Bifurcation in Onsager's model of the isotropic-nematic transition

In this paper Onsager's theory of the orientational order in a three-dimensional system of hard rods is reanalyzed as a nonlinear eigenvalue problem. Bifurcation is found and the equation of state is calculated from the orientational distribution function for a nematic phase. We also investigate the corresponding twodimensional system of hard lines. The existence and order of a phase transition are shown to depend on both the direction of bifurcation and on properties of the global solutions. The analysis can be adapted to other nonlinear equations obtained in theories of liquid crystals.