The Stationary Arrival Process of Independent Diffusers from a Continuum to an Absorbing Boundary Is Poissonian

Abstract

We consider the arrival process of infinitely many identical independent diffusion processes from an infinite bath to an absorbing boundary. Previous results on this problem were confined to independent Brownian particles arriving at an absorbing sphere. The present paper extends these results to general diffusion processes, without any symmetries and without resorting to explicit expressions for solutions to the relevant equations. It is shown that for general absorbing boundaries and force fields, the steady stream of arrivals is Poissonian with rate equal to the total flux on the absorbing boundary, as calculated from the continuum theory of diffusion with transport. The considered arrival problem arises in the theory of Langevin simulations of ions in electrolytic solutions. In a Langevin simulation ions enter and exit the simulation region, and it is necessary to compute the probability laws for their entrance times into the simulation. While the simulated ions inside the small simulation region interact with each other and with the far field of the surrounding bath and the applied voltage, the physical chemistry continuum description of the surrounding bath implies independent diffusion in a mean field. Under these conditions the result of this paper applies to the stream of new ions that arrive from the continuum bath into the discrete simulation region. The recirculation problem, of ions that have already visited and exited the simulation region, as well as the integration of these results into a simulation of interacting ions will be studied in separate papers.