Change in distributions of particle positions by Brownian diffusion in a non-uniform external field

General expressions for the particle position distribution in non-uniform external fields are provided based on an investigation of distribution function. For a case of uniform or vanishing external fields, the particle position distribution with time results in a normal distribution with a variance as obtained for pure Brownian motion. To account for the non-uniformity of external fields, simple linearization methods were applied and the change in particle position distribution was investigated. A procedure to determine time evolution of change in variance of particle position distribution in non-uniform external fields is presented. For any non-uniform external field described by an analytic function, this simple procedure can accurately calculate the spreading characteristics of nanoparticles in the diffusional regime. This approach is used to predict particle diffusional spreading in a coaxial DMA, and the results are seen to closely match Monte Carlo simulations.