Onsager's Hard Rod Fluid

A fluid consisting of long thin rods without any forces between them undergoes a transition from an isotropic state to an ordered nematic phase above a critical density. This intuitive result was first derived by Lars Onsager. It explains the transition from an isotropic to an ordered state in lyotropic liquid crystals as the increase of positional entropy is compensated by the decrease of orientational entropy.

I will sketch a simplified version of Onsager's original approach which was developped by Zwanzig. Both theories differ in the treatment of the orientational degrees of freedom. While Onsager assumes them as continuous variables, the alternative approach of Zwanzig allows only three different orientations. This simplification proves to be quite advantageous for the mathematical analysis as it permits a more accurate calculation of the virial coefficients while Onsager's model involves non-linear integral equations.