Onsager's missing steps retraced

TL;DR

This paper revisits Onsager's theory of phase transitions in anisotropic colloidal systems, clarifying its validity range and providing a more rigorous derivation based on Penrose's tree identity and mean-field approaches.

Contribution

It offers a revised theoretical framework that extends Onsager's original theory to realistic densities using advanced mathematical tools.

Findings

Onsager's theory is valid for a reasonable density range

A new derivation clarifies the original theory's limitations

The approach uses Penrose's tree identity and mean-field methods

Abstract

Onsager's paper on phase transition and phase coexistence in anisotropic colloidal systems is a landmark in the theory of lyotropic liquid crystals. However, an uncompromising scrutiny of Onsager's original derivation reveals that it would be rigorously valid only for ludicrous values of the system's number density (of the order of the reciprocal of the number of particles) Based on Penrose's tree identity and an appropriate variant of the mean-field approach for purely repulsive, hard-core interactions, our theory shows that Onsager's theory is indeed valid for a reasonable range of densities.