A Stochastic Approach to Maxwell Velocity Distribution via Central Limit Theorem and Boltzmann's Entropy Formula

TL;DR

This paper introduces a new derivation of Maxwell's velocity distribution using the central limit theorem and Boltzmann's entropy formula, offering deeper insight into its universality and the role of Boltzmann's constant.

Contribution

It presents a novel, general derivation of Maxwell's distribution based on probability theory and entropy, avoiding the ideal gas law explicitly.

Findings

Derivation using CLT explains Gaussian velocity distribution.

Shows the connection between entropy and velocity distribution.

Proves Maxwell distribution for harmonic oscillators.

Abstract

Maxwell's velocity distribution is known to be universally valid across systems and phases. Here we present a new and general derivation that uses the central limit theorem (CLT) of the probability theory. This essentially uses the idea that repeated intermolecular collisions introduce randomness in the velocity change in the individual components of the velocity vector, leading to, by the CLT, a Gaussian distribution. To complete the derivation, we next show that the mean-square velocity or the standard deviation follows exactly from Boltzmann's formula relating entropy to the density of states, thus avoiding the use of the ideal gas equation of state explicitly. We furthermore prove the Maxwell velocity distribution for a system of harmonic oscillators. This derivation provides a further insight into the origin of Boltzmann's constant in the Maxwell velocity distribution and also in the equipartition theorem. We propose that this derivation provides an approach that explains the universality of Maxwell Gaussian distribution.