Kinetic Theory
for Gases and Plasmas:
Lecture 2: Plasma Kinetics
Russel Caflisch
IPAM
Mathematics Department, UCLA
IPAM Plasma Tutorials 2012
Review of First Lecture
•
•
•
•
•
•
•
Velocity distribution function
Molecular chaos
Boltzmann equation
H-theorem (entropy)
Maxwellian equilibrium
Fluid dynamic limit
DSMC
– DSMC becomes computationally intractable near fluid
regime, since collision time-scale becomes small
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Temp. (eV)
Where are collisions signifiant in plasmas?
Example: Tokamak edge boundary layer
Scrape-off
layer
1000
500
Kinetic
Effects
0
Schematic views of divertor tokamak and edge-plasma region (magnetic
separatrix is the red line and the black boundaries indicate the shape of
magnetic flux surfaces)
From G. W. Hammett, review talk 2007
APS Div Plasmas Physics
Annual Meeting, Orlando, Nov. 12-16.
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R (cm)
Edge pedestal temperature profile near the
edge of an H-mode discharge in the DIII-D
tokamak. [Porter2000]. Pedestal is shaded
region.
Basics of
mathematical (classical)
plasma physics
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Gas or Plasma Flow: Kinetic vs. Fluid
Kinetic description
Fluid (continuum) description
• Discrete particles
• Density, velocity, temperature
• Motion by particle velocity
• Interact through collisions
• Evolution following fluid eqtns
(Euler or Navier-Stokes or MHD)
When does continuum description fail?
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Debye Length
• Charged particles rather than neutrals
Electrons: eFACM 2010
Debye Length
• Quasi-neutrality: nearly equal number of oppositely charged particles
Electrons: eFACM 2010
Ions: H+
Debye Length
• Pick out a distinguished particle
Electrons: eFACM 2010
Ions: H+
Debye Length
• Debye length = range of influence, e.g., for single electron
λD
Electrons: eFACM 2010
Ions: H+
Debye Length
• In neighborhood of an electron there is deficit of other electrons,
suplus of positive ions
Electrons: eFACM 2010
Ions: H+
Debye Length
• Replace positive charged particles by continuum, for simplicity
Electrons: e; test particle ; Ions:
smoothedFACM 2010
Debye Length: Derivation
• Distribution of electrons and ions
–
–
–
–
charge q; temperature T; dielectric coeff ε0;
potential φ, energy is -q φ
q  k BT
n
e
electrons in Gibbs distribution (in space)
e
ni  ne
Uniform ions distribution
• Poisson equation (linearized)
– Single electron at 0
2  (q   0 )( ( x)  ne eq  kBT  ne )
 (q   0 ) ( x)  (ne q 2   0 kBT )
• Solution
  (4 ) 1 (q /  0 )r 1e  r  
D
With length scale λD = Debye length:
D2  ne q 2 ( 0 k BT ) 1
FACM 2010
Interactions of Charged Particles
in a Plasma
• Plasma parameter g = (n λD3)-1
– Plasma approximation g<<1
– Many particles in a Debye sphere
– Otherwise, the system is an N-body problem
• Long range interactions
– r > λD
(λD = Debye length)
– Individual particle interactions are not significant
– Interaction mediated by electric and magnetic fields
• Short range interactions
– r < λD
– Coulomb interactions
FACM 2010
Levels of Description
• Magneto-hydrodynamic (MHD) equations
– Equilibrium
– Continuum
• Vlasov-Maxwell equations
– Nonequilibrium
– No collisions
• Landau-Fokker-Planck equations
– Nonequilibrium
– Collisions
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MHD Equations
t     u  0
 t u   (u )u  p  c J  B
1
E uB  0
Fluid equations
with Lorenz force
Ohm’s law
  B  4 c J
1
1
  E  c  t B
Maxwell’s equations
B  0
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Plasma kinetics
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Vlasov Equations
• Velocity distribution function
–
–
–
–
for each species
Convection
Lorentz force
Collisionless
f
 v  x f  m 1 FEM  v f  0
t
v B 

FEM  q  E 

c


m=mass, q=charge
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Landau Fokker Planck Equation
• Velocity distribution function
–
–
–
–
for each species
m=mass, q=charge
Convection
Electromotive force
Collisions
f
f
1
 v  x f  m FL  v f  ( )col
t
t
v B 

FL  q  E 

c


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Coulomb Collisions
• Collision of 2 charged particles (i=1,2) with
– Position xi, mass mi, charge qi
q q (x  x )
d
m1 x1  1 2 1 3 2
dt
x1  x2
q1q2 ( x2  x1 )
d
m2 x2 
3
dt
x1  x2
has solution
A  q1q2 / 
r 1  B cos(   )  Ab 2v02
  m1m2 / (m1  m2 )
tan   v02b / A
in which
– (r,θ) are polar coordinates for x1-x2
– v0 is incoming relative velocity,
– b is impact parameter
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B  b 2  A2v04b 4
Derivation of Fokker-Planck Eqtn
• Binary Coulomb collision
–
–
–
–
(with m1=m2, q1=q2)
relative velocity v0 , displacement b before collision
2
2
q
deflection angle θ
tan( / 2)  2
v0 b
scattering cross section (Rutherford)


q
 R ( )   2 2

v
sin
(

/
2)
 0

2
2
θ
b
v0
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Landau-Fokker-Planck equation
for collisions
• Coulomb interactions
– collision rate ≈ u-3 for two particles with
relative velocity u
• Fokker-Planck equation
f

1 2
( )col  
Fd ( v) f ( v) 
: D( v) f ( v)
t
v
2 vv
Fd ( v)  c1
H

f ( v ')
 c1 2 
dv '
v
v | v  v ' |
 2G
2
D( v)  c2
 c2
 f ( v ') | v  v ' | dv '
vv
vv
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Derivation of Fokker-Planck Eqtn
• Coulomb collisions are predominantly grazing
– Differential collision rate is singular at θ≈0 since
3
 R v  w sin    sin  
– Total collision rate
  v  w sin  d  
• Aggregate effect of the collisions
– measured by the momentum transfer, is
1
 (v ' v) R v  w sin  d  (w  v) 2 v  w   R (1  cos ) sin  d
– integrand is only marginally singular
 R (1  cos  )sin    sin  
1
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Derivation of Fokker-Planck Eqtn
• Debye cutoff
– Screened potential is
  (4 ) 1 (q /  0 )r 1e  r  D
– Approximate the effect of screening by cutoff in angle
   D   1  r0 / 2D
• Cross section for momentum transfer is

3
v  w   R (1  cos  ) sin  d  c v  w log 
D
• Corresponding Boltzmann collision operator Qλhas
collision rate
4
q
3
v  w  R ( )1 D sin   3
v  w 1 D
sin ( / 2)
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Derivation of Fokker-Planck Eqtn
• Analysis of Alexandre & Villani
– “On the Landau approximation in plasma physics” Ann. I. H.
Poincaré – AN 21 (2004) 61–95.
– Boltzmann eqtn without Lorentz force
f
 v  x f  Q ( f , f )
t
– Rescale (x,t) → (c/log Λ) (x’,t’) and drop ’,
Q ( f , f )
– to obtain f
 v  x f 
t
log 
– As Λ→∞,
• total angular cross section for momentum transfer goes to c|v-w|-3
• all collisions become grazing collisions
• the scaled Boltzmann collision operator converges to the LandauFokker –Planck collision operator
IPAM 31 March 2009
Derivation of Fokker-Planck Eqtn
• Scaling of Alexandre & Villani
– They find that the relevant time scale T and space scale X are
is
2
T
 r0 
vth log 
X  vthT
– On this time and space scale, they prove that
solution of the Boltzmann equation (without Lorentz force)
Q ( f , f )
f
 v  x f 
t
log 
converges to a solution of the LFP equation
f
 v  x f  QL ( f , f )
t

1 2

Fd ( v) f ( v) 
: D( v) f ( v)
v
2 vv
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Derivation of Fokker-Planck Eqtn
• Scaling difficulty
– Alexandre and Villani are unable to find a scaling such that
both the LFP collision operator and the Lorentz force terms are
significant
f
 v  x f  m 1 FL  v f  QL ( f , f )
t
– On a scale for which the Lorentz force is O(1), the collision
term is insignificant
IPAM 31 March 2009
Collisions in Gases vs. Plasmas
• Collisions between velocities v and v*
• Gas collisions
– hard spheres, σ = cross section area of sphere
– collision rate is σ | v - v* |
– any two velocities can collide → smoothing in v
• Plasma (Coulomb) collisions
–
–
–
–
–
–
very long range, potential O(1/r)
collisions are grazing, localized as in Landau eqtn
Collision rate | v - v* |-3 small for well separated velocities
differential eqtn in v, as well as x,t
waves in phase space
Landau damping (interaction between waves and particles)
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Comparison F-P to Boltzmann
• Boltzmann
– collisions are single physical collisions
– total collision rate for velocity v is
∫|v-v’| σ(|v-v’| ) f(v’) dv’
f (v)
(
)col     f (v1 ') f (v ')  f (v1 ) f (v) v  v1  (v  v1 )d dv1
t
• FP
– actual collision rate is infinite due to long range
interactions: σ = (sin θ)-4
– FP “collisions” are each aggregation of many small
deflections
– described as drift and diffusion in velocity space
f

1 2
( )col  
Fd ( v) f ( v) 
: D( v) f ( v)
t
v
2 vv
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Simulation methods
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Monte Carlo Particle Methods
for Coulomb Interactions
• Particle-field representation
– Mannheimer, Lampe & Joyce, JCP 138 (1997)
– Particles feel drag from Fd = -fd (v)v and diffusion of
strength σ = σ(D)
dv  Fd dt  σdb
– numerical solution of SDE, with Milstein correction
• Lemons et al., J Comp Phys 2008
• Particle-particle representation
– Takizuka & Abe, JCP 25 (1977), Nanbu. Phys. Rev. E. 55
(1997) Bobylev & Nanbu Phys. Rev. E. 61 (2000)
– Binary particle “collisions”, from collision integral
interpretation of FP equation
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Binary Collision Methods for LFP
• Bobylev-Nanbu (PRE 2000)
– Implicit-like transformation of LFP over a
single time step
– Expansion of scattering operator in spherical
harmonics
– Approximation at O(Δt) with tractable binary
collision interpretation
– Resulting binary collisions
• Every particle collides once per time step
• Collisions depend on Δt
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Bobylev-Nanbu Analysis
• Boltzmann eqtn, as scattering operator
 t f (v)   JF (U , u )dw
in which
U  (v  w) / 2 u  (v  w) / 2
F  f (v) f ( w)
JF (U , u  )   2 g ( u ,   n)  F (U , u n)  F (U , u  )  dn
S
J ( )   2 g (  n)  (n)  ( )  dn
S
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Implicit-like transformation
• First order approximation
f (v, t  t )  f (v, t )   JF (U , u )dw
• “Implicit” approximation
1  J
J  (e  I )
• Optimal choice of ε
   t
• Result
f (v, t  t )   1  e  t J f (v, t ) f ( w, t )dw
– Every particle collides once in every time step
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Transformation to Tractable Binary Form
• Implicit-like formulation
f (v, t  t )  
1
e
  D( u u  n, a u
 t J
3
f (v, t ) f ( w, t )dw
t ) f (v ', t ) f (w ', t )dndw
– with (for Landau-Fokker-Planck)
a  c log 
– D is an expansion in Legendre polynomials in |u| u·n
D(  , )   (4 ) 1 (2  1) P (  )e
( 1)
– D can be greatly simplified by approximation at O(Δt)
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Takizuka & Abe Method
• T. Takizuka & H. Abe, J. Comp. Phys. 25 (1977).
• T & A binary collision model is equivalent to the collision term in LandauFokker-Planck equation
– The scattering angle θ is chosen randomly from a Gaussian random variable δ
  tan( 2)
– δ has mean 0 and variance
– Parameters
2
 2  (e2 e2 nL log 8 02 m
u 3 )t
• Log Λ = Coulomb logarithm
• u = relative velocity
• Simulation
– Every particle collides once in each time interval
• Scattering angle depends on dt
• cf. DSMC for RGD: each particle has physical number of collisions
– Implemented in ICEPIC by Birdsall, Cohen and Proccaccia
– Numerical convergence analysis by Wang, REC, etal. (2007) O(dt1/2).
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Nanbu’s Method
• Combine many small-angle collisions into one aggregate collision
– K. Nanbu. Phys. Rev. E. 55 (1997)
• Scattering in time step dt
– χN = cumulative scattering angle after N collisions
– N-independent scattering parameter s
sin 2 (  N / 2  (1  e  s ) / 2
-- simulation
- theory
s  N 2 /2
– Aggregation is only for collisions between two given particle velocities
• Steps to compute cumulative scattering angle:
– At the beginning of the time step, calculate s
s  c3 u 3 (ln )t
– Determine A from
coth A  A1  e  s
– Probability that postcollison relative velocity is scattered into dΩ is
A
f (  )d  
e A cos  d 
4 sin hA
– Implemented in ICEPIC by Wang & REC
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Simulation for Plasmas:
Test Cases
•
•
•
•
•
Relaxation of an anisotropic Maxwellian
Bump-on-tail
Sheath
Two stream instability
Computations using Nanbu’s method and
hybrid method
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Numerical Test Case:
Relaxation of Anisotropic Distribution
• Specification
– Initial distribution is Maxwellian with
anisotropic temperature
– Single collision type: electron-electron
(e-e) or electron-ion (e-i).
– Spatially homogeneous.
• The figure at right shows the time
relaxation of parallel and transverse
temperatures.
– All reported results are for e-e; similar
results for e-i.
• Approximate analytic solution of
Trubnikov (1965).
IPAM 31 March 2009
Hybrid Method for Bump-on-Tail
FACM 2010
Hybrid Method Using Fluid Solver
• Improved method for spatial inhomogeneities
– Combines fluid solver with hybrid method
• previous results used Boltzmann type fluid solver
– Euler equations with source and sink terms from
therm/detherm
– application to electron sheath (below)
• potential (left), electric field (right)
FACM 2010
Conclusions and Prospects
• Landau Fokker Planck collision operator
• Infinite rate of grazing interactions
→ finite rate of aggregate collisions
• Monte Carlo simulation methods for kinetics
have trouble in the fluid and near-fluid regime
• Math leading to new methods that are robust in
fluid limit
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