For each surface on a unit cube, the stress on that surface can point in each of the three directions.
$F_x=\sum_i \partial \sigma_{i1} / \partial x$
The second moment of the particle distribution function, describing the flow of momentum in
the laboratory frame, is called the stress tensor,
\[ \boldsymbol{P} = \int m \vec{v}\vec{v} f(\vec{r},\vec{v},t) d^3\vec{v} .\]
The third order moment of the particle distribution function measuring the energy flux density is denoted by
\[ \boldsymbol{Q(\vec{r},t)} = \int \frac{1}{2} m v^2 \vec{v} f(\vec{r},\vec{v},t) d^3\vec{v} .\]
The stress tensor measured in the rest-frame is called the pressure tensor, $\boldsymbol{P}$, whereas the energy flux density becomes the heat flux density, $\boldsymbol{q}$. We introduce the relative velocity,
\[ \vec{w} = \vec{v} - \vec{V} .
Then, we get
\[ \boldsymbol{p(\vec{r},t)}= \int m \vec{w}\vec{w} f(\vec{r},\vec{v},t) d^3\vec{v} .\]
\[ \boldsymbol{q(\vec{r},t)}= \int \frac{1}{2}m w^2 \vec{w} f(\vec{r},\vec{v},t) d^3\vec{v} .\]
The trace of the pressure tensor measures the ordinary (scalar) pressure,
\[ p = \frac{1}{3}Tr(\boldsymbol{p}) =\frac{1}{3}(p_{xx}+p_{yy}+p_{zz}).\]
Note that the kinetic energy density is $3/2p$,
\[ \frac{3}{2}p = \int \frac{1}{2} m w^2 f(\vec{r},\vec{v},t) d^3\vec{v} .\]
The moments measures in the two different frames are related.
\[ \boldsymbol{P} = \boldsymbol{p} + mn\vec{V}\vec{V} \]
\[ \boldsymbol{Q} = \boldsymbol{q}+\boldsymbol{p}\cdot \vec{V}+ \frac{3}{2}p\vec{V}+\frac{1}{2}mnV^2\vec{V}.\]
