Czjzek distribution

A Czjzek distribution model [1] is a random distribution of the second-rank traceless symmetric tensors about a zero tensor. An explicit form of a traceless symmetric second-rank tensor, \({\bf S}\), in Cartesian basis, follows,

(62)\[\begin{split}{\bf S} = \left[ \begin{array}{l l l} S_{xx} & S_{xy} & S_{xz} \\ S_{xy} & S_{yy} & S_{yz} \\ S_{xz} & S_{yz} & S_{zz} \end{array} \right],\end{split}\]

where \(S_{xx} + S_{yy} + S_{zz} = 0\). The elements of the above Cartesian tensor, \(S_{ij}\), can be decomposed into second-rank irreducible spherical tensor components [3], \(R_{2,k}\), following

(63)\[\begin{split}S_{xx} &= \frac{1}{2} (R_{2,2} + R_{2,-2}) - \frac{1}{\sqrt{6}} R_{2,0}, \\ S_{xy} &= S_{yx} = -\frac{i}{2} (R_{2,2} - R_{2,-2}), \\ S_{yy} &= -\frac{1}{2} (R_{2,2} + R_{2,-2}) - \frac{1}{\sqrt{6}} R_{2,0}, \\ S_{xz} &= S_{zx} = -\frac{1}{2} (R_{2,1} - R_{2,-1}), \\ S_{zz} &= \sqrt{\frac{2}{3}} R_{20}, \\ S_{yz} &= S_{zy} = \frac{i}{2} (R_{2,1} + R_{2,-1}).\end{split}\]

In the Czjzek model, the distribution of the second-rank traceless symmetric tensor is based on the assumption of a random distribution of the five irreducible spherical tensor components, \(R_{2,k}\), drawn from an uncorrelated five-dimensional multivariate normal distribution. Since \(R_{2,k}\) components are complex, random sampling is performed on the equivalent real tensor components, which are a linear combination of \(R_{2,k}\), and are given as

(64)\[\begin{split}U_1 &= \frac{1}{\sqrt{6}} R_{2,0}, \\ U_2 &= -\frac{1}{\sqrt{12}} (R_{2,1} - R_{2,-1}), \\ U_3 &= \frac{i}{\sqrt{12}} (R_{2,1} + R_{2,-1}), \\ U_4 &= -\frac{i}{\sqrt{12}} (R_{2,2} - R_{2,-2}), \\ U_5 &= \frac{1}{\sqrt{12}} (R_{2,2} + R_{2,-2}),\end{split}\]

where \(U_i\) forms an ortho-normal basis. The components, \(U_i\), are drawn from a five-dimensional uncorrelated multivariate normal distribution with zero mean and covariance matrix, \(\Lambda=\sigma^2 {\bf I}_5\), where \({\bf I}_5\) is a \(5 \times 5\) identity matrix and \(\sigma\) is the standard deviation.

In terms of \(U_i\), the traceless second-rank symmetric Cartesian tensor elements, \(S_{ij}\), follows

(65)\[\begin{split}S_{xx} &= \sqrt{3} U_5 - U_1, \\ S_{xy} &= S_{yx} = \sqrt{3} U_4, \\ S_{yy} &= -\sqrt{3} U_5 - U_1, \\ S_{xz} &= S_{zx} = \sqrt{3} U_2, \\ S_{zz} &= 2 U_1, \\ S_{yz} &= S_{zy} = \sqrt{3} U_3,\end{split}\]

and the explicit matrix form of \({\bf S}\) is

(66)\[\begin{split}{\bf S} = \left[ \begin{array}{l l l} \sqrt{3} U_5 - U_1 & \sqrt{3} U_4 & \sqrt{3} U_2 \\ \sqrt{3} U_4 & -\sqrt{3} U_5 - U_1 & \sqrt{3} U_3 \\ \sqrt{3} U_2 & \sqrt{3} U_3 & 2 U_1 \end{array} \right].\end{split}\]

In a shorthand notation, we denote a Czjzek distribution of second-rank traceless symmetric tensor as \(S_C(\sigma)\).

Extended Czjzek distribution

An Extended Czjzek distribution model [2] is a random perturbation of the second-rank traceless symmetric tensors about a non-zero tensor, which is given as

(67)\[S_T = S(0) + \rho S_C(\sigma=1),\]

where \(S_T\) is the total tensor, \(S(0)\) is the non-zero dominant second-rank tensor, \(S_C(\sigma=1)\) is the Czjzek random model attributing to the random perturbation of the tensor about the dominant tensor, \(S(0)\), and \(\rho\) is the size of the perturbation. In the above equation, the \(\sigma\) parameter from the Czjzek random model, \(S_C\), has no meaning and is set to one. The factor, \(\rho\), is defined as

(68)\[\rho = \frac{||S(0)|| \epsilon}{\sqrt{30}},\]

where \(\|S(0)\|\) is the 2-norm of the dominant tensor, and \(\epsilon\) is a fraction.