6.5 A plasma is made up of a mixture of various particle species, the type a species having mass mrt, number density na. average macroscopic velocity uQ, random velocity cQ = v - ua, scalar pressure pa = nQkTQ, temperature TQ = (mQ/3k) < >, pressure dyad Va = nQmn < cQcQ >, and heat flow vector qQ = (nQmQ/2) < c^ca >. Similar quantities can be defined for the plasma as a whole, for example, we can define the total number density by
I he tin niiji ui;\» by
mo = — VnQmQ
a
and the average flow velocity by
= —— ^2 n*maUa
u0
n{]rn{)
We can also define an alternative random velocity for the type a species, with reference to Uo, as cq0 = v - Uo, as well as an alternative, absolute temperature by
mQ < c2q0 >
Tq0~ Ü
a corresponding pressure dyad by
Vq0 = nQma < cq0cq0 >
and heat flow vector by
QqO = ^nQmQ < cl0cQ0 > (a) Show that, for the plasma as a whole, the total pressure dvad is given
by
and the total scalar pressure by
PO =        (Po + 3n<*m<*Wl) a
where wa = ua - Uo is the macroscopic diffusion velocity.
(b) If ca is isotropic, that is, < c£t >= (1/3) < c\ >, for i = x,y,z, show that the total heat flow vector is given by
qo =      (q° + §P<»wQ + ^nQmawlwa)
a
(c) If an average temperature T0, for the plasma as a whole, is defined by requiring that po = no/cTo, show that
a
(d) Verify that
so that there is an average thermal energy of kTo/2 per degree of freedom.
6.1 Consider a system of particles characterized by the distribution function given in problem 5.1 (in Chapter 5).
(a) Show that the absolute temperature of the system is given by
T= ™% 3k
where m is the mass of each particle and k is Boltzmann's constant.
(b) Obtain the following expression for the pressure dyad
V = \pmvl 1
where pm = nm and 1 is the unit dyad.
(c) Verify that the heat flow vector q = 0.
From 5.1:    f(v) = K0    for    \vt\ <v0    (i = x,y,z)
f(v) = 0 otherwise, where Kq is a nonzero positive constant. Determine the value of Kq