Note on the number of steady states for a two-dimensional Smoluchowski equation

Abstract

Dynamics of concentrated polymer solutions are modelled by a Smoluchowski equation. At high concentrations, such solutions form liquid crystalline polymers of nematic structure. We prove that at high intensities the twodimensional Smoluchowski equation possesses exactly two steady states, corresponding to the isotropic and the nematic phases.

A Note on the Onsager Model of Nematic Phase Transitions

“…Constantin et al, obtain the general form of solutions (though in a different way), but do not make a precise statement about the number of critical states (after submission of this manuscript it was communicated to us by P. Constantin, that a simple way to account for all critical points was also reported in [4]). C. Luo et al, analyze the structure of critical points in Fourier representation, obtain the critical temperature τ c , and classify all critical points.…”

Steady states and their stability of homogeneous, rigid, extended nematic polymers under imposed magnetic fields

“…Although the Maier-Saupe and Onsager potential give the same behavior qualitatively, theoretical analysis with an Onsager potential would require a completely new set of tools which are yet to be discovered, because the Onsager potential requires a complete set of spherical harmonic modes, while the Maier-Saupe only needs the first few. It is this property of the Maier-Saupe potential that results in our reduced order method which has led to many in-depth theoretical results recently [5,6,7,10,34,35,38,20,41,42].…”

“…In the past, it has been perceived as colossal and complicated-hardly accessible to theoretical analysis. Given the rising interest in kinetic theory in the mathematics community these days, various attempts have been made to analyze the properties of the partial differential equations in the kinetic theories and obtain their solutions semianalytically and numerically [23,11,12,13,14,15,16,17,18,5,6,7,10,23,34,35,26,40,38,20]. A recent review of the state of the art in the mathematical and numerical analysis of multi-scale models of complex fluids is given by Li and Zhang [25].…”

Nonparallel solutions of extended nematic polymers under an external field

“…It has been shown that the equilibrium solution of the Smoluchowski equation is given by the Boltzmann distribution [6,7,8,12,21,22,26,11,25] ρ(m) = 1…”