For each surface on a unit cube, the stress on that surface can point in each of the three directions.
Fx=∑i∂σi1/∂x
The second moment of the particle distribution function, describing the flow of momentum in
the laboratory frame, is called the stress tensor,
P=∫m→v→vf(→r,→v,t)d3→v.
The third order moment of the particle distribution function measuring the energy flux density is denoted by
Q(→r,t)=∫12mv2→vf(→r,→v,t)d3→v.
The stress tensor measured in the rest-frame is called the pressure tensor, P, whereas the energy flux density becomes the heat flux density, q. We introduce the relative velocity,
\[ \vec{w} = \vec{v} - \vec{V} .
Then, we get
p(→r,t)=∫m→w→wf(→r,→v,t)d3→v.
q(→r,t)=∫12mw2→wf(→r,→v,t)d3→v.
The trace of the pressure tensor measures the ordinary (scalar) pressure,
p=13Tr(p)=13(pxx+pyy+pzz).
Note that the kinetic energy density is 3/2p,
32p=∫12mw2f(→r,→v,t)d3→v.
The moments measures in the two different frames are related.
P=p+mn→V→V
Q=q+p⋅→V+32p→V+12mnV2→V.
