1. Velocity Distribution and Temperature
f(v,t) dv is the number of particles per unit volume in the velocity space interval v, v + dv.
n(t) = ∫ f(v,t) dv ( )
For thermal equilibrium, this distribution is the so-called Maxwellian distribution
f(v) = A.exp(-1/2mv2/(kBT)) ( )
where kB is Boltzmann’s constant = 1.38x10-23 JK-1. Integrating as in ( )
A = n(m/(2kBT) ( )
so the Maxwellian distribution is
f(v) = n(m/(2kBT)1.5.exp(-1/2mv2/(kBT)) ( )
Now the average kinetic energy of the particles in the distribution is
Eav = ∫ f(v) (1/2mv2) dv/∫ f(v) dv = 3kBT/2 ( )
This identifies T as the temperature, so that the average energy is kBT/2 per degree of freedom. In plasma physics, it is customary to quote temperatures in energy units corresponding to kBT e.g. for kBT = 1 eV = 1.6x10-19 J. Hence T = 1.6x10-19/(1.38x10-23) = 11,600 K.

2. Debye Shielding 1/2
Poisson’s equation in one dimension is (Z = 1 i.e. singly charged ions):
0d2/dx2 = -e(ni-ne) ( )
If the density far away from the grid is n then
ni = n ( )
In the presence of the potential the electrons have a velocity distribution
f(u) = A.exp(-(1/2mu2 +q)/(kBTe)) ( )
This is intuitively clear as this means that there are fewer electrons where the potential energy (q) is large, since not all electrons have the energy to get there. Integrating this distribution over u and setting q = -e, and noting that ne  n. as x  ,
ne = n.exp(e)/(kBTe)) ( )
Substituting for ni and ne into (1), we have;
0d2/dx2 = e n.(exp(e)/(kBTe)) – 1) ( )

3. (x) = 0exp( -x/D) (1.1.2-7)
Debye Shielding 2/2
Expanding the exponential in a Taylor series and keeping only the linear term (which can be justified only a posteriori because very near the grid (e)/(kBT) can be large. (the contribution of this region is small as the potential falls rapidly there), and defining a length D (the Debye Length) as;
D = (0 kBTe/ (nee2))0.5 ( )
then the solution of the potential ( ) is
(x) = 0exp( -x/D) ( )
Thus the characteristic dimension of the sheath is the Debye length.
We are now in a position to define “quasi-neutrality”. This obtains if the characteristic dimensions L of a system are much larger than the Debye length,
D << L ( )

4. Collective behaviour requires ND >> 1 (1.1.3-2)
Criteria for Plasmas
Quasineutrality
D << L ( )
Collective behaviour requires
ND >> 1 ( )
where ND = n4/3D3 ( )
Electromagnetic effects dominate collisions with neutrals
 >> ( )
where  is the mean time for collisions of charged particles with neutrals and  is the characteristic frequency of typical plasma oscillations