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 ofthe simulation and buffer regions, it can be approximated by a Poisson process. This seems to be the first result on the time course ofa discrete simulation ofa test volume embedded in a continuum.