On an integral equation in the theory of phase transitions for a system of magnetic rods

We study a nonlinear integral equation for the orientational distribution function (ODF) describing anisotropic nematic ordering in a system of magnetic rods. For highly elongated rods, we give a classification of bifurcation points and find their asymptotic behavior in the limits of small and large magnetic moments of the rods. We develop an algorithm to construct a nearly-isotropic ODF in the vicinity of the bifurcation point. We show that for both small and large magnetic moments, the ODF obtained has a left direction of bifurcation. However, for intermediate values of the magnetic moments, solutions with a right direction of bifurcation exist along with those with the left direction.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 3, pp. 427–440, December, 1996.