### 16.3 Computing the moments of the velocity distribution function

Once we are able to measure the density particle distribution function , 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 the density particle distribution function, we define as moment of order of the distribution the quantity , i.e.,

It follows that the first 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 and 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 we extract the temperature tensor from Equations (109, 105). Finally, the scalar pressure and temperature can be obtained from the trace of the relative tensors

and