CollisionsandtransportProcess

Twobodycollisions

xs

⌧E,e : ⌧E,p : ⌧EQ =me

mp:r

me

mp: 1

Fokker-Planckequa:on@f

@t+ v ·rf +

q

m

✓E+

v ⇥B

c

◆·rvf + g ·rvf = ⌃

✓@f

@t

◆

c

Withoutcollisionterm,Vlasovequa:onWithtwobodycollisions,Fokker-Planckequa:on

f(x,v,t)

Plasmaconduc:vity:unmagne:zedcase

•  Resis:vity

•  Spitzerresis:vity(Spitzer&Härm1953)

⌘ =⇡3/2pmeZe2

2�E(2kBT )3/2ln⇤ = 7.3⇥ 10�9 ln⇤

T 3/2

Conduc:vityσ=1/η,magne:cdiffusivityc2/4πσ

⌘ =me⌫cne2

Independentofdensity!

Lorenznumber

•  Thermalconduc:vity

K =

5

2⌧n

k2BT

me Electronmo:ondominates

Recall � =ne2⌧cme

K

�T=

⇡2

3

✓kBe

◆2

Independentofτc,similarformetals

c

�0 =ne2

me⌫c

Plasmaconduc:vity:magne:zedcase

�? =�0

1 + (⌦⌧)2

�H =�0⌦⌧

1 + (⌦⌧)2

�k =

j = � ·E� =

0

@�? ��H 0�H �? 00 0 �k

1

A

j = �kEk + �?E? � �H(E? ⇥B)/B

ForarbitraryBfield,

•  Ωτ<<1,isotropic•  Ωτ>>1,CurrentflowsonlyalongB•  Ωτ<1,PerpendicularcurrentdominatesHallone•  Ωτ>1,Hallcurrentdominates•  Ωτ=1,45degreeintheplaneperpendiculartoB

Hallcurrent

DiffusionacrosstheBfield⇢dv

dt= �rp+ nq

✓E+

v ⇥B

c

◆� ⇢v/⌧

v

y

(1 + ⌦2⌧

2) = �µE

y

� D

n

@n

@y

� ⌦2⌧

2c

E

x

B

� ⌦2⌧

2c

k

B

T

neB

@n

@x

Mobilitycoefficient

Diffusioncoefficient

ExBdrif

Diamagne:cdrif

v? = �µ?E�D?rn

n+

vE + vD

1 + (⌦⌧)�2

D? =D

1 + ⌦2⌧2

Collisionsenhancecrossfielddiffusion!D =

kBT ⌧

m

µ =e⌧

m

Braginski’stransportcoefficients

ik =

3.9niTi⌧iM

ek =

3.16neTe⌧eme

⌘0 = 0.96niTi⌧i

i? =

2niTi

M⌦2⌧i

(TinunitofeV)

Viscosity:

Thermalconduc:vity