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

TL;DR

This paper derives a probability distribution for the unbalanced impulse in Langevin Brownian motion, showing it varies from near half-Gaussian at low temperatures to uniform at high temperatures, aligning with theoretical and experimental expectations.

Contribution

It introduces a model based on Maxwell-Boltzmann statistics to derive the impulse distribution, revealing temperature-dependent behavior of the unbalanced force in Brownian motion.

Findings

Distribution approaches half-Gaussian at low temperatures for large particles

Distribution tends toward uniform at high temperatures for small particles

Results align with established theoretical and experimental insights

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.