Abstract
We study the bifurcation diagram of the Onsager free-energy functional for liquid crystals with orientation parameter on the sphere. In particular, we concentrate on the bifurcations from the isotropic solution for a general class of two-body interaction potentials including the Onsager kernel. Reformulating the problem as a non-linear eigenvalue problem for the kernel operator, we prove that spherical harmonics are the corresponding eigenfunctions and we present a direct relationship between the coefficients of the Taylor expansion of this class of interaction potentials and their eigenvalues. We find explicit expressions for all bifurcation points corresponding to bifurcations from the isotropic state of the Onsager free-energy functional equipped with the Onsager interaction potential. A substantial amount of our analysis is based on the use of spherical harmonics and a special algorithm for computing expansions of products of spherical harmonics in terms of spherical harmonics is presented. Using a Lyapunov–Schmidt reduction, we derive a bifurcation equation depending on five state variables. The dimension of this state space is further reduced to two dimensions by using the rotational symmetry of the problem and the invariant theory of groups. On the basis of these results, we show that the first bifurcation from the isotropic state of the Onsager interaction potential is a transcritical bifurcation and that the corresponding solution is uniaxial. In addition, we prove some global properties of the bifurcation diagram such as the fact that the trivial solution is the unique local minimiser if the bifurcation parameter is high, that it is not a local minimiser if the bifurcation parameter is small, the boundedness of all equilibria of the functional and that the bifurcation branches are either unbounded or that they meet another bifurcation branch.
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Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory, Vol. 50. American Mathematical Society, Providence, 2002
Abud M., Sartori G.: The geometry of spontaneous symmetry breaking. Ann. Phys. 150, 307–372 (1983)
Ambrosetti, A., Prodi, G.: A primer of nonlinear analysis. Cambridge Studies in Advanced Mathematics, Vol. 34. Cambridge University Press, Cambridge, 1993
Atkinson, K., Han, W.: Spherical harmonics and approximations on the unit sphere: an introduction. Lecture Notes in Mathematics, Vol. 2044. Springer, Heidelberg, 2012. doi:10.1007/978-3-642-25983-8
Bailey, W.N.: Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Generalized Hypergeometric Series. Stechert-Hafner Service Agency, New York, 1964
Bröcker, T., Tom Dieck, T.: Representations of compact Lie groups. Graduate Texts in Mathematics, Vol.98. Springer, New York, 1985
Chen W., Li C., Wang G.: On the stationary solutions of the 2D Doi–Onsager model. Nonlinear Anal. Theor. 73(8), 2410–2425 (2010)
Chen, Y.C.: Singularity theory and nonlinear bifurcation analysis. (Eds. Fu Y.B. and Ogden R.W.) Nonlinear Elasticity: Theory and Applications, London Mathematical Society Lecture Note Series, Vol. 283. Cambridge University Press, Cambridge, 305–344, 2001
Chossat, P., Lauterbach, R.: Methods in equivariant bifurcations and dynamical systems. Advanced Series in Nonlinear Dynamics, Vol. 15. World Scientific Publishing Co. Inc., River Edge, 2000
Constantin P., Kevrekidis I.G., Titi E.S.: Asymptotic states of a Smoluchowski equation. Arch. Ration. Mech. Anal. 174(3), 365–384 (2004)
Elliott, C.: Theory of PDEs. Lecture Notes. University of Warwick, Coventry
Fatkullin I., Slastikov V.: Critical points of the Onsager functional on a sphere. Nonlinearity 18(6), 2565–2580 (2005)
Freiser M.J.: Ordered states of a nematic liquid. Phys. Rev. Lett. 24(2), 1041–1043 (1970)
Friedrich, T.: On types of non-integrable geometries. Proceedings of the 22nd Winter School “Geometry and Physics” (Srní, 2002), Vol. 71, 99–113, 2003
Garrett, P.: Lecture notes on harmonic analysis on spheres 1, 2011. http://www.math.umn.edu/~garrett/m/mfms/notes_c/spheres_I.pdf
Garrett, P.: Lecture notes on harmonic analysis on spheres 2, 2011. http://www.math.umn.edu/~garrett/m/mfms/notes_c/spheres_II.pdf
Hussain, S.A.: Properties of spherical harmonics, 2014. https://sahussaintu.files.wordpress.com/2014/03/spherical_harmonics.pdf
Kayser R.F., Raveché H.J.: Bifurcations in Onsager’s model of isotropic-nematic transition. Phys. Rev. A 17, 2067–2072 (1978)
Liu H., Zhang H., Zhang P.: Axial symmetry and classification of stationary solutions of Doi–Onsager equation on the sphere with Maier–Saupe potential. Commun. Math. Sci. 3(2), 201–218 (2005)
Lucia M., Vukadinovic J.: Exact multiplicity of nematic states for an Onsager model. Nonlinearity 23(12), 3157–3185 (2010)
Maier W., Saupe A.: Eine einfache molekulare Theorie des nematischen kristallinflüssigen Zustandes. Zeitschrift für Naturforschung 13, 564 (1958)
Marinucci, D., Peccati, G.: Random fields on the sphere. London Mathematical Society Lecture Note Series, Vol. 389. Cambridge University Press, Cambridge, 2011
Niksirat, M.A., Yu, X.: On stationary solutions of the 2D Doi-Onsager model. ArXiv e-prints 2015
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge, 2010
Onsager, L.: The Effects of Shape on the Interaction of Collodial Particles. Sterling Chemistry Laboratory, Yale University, New Haven, 1949
Palais R.S.: The principle of symmetric criticality. Commun. Math. Phys. 69, 19–30 (1979)
Rabinowitz P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)
Schwarz G.W.: Smooth functions invariant under the action of a compact Lie group. Topology 14, 63–68 (1975)
Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 3rd edn. Springer, New York, 2004
Vollmer, M.A.C.: Critical points and bifurcations of the three-dimensional Onsager model for liquid crystals, 2015. http://arxiv.org/abs/1509.02469
Wachsmuth, J.: Suspensions of rod-like molecules: the istropic–nematic phase transition and flow alignment in 2-d. Master’s thesis, University of Bonn 2006. Unpublished
Wang H., Zhou H.: Multiple branches of ordered states of polymer ensembles with the Onsager excluded volume potential. Phys. Lett. Sect. A 372(19), 3423–3428 (2008)
Wang H., Zhou H.: Phase diagram of nematic polymer monolayers with the Onsager interaction potential. J. Comput. Theor. Nanosci. 7(4), 738–755 (2010)
Weisstein, E.W.: Associated Legendre polynomials. http://mathworld.wolfram.com/AssociatedLegendrePolynomial.html. From MathWorld-A Wolfram Web Resource
Weisstein, E.W.: Spherical harmonics. http://mathworld.wolfram.com/SphericalHarmonic.html. From MathWorld-A Wolfram Web Resource
Weisstein, E.W.: Spherical harmonics. http://mathworld.wolfram.com/LegendrePolynomial.html. From MathWorld-A Wolfram Web Resource
Weisstein, E.W.: CRC Concise Encyclopedia of Mathematics, 2nd edn. Chapman Hall/CRC, Boca Raton, 2002
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge, 1996
Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos. In:Marsden, J.E., Sirovich, L.,Antman, S.S. (eds) Texts in Applied Mathematics, Vol. 2. Springer, New York, 1990
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Communicated by E. G. Virga
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Vollmer, M.A.C. Critical Points and Bifurcations of the Three-Dimensional Onsager Model for Liquid Crystals. Arch Rational Mech Anal 226, 851–922 (2017). https://doi.org/10.1007/s00205-017-1146-8
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DOI: https://doi.org/10.1007/s00205-017-1146-8