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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.,

integral Mn = vnf (v)d3w. (104)
It follows that the first 4 moments of the distribution are the following:

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

integral P = m f (v)(v - V)(v - V)d3w, (109)
and
integral H = m- f(v)|v - V |2(v - V)d3w. (110) 2

The new Equations (109View Equation, 110View Equation) represent the pressure tensor and the heat flux vector, respectively. Moreover, using the relation P = nKT we extract the temperature tensor from Equations (109View Equation, 105View Equation). 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


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