16.3 Computing the moments of the velocity distribution function

Once we are able to measure the density particle distribution function $ff,h,v$ , we can compute the most used moments of the distribution in order to obtain the particle number density, velocity, pressure, temperature, and heat-flux (Paschmann et al., 1998 ).

If we simply indicate with $f(v)$ the density particle distribution function, we define as moment of order $n$ of the distribution the quantity $Mn$ , i.e.,

It follows that the first $4$ moments of the distribution are the following:

the number density
the number flux density vector
the momentum flux density tensor
the energy flux density vector

Once we have computed the zero-order moment, we can obtain the velocity vector from Equation (106 ). Moreover, we can compute $TT$ and $Q$ in terms of velocity differences with respect to the bulk velocity, and Equations (107 , 108 ) become

and

The new Equations (109 , 110 ) represent the pressure tensor and the heat flux vector, respectively. Moreover, using the relation $P = nKT$ we extract the temperature tensor from Equations (109 , 105 ). Finally, the scalar pressure $P$ and temperature $T$ can be obtained from the trace of the relative tensors

T-r(Pij) P = 3

and

T = Tr(Tij)-. 3