6.4 Homogeneous Distribution for Second-Rank Tensors

The usual definition of the norm of a tensor provides the only natural definition of distance in the space of all possible tensors. This shows that, when using a Cartesian system of coordinates, the components of a tensor are the Cartesian coordinates in the 6D space of symmetric tensors. The homogeneous distribution is then represented by a constant (nonnormalizable) probability density:

Instead of using the components, we may use the three eigenvalues { ? ₁, ? ₂, ? ₃} of the tensor and the three Euler angles { ?, ?, ?} defining the orientation of the eigendirections in the space. As the Jacobian of the transformation

is

the homogeneous probability density (6.45) transforms into

Although this is not obvious, this probability density is isotropic in spatial directions (i.e., the 3D referential defined by the three Euler angles are isotropically distributed). In this sense, we recover isotropy as a special case of homogeneity.

Rule 6.8, imposing that any probability density of the variables { ? ₁, ? ₂, ? ₃, ?, ?, ?} has to tend to the homogeneous probability density (6.48) when the dispersion parameters tend to infinity imposes a strong constraint on the form of acceptable probability densities that is, generally, overlooked.

For instance, a Gaussian model for the variables { ? _xx , ? _yy , ? _zz , ? _xy , ? _yz , ? _zx }