Probability density function of the unbalanced impulse in Langevin theory of Brownian motion

Abstract

This paper attempts to find a probability distribution for the white noise (rapidly fluctuating unbalanced force) in the Langevin Equation. Unbalanced force is the resultant impulse provided to the brownian particle by the colliding fluid molecules. Therefore, a probability distribution of the speed of the particles after each impact will have the same probability distribution of the white noise. Such a distribution is discovered in this work by constructing a simple model based on thermal molecules colliding with the particle from all directions. The molecules obey Maxwell-Boltzmann speed distribution law. At low temperatures, for bigger brownian particles, existence of some non-random distribution for the unbalanced impulse, in itself is an interesting result. The distribution takes a near half gaussian form at these limits. At high temperatures, for small brownian particles(e.g: pollen grains), the distribution is shown to approach uniform distribution, and hence consistent with bulk of well established theoretical assumptions and experimental results in the literature that claims the unbalanced force to be a random white noise.