Virial theorem for the anisotropic Ginzburg-Landau theory

The scalar virial theorem for the Ginzburg-Landau theory is generalized to a vector virial theorem and follows from similar scaling properties of the Gibbs free-energy density. All the components of the magnetic field H are determined in terms of average values of the kinetic and field tensor components of the Helmholtz free-energy density. We consider two frames, the crystal's and the magnetic induction's B, where the scaling properties yield useful relations due to anisotropy. In the last case the scaling relations do not completely determine H; instead, they provide useful identities that reflect collective properties of the vortex state. We compare both the scaling and the thermodynamic methods for the particular case of straight tilted parallel vortex lines in the London limit.