Consider a distribution made of a sum of Maxwellians of temperatures and densities , at z = 0. Eq.(4) shows that the distribution at distance z is a sum of Maxwellians of temperatures and densities
The temperature of each individual Maxwellian (which, as seen in Sect.4, is independent of q and z) is related to its moment by Eq.(6), i.e.
where, by definition, . The temperature of the whole distribution at distance z is obtained by substituting in (6) its moment with the above expressions of . This gives
To find the sense of variation of , we have to calculate the sign of the derivative :
The bracket can be rearranged by dropping the terms that cancel out in the summations, and symmetrizing over the dummy indices. This gives:
which is a sum of positive terms, so that the bracket in Eq.( 31) is positive. Hence, with a monotonic attractive potential ( ), the temperatures all increase with z. This holds for a velocity distribution which can be modelled by a sum of several Maxwellians (a single Maxwellian giving ).