1. What is the particle distribution for a plasma in force-balance?
(Credit:
What is the particle distribution for a plasma in force-balance?
From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials
Oliver Allanson
Thomas Neukirch, Sascha Troscheit & Fiona Wilson

2. ~ 100,000K most matter is ionised
What is a plasma?
1/17
(Credit: Tetronics.com)
~ 100,000K most matter is ionised
(N. A. Krall and A. W. Trivelpiece, Principles of plasma physics, 1973)
100,000K: Who cares?

3. More than 99% of the visible universe is in the plasma state of matter
(Credit: NASA)
2/17
More than 99% of the visible universe is in the plasma state of matter
(W. Baumjohann and R.A. Treumann, Basic Space Plasma Physics, 1997)

4. Electrodynamics dominate Collective behaviour
3/17
(credit:

5. Statistical description of plasma: Maxwell-Boltzmann
4/17
1860s/70s
Self-consistent electromagnetic fields

6. Particle Collisions -> The Maxwellian distribution
5/17
(Credit: Ian Hutchinson, MIT)

7. Collisionless plasmas
6/17
practicalphysics.org
E. Marsch. “Kinetic Physics of the Solar Corona and Solar Wind”. Living Reviews in Solar Physics, 2006
Hot and/or diffuse plasmas
E.g. in the solar corona
Collisions not able to drive plasma towards thermal equilibrium
Statistical equilibrium need not be a Maxwellian
Infinite (mathematical) possibilities

8. K. Schindler. Physics of Space Plasma Activity,2007.
7/17
A. A. Vlasov, “The vibrational properties of an electron gas”. Physics-Uspekhi, 1968
K. Schindler. Physics of Space Plasma Activity,2007.
Pressure
Bulk Flow
(Credit: formula1.com)

9. The forward and inverse approaches
8/17
(credit: M.G. Harrison PhD thesis)
Forward: Differential Equations
Inverse: Integral Equations

10. Jeans’ Theorem Unknown function
9/17
Any differentiable function of constants of motion will solve the Vlasov equation
J. H. Jeans. “On the theory of star-streaming and the structure of the universe”. Monthly Notices of the Royal Astronomical Society 1915
Noether: In a one-dimensional (z) plasma we immediately have 3 constants of motion
Assume
Unknown function

11. The inverse problem Weierstrass integral transform
10/17
P. J. Channell, Physics of Fluids, 1976
Weierstrass integral transform
Green’s function for Heat Equation
G. G Bilodeau. “The Weierstrass transform and Hermite polynomials”. Duke
Mathematical Journal, 1962

12. Solution methods Given a Maclaurin expansion of LHS
11/17
Given a Maclaurin expansion of LHS
The formal solution of I.C. is given in Hermite polynomials

13. What are Hermite Polynomials?
12/17
What are Hermite Polynomials?
Complete, orthogonal set of functions
G. B. Arfken and H. J. Weber. Mathematical methods for physicists, 2001.

14. The “solution” Convergent? Non-negative? Bounded? 13/17
O. Allanson et al. “From one- dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials”, Journal of Plasma Physics, 2016
Bounded?

15. Convergence & “Boundedness”
14/17
Velocity moments of all order

16. Non-negativity (I) : Intuition/physics
15/17
Non-negativity (I) : Intuition/physics
>0
>0
I.C.? Sign?
The “initial condition” is time-dependent ….. hmmmm
“Now/ Final condition” is positive for all Aj / “space”
“If we have a positive heat distribution now, could we have had a distribution somewhere negative in the past?

17. Non-negativity (II) : Detail/maths
16/17
Non-negativity (II) : Detail/maths
>0
>0
I.C.? Sign?
Finitely bounded from below?
Yes, for all but the most awkward functions
Prove this by contradiction, with an “envelope”
If there exists a finite lower bound for g, then we can prove that …

18. 17/17
Summary
Collisionless plasmas: -> infinite class of equilibrium distribution functions
For a given force-balance macroscopic equilibrium: -> inverse problem
Expansions in Hermite polynomials can give 1-1 correspondence
Sufficient criteria for convergence and “boundedness”
On our way towards firm non-negativity results. Any suggestions?
Many `physical’ questions remain (stability, physical suitability …)
Thank you!

19. Can not solve equation of motion for all particles!
Particle dynamics
4/17
Can not solve equation of motion for all particles!
T. G. Northrop. “Adiabatic Charged-Particle Motion”. Reviews of Geophysics and Space Physics, 1963.

20. Statistical description of plasma (I): Klimontovich
4/17
“Pathologically jagged”
Don’t care about individual particles
(credit for both pics: Ben McMillan, Warwick)

21. Magnetic fields dominate the collective behaviour
(Credit: NASA SDO)
4/17
Magnetic fields dominate the collective behaviour

23. Vacuum range
Molecules/cm^3
Mean free path
Ambient pressure
10^19
Nanometres 10^-9
Low vacuum
10^16
Micrometers 10^-6
Medium vacuum
10^13
Millimetres 10^-3
High vacuum
10^9
Metres
Ultra-high vacuum
10^4
Megametres 10^6
Extremely high vacuum
<10^4
> 100 megametres 10^8

24. : “that which is formed or moulded”
Irving Langmuir – 1927
cf. blood plasma
: “that which is formed or moulded”