# Maxwell and the normal distribution: A colored story of probability, independence, and tendency toward equilibrium

## TL;DR

This paper reevaluates Maxwell's 1860 work, emphasizing the role of molecular collision models in deriving the normal distribution of particle velocities and his early proof of a tendency toward equilibrium, challenging common interpretations.

## Contribution

It highlights Maxwell's use of collision models in deriving the normal distribution and his early proof of equilibrium tendency, clarifying historical and conceptual misunderstandings.

## Key findings

- Maxwell's collision model is crucial for deriving the normal distribution.
- Maxwell's Proposition VI provides an early proof of tendency toward equilibrium.
- Potential conflation of probabilistic and value independence affected Maxwell's assumptions.

## Abstract

We investigate Maxwell's attempt to justify the mathematical assumptions behind his 1860 Proposition IV according to which the velocity components of colliding particles follow the normal distribution. Contrary to the commonly held view we find that his molecular collision model plays a crucial role in reaching this conclusion, and that his model assumptions also permit inference to equalization of mean kinetic energies (temperatures), which is what he intended to prove in his discredited and widely ignored Proposition VI. If we take a charitable reading of his own proof of Proposition VI then it was Maxwell, and not Boltzmann, who gave the first proof of a tendency towards equilibrium, a sort of H-theorem. We also call attention to a potential conflation of notions of probabilistic and value independence in relevant prior works of his contemporaries and of his own, and argue that this conflation might have impacted his adoption of the suspect independence assumption of Proposition IV.