Let us first consider the classical case where 
is a Maxwellian
of
temperature 
. One sees from
(4) that the distribution remains Maxwellian for 
,
with the same temperature, and, from (6), that all the
moments 
vary with z as
. This would justify the
widely-used assumption of constant temperatures along magnetic
field lines, if the particle velocity distributions were
actually Maxwellian; of course, in this case, the density and
temperature are not anticorrelated.
If the distribution is now a linear combination of
Maxwellians, so that there is no more thermal equilibrium, the temperatures
are no longer equal to each
other, nor independent of z (although the temperature of each
Maxwellian is independent of z). The generally increase with
q,
since higher-order moments favor components of higher
temperatures. In particular, the effective temperature given by
Eq.(1) is then 
.
In the Appendix, we
show analytically that with such a distribution and a monotonic
potential which attracts particles to z = 0, all the generalized
temperatures increase with z. Hence, since the
density decreases with z, all the temperatures vary in
anticorrelation with 
. An important consequence is that if
a
polytrope law 
does exist, its
index is necessarily
smaller than one (or just equal to one in the limiting case of a
Maxwellian distribution.)
This generalizes to the temperatures the anticorrelation
between density and temperature first shown by Scudder
(1992a) in a general context, for the traditional temperature,
using graphical arguments; this is a generic property of
non-thermal distributions.