Kinetic Theory for Gases and Plasmas Lecture 1. Gas Kinetics

IPAM Plasma Tutorials 2012

Kinetic Theory for Gases and PlasmasLecture 1. Gas Kinetics

Russel CaflischIPAM

Mathematics Department, UCLA


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


Plasmas

• Plasma– gas of ionized

particles

– 99% of visible matter

• Examples– fluorescent lights

– sun

– fusion energy plasmas


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?


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


Rarefied vs. Continuum Flow:Knudsen number Kn


Collisional Effects in the Atmosphere


Collisional Effects in MEMS and NEMS

Gas kinetics



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


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


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


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


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


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

( , ) ( , ) 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


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


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


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


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


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


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


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


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


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


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)

Kinetic Theory for Gases and Plasmas Lecture 1. Gas Kinetics

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