Multiple branches of ordered states of polymer ensembles with the Onsager excluded volume potential

We study the branches of equilibrium states of rigid polymer rods with the Onsager excluded volume potential in two-dimensional space. Since the probability density and the potential are related by the Boltzmann relation at equilibrium, we represent an equilibrium state using the Fourier coefficients of the Onsager potential. We derive a non-linear system for the Fourier coefficients of the equilibrium state. We describe a procedure for solving the non-linear system. The procedure yields multiple branches of ordered states. This suggests that the phase diagram of rigid polymer rods with the Onsager potential has a more complex structure than that with the Maier-Saupe potential. A study of free energy indicates that the first branch of ordered states is stable while the subsequent branches are unstable. However, the instability of the subsequent branches does not mean they are not interesting. Each of these unstable branches, under certain external potential, can be made metastable, and thus may be observed.