AS554 Irreversible Processes in Plasmas March 9, 1999

Problem Set #5 (due Mar. 23, 1999)

Question 1

Determine the electrical conductivity of a Lorentz plasma. [Take Z_{i} large so that only electron-ion collisions need to be considered. Write f_{e} = f_{M} + f_{1} where f_{M} is a Maxwellian. Linearize the Fokker-Planck equation (including an external electric field) in E, assuming f_{1} = O(E). Solve the resulting equation.]

Question 2

Using the Rosenbluth potentials, evaluate the collisional flux J^{a/b} when f_{b} is a Maxwellian. Express in terms of the relaxation rates ν_{s}^{a/b}, ν_{∥}^{a/b}, and

ν_{⊥}^{a/b} (see Plasma Formulary). Put the answer in a form similar to the one in the Formulary (under Fokker–Planck Equation) but note that the answer in older editions of the Formulary is wrong! Useful formulas:

J^{a/b} = 4πν_{0}^{a/b}v^{3}[−\frac{m_{a}}{m_{b}} (∇_{v}ϕ_{b})f_{a} + (∇_{v}∇_{v}ψ_{b}) · ∇_{v}f_{a}]

where

ν_{0}^{a/b} = \frac{e_{a}^{2}e_{b}^{2} log Λn_{b}}{4πϵ_{0}^{2}m_{a}^{2}v^{3}} ,

and

∇_{v}^{2}ϕ_{b} = f_{b},  ∇_{v}^{2}ψ_{b} = ϕ_{b},  ∫ f_{b} d^{3}v = 1.

If f_{b} is spherically symmetric, ϕ_{b} and ψ_{b} are found by straightforward integration. In particular if f_{b} is a Mawellian with temperature T_{b} then

\frac{dϕ_{b}}{dv} = \frac{1}{4πv^{2}} (erf(u) − u erf′(u)) = \frac{1}{4πv^{2}} \frac{ν_{s}^{a/b}}{(1 + m_{a}/m_{b})ν_{0}^{a/b}}

\frac{dψ_{b}}{dv} = −\frac{1}{16π} ((2 − \frac{1}{u^{2}} ) erf(u) + \frac{erf′(u)}{u}) = −\frac{1}{16π} \frac{ν_{⊥}^{a/b}}{ν_{0}^{a/b}}

\frac{d^{2}ψ_{b}}{dv^{2}} = −\frac{1}{8πv} (\frac{erf(u)}{u^{2}} − \frac{erf′(u)}{u}) = −\frac{1}{8πv} \frac{ν_{∥}^{a/b}}{ν_{0}^{a/b}}
erf(u) = \frac{2}{\sqrt{π}} ∫_{0}^{u} exp(−x^{2}) dx,  erf′(u) = \frac{2}{\sqrt{π}} exp(−u^{2})
u = v/\sqrt{2T_{b}/m_{b}}