Random Fields Related to the Symmetry Classes of Second-Order Symmetric Tensors

Abstract

Under the change of basis in the three-dimensional space by means of an orthogonal matrix g, a matrix A of a linear operator is transformed as $A\mapsto gAg^{-1}$. Mathematically, the stationary subgroup of a symmetric matrix under the above action can be either $D_2\times Z^c_2$, when all three eigenvalues of A are different, or $\mathrm {O}(2)\times Z^c_2$, when two of them are equal, or $\mathrm {O}(3)$, when all three eigenvalues are equal. Physically, one typical application relates to dependent quantities like a second-order symmetric stress (or strain) tensor. Another physical setting is that of dependent fields, such as conductivity with such three cases is the conductivity (or, similarly, permittivity, or anti-plane elasticity) second-rank tensor, which can be either orthotropic, transversely isotropic, or isotropic. For each of the above symmetry classes, we consider a homogeneous random field taking values in the fixed point set of the class that is invariant with respect to the natural representation of a certain closed subgroup of the orthogonal group. Such fields may model stochastic heat conduction, electric permittivity, etc. We find the spectral expansions of the introduced random fields.