Kinetic Theory for Gases and Plasmas Lecture 1. Gas Kinetics

1. Kinetic Theory for Gases and PlasmasLecture 1. Gas Kinetics Russel Caflisch IPAM Mathematics Department, UCLA IPAM Plasma Tutorials 2012

2. 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

3. Plasmas • Plasma • gas of ionized particles • 99% of visible matter • Examples • fluorescent lights • sun • fusion energy plasmas IPAM Plasma Tutorials 2012

4. 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? IPAM Plasma Tutorials 2012

5. 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 IPAM Plasma Tutorials 2012

6. Rarefied vs. Continuum Flow:Knudsen number Kn IPAM Plasma Tutorials 2012

7. Collisional Effects in the Atmosphere IPAM Plasma Tutorials 2012

8. Collisional Effects in MEMS and NEMS IPAM Plasma Tutorials 2012

9. Gas kinetics IPAM Plasma Tutorials 2012

10. 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

11. Macroscopic Variables and Equilibrium • Macroscopic variables • Density ρ, velocity u, temperature T • Maxwellian equilibrium IPAM Plasma Tutorials 2012

12. 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 v’ w’ v w 2θ v-w IPAM Plasma Tutorials 2012

13. 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 2θ v-w IPAM Plasma Tutorials 2012

14. 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 Boltzmann’s grave IPAM Plasma Tutorials 2012

15. Boltzmann Equation • Evolution of velocity distribution function f(x,v,t) due to • Convection of particles by velocity • Collisions between particles IPAM Plasma Tutorials 2012

16. Collision Invariants By symmetries of collision operator Q in which IPAM Plasma Tutorials 2012

17. 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

18. Entropy • Let φ=log f, then • It follows that Boltzmann’s H-theorem IPAM Plasma Tutorials 2012

19. Entropy and Equilibration • Since with equality iff ; i.e., Then IPAM Plasma Tutorials 2012

20. Fluid Dynamic Limit of Boltzmann • The limit of small mean free path ε • Hilbert expansion • Leading order ε-1 IPAM Plasma Tutorials 2012

21. Fluid Dynamic Limit of Boltzmann • Order ε0 • Conservation laws • Use to get IPAM Plasma Tutorials 2012

22. 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 IPAM Plasma Tutorials 2012

23. 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) IPAM Plasma Tutorials 2012

24. Simulation for Gases:Test Cases • Relaxation to equilibrium • Shock • Flow past a plate IPAM Plasma Tutorials 2012

25. 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

26. Comparison of DSMC (blue) and IFMC (red) for a shock with Mach=1.4 and Kn=0.019 Direct convection of Maxwellians u ρ T HKUST, 9 Dec 2007

27. Comparison of DSMC (contours with num values) and IFMC (contours w/o num values) for the leading edge problem. ρ T v u HKUST, 9 Dec 2007

28. 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