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Kinetic Theory
for Gases and Plasmas
Lecture 1. Gas Kinetics
Russel Caflisch
IPAM
Mathematics Department, UCLA
IPAM Plasma Tutorials 2012
Outline
• Gas kinetics
–
–
–
–
Macroscopic variables ρ,u, T
Velocity distribution function f(x,v,t)
Hard sphere collisions (short range)
Boltzmann equation
• Plasma kinetics
– Debye length λD
– Coulomb interactions (long-range)
– Landau-Fokker-Planck equation
• Simulation methods
– Direct Simulation Monte Carlo (DSMC) for gases
– Binary collision methods for plasmas
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Plasmas
• Plasma
– gas of ionized
particles
– 99% of visible
matter
• Examples
– fluorescent lights
– sun
– fusion energy
plasmas
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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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When Does the Continuum
Description Fail?
• Knudsen number Kn=ε
– ε = (mean free path)/(characteristic distance)
– measures significance of collisions
– mean free path = distance traveled by a particle
between collisions
• Rarefied gases and plasmas
– Ε not negligibly small
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Rarefied vs. Continuum Flow:
Knudsen number Kn
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Collisional Effects in the Atmosphere
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Collisional Effects in MEMS and NEMS
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Gas kinetics
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Velocity Distribution Function
• A typical gas or plasma consists of too many particles for a
tractable direct description
– typical number of 1020 particles
• Statistical description
– Probability density function f=f(x,v,t)
– phase space (position x, velocity v)
– Maxwell & Boltzmann
IPAM Plasma Tutorials 2012
Macroscopic Variables and
Equilibrium
• Macroscopic variables
– Density ρ, velocity u, temperature T




1
  




v
 f (v)dv
 u    
 T 
1

2


 vu 
3

• Maxwellian equilibrium
M ( v)   (2 T )3 2 exp(( v  u)2  2T )
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Collisions
• Velocities
v’
v
-v,w before collision
-v’, w’ after collision
• Momentum and energy conservation
w’
w
- v + w = v’ + w’
- |v|2 + |w|2 = |v’|2 + |w’|2
-Two free parameters
- ω = (ε,θ) on sphere
- θ = scattering angle
- ε = angle of collision plane
2θ
-Explicit formulas
v '  v    (v  w)
w '  w    (v  w)
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v-w
Collision Cross-Section
•The differential collision cross section (per particle)
   ( , v )
- the area (per angle and per particle) governing the collision rate
-for particles with velocity difference v
• The rate of collisions between velocities v and w with collision
parameters in dω is
 ( , v  w ) v  w f2 (v, w)ddv
-Factor of |v-w| accounts for flux
-f2 is two particle distribution function
-Hard spheres2of radius r
  r cos 
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2θ
v-w
Molecular Chaos
• Boltzmann assumed particles are
independent before a collision
– Before collision of v and w
f2 (v, w)  f (v) f (w)
– Particles not independent after collision
– Proof by Lanford (1976) as n→∞
– Origin of irreversibility
• For a given velocity v
– Rate of collisions v → v’
 ( , v  w ) v  w f (v) f (w)dw
– Rate of collisions v’ → v
 ( , v  w ) v  w f (v ') f (w')dw
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Boltzmann’s grave
Boltzmann Equation
• Evolution of velocity distribution function
f(x,v,t) due to
– Convection of particles by velocity
– Collisions between particles
f
 v  x f  Q ( f , f )
t
Q( f , f )(v)     B( , v  w )  f (v ') f (w ')  f (v) f (w)  d d  dw
B( , v )   ( , v ) v sin 
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Collision Invariants
By symmetries of collision operator Q

 (v) f (v)dv    (v)Q( f , f )dv

t
  B ( , v  w ) (v)  f (v ') f ( w ')  f (v) f ( w)  d d  dwdv
  B  f ' f1 ' f f1  d d  dwdv
  B1  f ' f1 ' f f1  d d  dwdv
   B '  f ' f1 ' f f1  d d  dwdv
   B1 '  f ' f1 ' f f1  d d  dwdv
in which
1
  B   1   ' 1 ' f ' f1 ' f f1  d d  dwdv
4
 f , f1, f ', f1 '   f (v), f (w), f (v '), f (w ')
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Collision Invariants
• By symmetries of collision operator Q
0    (v)Q( f , g )dv  0
  B(  1   ' 1 ')  f ' f1 ' fg  d d dwdv
• This is 0 for all f and g iff
  1   ' 1 '  0
for all v, w, θ, ε, which implies
 (v)  c0  c v  c4 v
2
c  (c1 , c2 , c3 )
• Corresponds to conservation of mass,
momentum and energy
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Entropy
• Let φ=log f, then
(  1   ' 1 ')  log f  log f1  log f ' log f1 '
 log
f f1
f ' f1 '
• It follows that
 t  f log f dv    (v)Q( f , g )dv
 f f1 
1
   f ' f1 ' f f1  log 
 d d  dwdv
4
 f ' f1 ' 
0
Boltzmann’s H-theorem
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Entropy and Equilibration
• Since
 t  f log f dv  0
with equality iff
log f (v)  c0  c v  c4 v
2
; i.e.,
f (v)  M ( v)   (2 T )3 2 exp(( v  u)2  2T )
Then
f  M as t  
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Fluid Dynamic Limit of
Boltzmann
• The limit of small mean free path ε
f
1
 v  x f  Q ( f , f )
t

• Hilbert expansion
f  f0   f1 
• Leading order ε-1
Q( f , f )  0  f0  M (  , u, T ; v)
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Fluid Dynamic Limit of Boltzmann
• Order ε0
 t  v x  f0  2Q( f0 , f1 )
• Conservation laws
 1 
 1 
 
 
  v  Q( f , f )dv  0    v    t  v   f 0dv  0
 v2
 v2
 
 
• Use f0  M (  , u, T ; v) to get
t     ( u)  0
 t   u       u  u   p  0
 t E    (uE  up )  0
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1
2
 ( u  3T )
2
p  T
E
Dominant numerical method for dilute flows
• DSMC = Direct Simulation Monte Carlo
– Invented by Graeme Bird, early 1970’s
– Represents density f as collection of particles
N
f (v)     (v  vk (t )) ( x  xk (t ))
k 1
• γ = number of physical particles per numerical particle
>>1
– Perform randomly chosen collisions
• Velocity pairs vk and vℓ and collision angles (ε,θ)
• vk , vℓ → vk’ , vℓ’
– Particle advection
dxk / dt  vk
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DSMC Features
• Fidelity to physics
– Each collision is a (randomly chosen) physical collision
– Each particle has physically correct number of collisions
• Probability of collision (vk ,vℓ , ε, θ) proportional to B(θ, | vk - vℓ |)
– Total numerical collision rate
c  B( , vk  v )d
k,
= physical collision rate for N particles
= γ-1 (total physical collision rate)
• Convergence (Wagner 1992)
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Simulation for Gases:
Test Cases
• Relaxation to equilibrium
• Shock
• Flow past a plate
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Relxation to Equilibrium
• Spatially homogeneous, Kac model
• Similarity solution (Krook & Wu, 1976)
f
f
v
v
Comparison of exact solution (-), DSMC(+) and IFMC(◊)
At time t=1.5 (left) and t=3.0 (right).
HKUST, 9 Dec 2007
Comparison of DSMC (blue) and IFMC (red) for a
shock with Mach=1.4 and Kn=0.019
Direct convection of Maxwellians
ρ
u
T
HKUST, 9 Dec 2007
Comparison of DSMC (contours with num values)
and IFMC (contours w/o num values)
for the leading edge problem.
ρ
u
T
v
HKUST, 9 Dec 2007
Conclusions and Prospects
•
•
•
•
•
•
•
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
– math needed to design methods that overcome this
difficulty!
– Multiscale method combining fluid & particle descriptions
• RC, Ricketson, Rosen, Yann (UCLA), Cohen , Dimits (LLNL)
IPAM Plasma Tutorials 2012