496 Chapter 17. Dusty plasmas
tency cannot be resolved influid theory and it is necessary to revert to the more fundamental
Vlasov description.
Particles are characterized by a velocity distribution functionf(r,v)in the Vlasov de-
scription and the Vlasov equation describes the time evolution off.The test particle loca-
tion is defined to be at the origin of a spherical coordinate system and the system is assumed
to be spherically symmetric so that the only spatial dependence ofφandfis on the spheri-
cal radiusr. The nominal location of the last collision made by a particle incident uponthe
test particle is the collision mean free pathλmfpand since any particle incident upon the
test particle will have traveled many mean free paths, the velocity distribution of particles
entering a sphere of orderlmfpwill be Maxwellian. Because the incident particles are col-
lisionless within theλmfpsphere, their velocity distribution function must be a solution to
the collisionless Vlasov equation inside the sphere. The boundary condition this solution
must satisfy is that its largerlimit should correspond to the collisional (i.e., Maxwellian)
distribution.
Solutions to the collisionless Vlasov equation are functions of constants of themotion
as was shown in Sec.2.2 and the appropriate constant of the motion here is the particle
energyW=mσv^2 /2+qσφ(r). Hence the distribution function in the collisionless region
near the test particle must be
fσ(r,v)=nσ 0(
mσ
2 πκTσ) 3 / 2
exp(
−
mσv^2 /2+qσφ(r)
κTσ)
(17.58)
since this maps to a Maxwellian distribution at large distancesrwhereφ(r)→ 0.
A negatively charged particle such as a dust grain or an electron experiences a repulsive
force upon approaching the dust grain test particle and so slows down. Some approaching
negatively charged particles reflect and so the minimum velocity of electrons or dust grains
approaching the dust grain test particle is zero. The density of these particles will thus be
nσ =∫∞
0fσ(r,v)d^3 v= nσ 0 exp(
−
qσφ(r)
κTσ)
(17.59)
which is the same as thefluid theory Boltzmann relation. It is useful at this point to change
over to the non-dimensional scalarψdefined in Eq.(17.15). Since the dust grains are neg-
atively charged,ψis large and positive in the vicinity of a dust grain. The respective
normalized electron and field dust grain densities are thus
ne
ne 0= exp(−ψTi/Te) (17.60)
nd
nd 0= exp(
−Zψ ̄)
(17.61)
where
Z ̄=ZdTi/Td. (17.62)
Both the electron and field dust grain densities decrease in the vicinity of the dust grain test
particle. However, becauseZ ̄is typically very large andTi/Teis assumed to be very small,
the field dust density scale length is much shorter than the electron density scale length.