IPAM Plasma Tutorials 2012

Kinetic Theory for Gases and PlasmasLecture 1. Gas Kinetics

Russel CaflischIPAM

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

IPAM Plasma Tutorials 2012

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• Discrete particles• Motion by particle velocity• Interact through collisions

Fluid (continuum) description• Density, velocity, temperature• 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

Gas kinetics

IPAM Plasma Tutorials 2012

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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

• Maxwellian equilibrium3 2 2( ) (2 ) exp( ( ) 2 )M T T v v u

2

1

( )

1

3

u v f v dv

Tv u

IPAM Plasma Tutorials 2012

Collisions

• Velocities -v,w before collision-v’, w’ after collision

• Momentum and energy conservation- 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

-Explicit formulas

vv’ w’

w

' ( )

' ( )

v v v w

w w v w

v-w

2θ

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Collision Cross-Section

•The differential collision cross section (per particle)

- 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

-Factor of |v-w| accounts for flux-f2 is two particle distribution function

-Hard spheres of radius r

( , )v

2( , ) ( , )v w v w f v w d dv

2 cosr v-w

2θ

Molecular Chaos • Boltzmann assumed particles are

independent before a collision– Before collision of v and 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’

– Rate of collisions v’ → v

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2 ( , ) ( ) ( )f v w f v f w

( , ) ( ) ( )v w v w f v f w dw

( , ) ( ') ( ')v w v w f v f w dw

Boltzmann’s grave

Boltzmann Equation

• Evolution of velocity distribution function f(x,v,t) due to– Convection of particles by velocity– Collisions between particles

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( , )( ) ( , ) ( ') ( ') ( ) ( )

( , ) ( , ) sin

Q f f v B v w f v f w f v f w d d dw

B v v v

( , )x

fv f Q f f

t

Collision InvariantsBy symmetries of collision operator Q

in which

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1

1 1

1 1 1

1 1

1 1

1

1

1 1

( ) ( , )

( , ) ( ) ( ') ( ') ( ) ( )

( ) ( )

1' ' ' '

' '

' '

' ' '

' ' '

4

v Q f f dv

B v w v f v f w f v f w d d dwdv

B f f f f d d dwdv

B f f

v f v dvt

B f f f f d d d

f f d d dwdv

B f f f f d d dwdv

B f f f f d d dw

w v

dv

d

1 1, , ', ' ( ), ( ), ( '), ( ')f f f f f v f w f v f w

Collision Invariants

• By symmetries of collision operator Q

• This is 0 for all f and g iff

for all v, w, θ, ε, which implies

• Corresponds to conservation of mass, momentum and energy

IPAM Plasma Tutorials 2012

1 1 1

0 ( ) ( , ) 0

( ' ') ' '

v Q f g dv

B f f fg d d dwdv

1 1' ' 0

1 2 3

2

0 4 ( ,) , )(v c c v c v c c c c

Entropy

• Let φ=log f, then

• It follows that

Boltzmann’s H-theorem

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1 1 1 1

1

1

( ' ') log log log ' log '

log' '

f f f f

f f

f f

11 1

1

( ) ( , )

1' ' log

4 '

og

'

l

0

t v Q f g dv

f ff

f f d

f f f d d dwdvf f

v

Entropy and Equilibration

• Since

with equality iff ; i.e.,

Then

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log 0t f f dv

3 2 2( ) ( ) (2 ) exp( ( ) 2 )f v M T T v v u

f M as t

2

0 4log ( )f v c c v c v

Fluid Dynamic Limit of Boltzmann

• The limit of small mean free path ε

• Hilbert expansion

• Leading order ε-1

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1( , )x

fv f Q f f

t

0 1f f f

0( , ) , , )0 ( ;fQ M T vf uf

Fluid Dynamic Limit of Boltzmann• Order ε0

• Conservation laws

• Use to get

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0 0 12 ( , )t xv f Q f f

0 ( , , ; )f M u T v

0

2 2

1 1

( , ) 0 0tv Q f f dv v v f dv

v v

( ) 0

0

( ) 0

t

t

t

u

u u u p

E uE up

21( 3 )

2E u T

p T

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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

• 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

1

( ) ( ( )) ( ( ))N

k kk

f v v v t x x t

/k kdx dt v

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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

= physical collision rate for N particles

= γ-1 (total physical collision rate)

• Convergence (Wagner 1992)

,

( , )kk

c B v v d

Simulation for Gases:Test Cases

• Relaxation to equilibrium

• Shock

• Flow past a plate

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HKUST, 9 Dec 2007

Relxation to Equilibrium• Spatially homogeneous, Kac model• Similarity solution (Krook & Wu, 1976)

Comparison of exact solution (-), DSMC(+) and IFMC(◊)At time t=1.5 (left) and t=3.0 (right).

v v

f f

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 T

u v

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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)