title: Octupolar order in three dimensions.
abstract: Octupolar order in three space dimensions is described by a real-valued, fully symmetric and traceless, third-rank tensor A. The real generalized eigenvalues of A are also the critical values of a real-valued potential [Formula: see text] defined on the unit sphere [Formula: see text] by A. Generalized eigenvalues of A and critical points of [Formula: see text] are equivalent means to describe octupolar order in a molecular assembly according to Buckingham's formula for the probability density distribution. Intuition suggests that [Formula: see text] would generically have four maxima, corresponding to the most probable molecular orientations, so that a (possibly distorted) tetrahedron would effectively describe A. This paper shows that another generic octupolar state flanks the expected one, featuring three maxima of [Formula: see text]. The two generic states are divided by a separatrix manifold, which may physically represent an intra-octupolar transition.
authors.affiliation: Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, I-20133, Milano, Italy.
authors.forename: Giuseppe
authors.initials: G
authors.lastname: Gaeta
authors.affiliation: Dipartimento di Matematica, Università di Pavia, via Ferrata 5, I-27100, Pavia, Italy. eg.virga@unipv.it.
authors.forename: Epifanio G
authors.initials: EG
authors.lastname: Virga
keywords: Soft Matter: Liquid crystals