# Models¶

## 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.