1. Introduction to Plasma Physics and Plasma-based Acceleration
Fluid models of plasma II

2. Distribution function
For each particle species, a distribution function fα is introduced
Number of particles in volume element:
This number is changed by collisions:

3. Boltzmann equation
We have: but also:

4. Boltzmann equation
Boltzmann equation (Ludwig Boltzmann, ) Collisional term: No collisions: Vlasov equation (Anatoly Vlasov, )

5. Charge, current
Particle density: Average velocity: Charge density: Current density: Sources for Maxwell’s equations The Vlasov-Maxwell system is closed The Vlasov-Boltzmann system needs an additional collision model

6. Moment equations Vlasov equation hard to tackle analytically
Solution: use moments:
Multiply by some power of v
Integrate over all v
Use:
Define: for any function Ψ

7. Particle conservation
Use to obtain “zeroth” order moment equation: Note that this equation contains the first order moment u.

8. Momentum conservation
Use to obtain first order moment equation: Note that this equation contains the second order moment <v·v>.

9. Energy conservation
Use to obtain second order moment equation: Note that this equation contains the third order moment <v2v>.

10. Closing the system
Moment equation of order n contains moment of order n+1.
System is never closed!
Need to provide equation of state for highest order moment (or just drop it):
Neglect heat flux (fast processes)
Prescribe pressure
Add viscosity model

11. Warning
Closure: switching from infinite system to finite system: different system!
Moments from infinite system correspond to some distribution fα
Moments from finite system need not correspond to any distribution
Need not be a problem if moments show correct behaviour, but beware!

12. Grooming the equations
Relative velocity: Pressure tensor: Isotropic pressure: Kinetic temperature: Heat flux: Collisional exchanges:

13. Momentum revisited
Insert “physical” quantities and subtract u*(continuity equation): Flow equation of electron/ion fluid Left: advection Right: pressure, viscosity, Lorentz force, momentum exchange between species

14. Energy revisited
Insert “physical” quantities and subtract u2*(continuity equation) and u·(momentum equation): Left: Internal energy, work Right: heat flux, viscosity, heat exchange between species

15. Example: adiabatic compression
Need to provide expressions for viscosity, heat flux, friction, heat exchange between species, to close system. For adiabatic compression:

16. One-fluid model
So far, model allowed electrons and ions to move separately
Possible to derive “one-fluid model”, in which electrons and ions move (mostly) like a single fluid
No derivation; shown for its interesting consequences:

17. Ohm’s law for plasma
Follows from energy balance for quasi-neutral, magnetised plasma; valid for long time (t>tcoll) and length (l>c/ωp) scales:
Left: external fields, Hall term, pressure gradient
Right: resistive current (caused by small difference between electron and ion velocity)

18. Ideal Ohm’s law
For an “idealised” plasma (collisions neglected, gyration averaged away), e.g. a space plasma: In this case, magnetic fields are “frozen” with respect to the plasma

19. Summary Boltzmann and Vlasov equations for plasma
Moment equations (particle, momentum, energy conservation)
Closure of moment system
Connection with “old-school” thermo-dynamics
One-fluid model and Ohm’s law for plasma