Exact multiplicity of nematic states for an Onsager model

摘要

Isotropic-nematic phase transition due to excluded volume effects is considered for the 2D Onsager model of rigid rod-like polymers with a general interaction kernel involving a finite number of Fourier modes. Nematic phases appear above a concentration threshold. We prove the existence of continuous branches of nematic states bifurcating from the isotropic phase at concentrations proportional to the Fourier coefficients of the interaction kernel. The bifurcation structure is shown to depend on the size of the spectral gaps of the interaction operator. Exact multiplicity of nematic phases is proved for a class of two-mode trigonometric kernels. Our arguments use simple bifurcation and variational tools applied to an equivalent finite dimensional problem that involves multivariable Bessel functions.