Steady-State Random Walks with Application to Homogeneous Nucleation

Abstract

The kinetics of steady-state random walks on the positive integers, between a source of 0 and a sink at i*, are treated with elementary techniques from the theory of stochastic processes. It is shown that the steady-state flux of random walkers past any point i is equal to the steady-state number density at i multiplied by the probability that a random walker at i will never return to i after taking a forward step. The mean time it takes a random walker to traverse integers is derived and related to the flux. For physical processes such as membrane transport and nucleation, the steady-state number density at i is shown to be equal to the equilibrium number density at i multiplied by the probability that a randomly walking particle at i will return to zero. Homogeneous nucleation in the vapor phase is dealt with at length and serves to demonstrate the connection between physical processes and the present treatment of random walks. An exact, discrete expression for the Zeldovich factor is derived. It is shown that the traditional expression for the Zeldovich factor which uses continuous functions is quite accurate.