Generating Spin Polarization from Vorticity through Nonlocal Collisions

We derive the collision term in the Boltzmann equation using the equation of motion for the Wigner function of massive spin-1/2 particles. To next-to-lowest order in ℏ, it contains a nonlocal contribution, which is responsible for the conversion of orbital into spin angular momentum. In a proper choice of pseudogauge, the antisymmetric part of the energy-momentum tensor arises solely from this nonlocal contribution. We show that the collision term vanishes in global equilibrium and that the spin potential is, then, equal to the thermal vorticity. In the nonrelativistic limit, the equations of motion for the energy-momentum and spin tensors reduce to the well-known form for hydrodynamics for micropolar fluids.

Liquid crystal distortions revealed by an octupolar tensor

The classical theory of liquid crystal elasticity as formulated by Oseen and Frank describes the (orientable) optic axis of these soft materials by a director n. The ground state is attained when n is uniform in space; all other states, which have a nonvanishing gradient ∇n, are distorted. This paper proposes an algebraic (and geometric) way to describe the local distortion of a liquid crystal by constructing from n and ∇n a third-rank, symmetric, and traceless tensor A (the octupolar tensor). The (nonlinear) eigenvectors of A associated with the local maxima of its cubic form Φ on the unit sphere (its octupolar potential) designate the directions of distortion concentration. The octupolar potential is illustrated geometrically and its symmetries are charted in the space of distortion characteristics, so as to educate the eye to capture the dominating elastic modes. Special distortions are studied, which have everywhere either the same octupolar potential or one with the same shape but differently inflated.