Kinetic theory of gases
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1 Kinetic theory of gases Toan T. Nguyen Penn State University Graduate Student seminar, PSU Jan 19th, 2017 Fall 2017, I teach a graduate topics course: same topics! Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
2 Figure : 4 states of matter, plus Bose-Einstein condensate! Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
3 Figure : 4 states of matter, plus Bose-Einstein condensate! James Clerk Maxwell, in 1859, gave birth to kinetic theory: use the statistical approach to describe the dynamics of a (rarefied) gas. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
4 (x, p) Figure : Kinetic theory of gases: phase space! Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
5 (x, p) Figure : Kinetic theory of gases: phase space! Gas molecules are identical. One-particle phase space: position x Ω R d, momentum p R d Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
6 (x, p) Figure : Kinetic theory of gases: phase space! Gas molecules are identical. One-particle phase space: position x Ω R d, momentum p R d f (t, x, p) the probability density distribution. Mass: f (t, x, p) dxdp. Interest: the dynamics of f (t, x, p)? Non-equilibrium theory. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
7 I. Collisionless Kinetic Theory Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
8 Hamilton 1830 s classical mechanics 1 : one-particle Hamiltonian H(x, p) (induced from Lagrangian): ẋ = p H, ṗ = x H 1 In truth, it should be an N-particle Hamiltonian dynamics, then mean field limit or BBGKY hierarchy to kinetic theory. S1: Presentation on its formal derivation? Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
9 Hamilton 1830 s classical mechanics 1 : one-particle Hamiltonian H(x, p) (induced from Lagrangian): ẋ = p H, ṗ = x H Examples: Free particles: H(x, p) = 1 2m p 2 1 In truth, it should be an N-particle Hamiltonian dynamics, then mean field limit or BBGKY hierarchy to kinetic theory. S1: Presentation on its formal derivation? Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
10 Hamilton 1830 s classical mechanics 1 : one-particle Hamiltonian H(x, p) (induced from Lagrangian): ẋ = p H, ṗ = x H Examples: Free particles: Particles in a potential: H(x, p) = 1 2m p 2 H(x, p) = 1 2m p 2 + V pot (x) 1 In truth, it should be an N-particle Hamiltonian dynamics, then mean field limit or BBGKY hierarchy to kinetic theory. S1: Presentation on its formal derivation? Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
11 Also for plasmas: charged particles in electromagnetic fields H(x, p) = 1 2m p qa(x) 2 + φ(x) with electric and magnetic potentials φ, A. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
12 Also for plasmas: charged particles in electromagnetic fields H(x, p) = 1 2m p qa(x) 2 + φ(x) with electric and magnetic potentials φ, A. Force acting on particles: F = x H = E + q (p B) m which is the Lorentz force, with E = x φ and B = x A. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
13 t > 0 (x t, p t ) (x 0, p 0 ) Figure : Liouville s Theorem: volume in phase space remains constant (Exercise: prove this). Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
14 t > 0 (x t, p t ) (x 0, p 0 ) Figure : Liouville s Theorem: volume in phase space remains constant (Exercise: prove this). This yields the kinetic equation: t f = {H, f } (Poisson bracket), explicitly 0 = d dt f (t, x(t), p(t)) = tf + p H x f x H p f = t f + v x f + F force v f Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
15 Three examples of kinetic equations (remain very active): Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
16 Three examples of kinetic equations (remain very active): Vlasov dynamics (transport in phase space): t f + v x f + F v f = 0 Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
17 Three examples of kinetic equations (remain very active): Vlasov dynamics (transport in phase space): Vlasov-Poisson: F = x φ, t f + v x f + F v f = 0 x φ = σ f (t, x, v) dv, R 3 with σ = 1 (plasma physics: charged particles) or σ = 1 (gravitational case: stars in a galaxy). Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
18 Three examples of kinetic equations (remain very active): Vlasov dynamics (transport in phase space): Vlasov-Poisson: F = x φ, t f + v x f + F v f = 0 x φ = σ f (t, x, v) dv, R 3 with σ = 1 (plasma physics: charged particles) or σ = 1 (gravitational case: stars in a galaxy). Vlasov-Maxwell (plasma): Lorentz force F = E + v B with the electromagnetic fields E, B solving Maxwell (generated by charge and current density). Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
19 Can you solve Vlasov, the Cauchy problem? Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
20 Can you solve Vlasov, the Cauchy problem? In fact, it s simply ODEs: ẋ = v, v = F (t, x, v) Then, Vlasov f (t, x, v) = f 0 (x t, v t ). Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
21 Can you solve Vlasov, the Cauchy problem? In fact, it s simply ODEs: ẋ = v, v = F (t, x, v) Then, Vlasov f (t, x, v) = f 0 (x t, v t ). One needs Lipschitz bounds on F (t, x, v). Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
22 Can you solve Vlasov, the Cauchy problem? In fact, it s simply ODEs: ẋ = v, v = F (t, x, v) Then, Vlasov f (t, x, v) = f 0 (x t, v t ). One needs Lipschitz bounds on F (t, x, v). Vlasov-Poisson: F = x φ and φ = ± f (t, x, v) dv. Roughly speaking: x F = 2 x( x ) 1 ρ ρ, up to a log of derivatives of f. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
23 Can you solve Vlasov, the Cauchy problem? In fact, it s simply ODEs: ẋ = v, v = F (t, x, v) Then, Vlasov f (t, x, v) = f 0 (x t, v t ). One needs Lipschitz bounds on F (t, x, v). Vlasov-Poisson: F = x φ and φ = ± f (t, x, v) dv. Roughly speaking: x F = 2 x( x ) 1 ρ ρ, up to a log of derivatives of f. Global unique solutions (1991): Pfaffelmoser, Lions-Perthame, Schaeffer, Glassey s book. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
24 Can you solve Vlasov, the Cauchy problem? In fact, it s simply ODEs: ẋ = v, v = F (t, x, v) Then, Vlasov f (t, x, v) = f 0 (x t, v t ). One needs Lipschitz bounds on F (t, x, v). Vlasov-Poisson: F = x φ and φ = ± f (t, x, v) dv. Roughly speaking: x F = 2 x( x ) 1 ρ ρ, up to a log of derivatives of f. Global unique solutions (1991): Pfaffelmoser, Lions-Perthame, Schaeffer, Glassey s book. Vlasov-Maxwell: Outstanding open problem! Glassey Strauss s theorem. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
25 Several mathematical issues: t f = {H, f } (focus on previous examples). Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
26 Several mathematical issues: t f = {H, f } (focus on previous examples). Infinitely many conserved quantities, and infinitely many equilibria: D ϕ(f ) = 0, dt f = ϕ(h(x, p)) for arbitrary ϕ( ). Which one is dynamically stable? and shouldn t it be just Gaussian? Maxwell-Boltzmann distribution (next section)? Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
27 Several mathematical issues: t f = {H, f } (focus on previous examples). Infinitely many conserved quantities, and infinitely many equilibria: D ϕ(f ) = 0, dt f = ϕ(h(x, p)) for arbitrary ϕ( ). Which one is dynamically stable? and shouldn t it be just Gaussian? Maxwell-Boltzmann distribution (next section)? Hence, large time dynamics and stability of equilibria are very delicate! Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
28 VP: t f + v x f x φ v f = 0, φ = ρ 1. For instance, homogenous equilibria µ = µ(e), e := 1 2 v 2, φ 0 = 0. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
29 VP: t f + v x f x φ v f = 0, φ = ρ 1. For instance, homogenous equilibria µ = µ(e), e := 1 2 v 2, φ 0 = 0. Arnold: Look at casimir s invariant [1 ] A[f ] = 2 v 2 f + ϕ(f ) dvdx φ 2 dx. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
30 VP: t f + v x f x φ v f = 0, φ = ρ 1. For instance, homogenous equilibria µ = µ(e), e := 1 2 v 2, φ 0 = 0. Arnold: Look at casimir s invariant [1 ] A[f ] = 2 v 2 f + ϕ(f ) dvdx φ 2 dx. µ is a critical point iff ϕ (µ) = 1 2 v 2 or ϕ = µ 1. Convexity of A[f ] requires µ e < 0. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
31 VP: t f + v x f x φ v f = 0, φ = ρ 1. For instance, homogenous equilibria µ = µ(e), e := 1 2 v 2, φ 0 = 0. Arnold: Look at casimir s invariant [1 ] A[f ] = 2 v 2 f + ϕ(f ) dvdx φ 2 dx. µ is a critical point iff ϕ (µ) = 1 2 v 2 or ϕ = µ 1. Convexity of A[f ] requires µ e < 0. µ e < 0 implies Arnold s nonlinear stability. But, what s long-time dynamics? Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
32 µ(v) µ(v) v min v v min v Figure : Stable vs unstable equilibria. Beautiful Penrose s iff criteria: Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
33 µ(v) µ(v) v min v v min v Figure : Stable vs unstable equilibria. Beautiful Penrose s iff criteria: linearly unstable iff v min exists and µ (v)dv P.V. > 0. R v v min S2: Presentation on Penrose s criteria? Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
34 Mouhot-Villani: nonlinear Landau damping of Penrose stable homogenous equilibria of VP (earning Villani a Fields medal). S3: presentation on linearized case? Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
35 Mouhot-Villani: nonlinear Landau damping of Penrose stable homogenous equilibria of VP (earning Villani a Fields medal). S3: presentation on linearized case? Open: Are there stable inhomogenous equlibria (VP)? f = ϕ(h(x, p)). Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
36 Mouhot-Villani: nonlinear Landau damping of Penrose stable homogenous equilibria of VP (earning Villani a Fields medal). S3: presentation on linearized case? Open: Are there stable inhomogenous equlibria (VP)? f = ϕ(h(x, p)). Some of my works (available on my homepage): Stability of inhomogenous equilibria of VM (w/ Strauss). Various asymptotic limits of Vlasov (w/ Han-Kwan). Non-relativistic limits of VM (w/ Han-Kwan and Rousset). Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
37 Collisional kinetic theory II. Collisional Kinetic Theory Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
38 Collisional kinetic theory v v v v v v = 0 v v v Figure : Elastic collisions: momentum and energy are exchanged between particles, however conserved, after collision: Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
39 Collisional kinetic theory v v v v v v = 0 v v v Figure : Elastic collisions: momentum and energy are exchanged between particles, however conserved, after collision: v v + v = v + v v 2 + v 2 = v 2 + v 2 v v+v 2 Sphere S vv : v v Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
40 Collisional kinetic theory One has the Boltzmann equation: in which collision operator Q[f ]: t f {H, f } = Q[f ] Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
41 Collisional kinetic theory One has the Boltzmann equation: t f {H, f } = Q[f ] in which collision operator Q[f ]: [ ] Q[f ](x, v) = ω(v, v v, v ) R 9 }{{} f 2 (x, v, v ) f }{{} 2 (x, v, v ) }{{} dv dv dv Scattering Gain Loss with f 2 (x, v, v ) the probability distribution of finding two particles with respective momenta. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
42 Collisional kinetic theory One has the Boltzmann equation: t f {H, f } = Q[f ] in which collision operator Q[f ]: [ ] Q[f ](x, v) = ω(v, v v, v ) R 9 }{{} f 2 (x, v, v ) f }{{} 2 (x, v, v ) }{{} dv dv dv Scattering Gain Loss with f 2 (x, v, v ) the probability distribution of finding two particles with respective momenta. Molecular chaos assumption: f 2 (x, v, v ) = f (x, v)f (x, v ) asserting velocities are uncorrelated before the collision (unlike Hamiltonian, here it introduces a notion of time arrow!). Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
43 Collisional kinetic theory Conservation laws: v, v S vv and hence dv dv = dσ(s vv ) = v v dσ(n), n S 2. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
44 Collisional kinetic theory Conservation laws: v, v S vv and hence dv dv = dσ(s vv ) = v v dσ(n), n S 2. (symmetric) Scattering: Elastic hard-sphere collisions: ω(v, v v, v ) = n (v v ) v v (angle of collision) Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
45 Collisional kinetic theory Conservation laws: v, v S vv and hence dv dv = dσ(s vv ) = v v dσ(n), n S 2. (symmetric) Scattering: Elastic hard-sphere collisions: ω(v, v v, v ) = n (v v ) v v (angle of collision) The Boltzmann equation (1872) for hard-sphere gases: [ t f {H, f } = n (v v ) f f ff ]dndv. R 3 S 2 Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
46 Collisional kinetic theory Conservation laws: v, v S vv and hence dv dv = dσ(s vv ) = v v dσ(n), n S 2. (symmetric) Scattering: Elastic hard-sphere collisions: ω(v, v v, v ) = n (v v ) v v (angle of collision) The Boltzmann equation (1872) for hard-sphere gases: [ t f {H, f } = n (v v ) f f ff ]dndv. R 3 S 2 Is it solvable? Di Perna-Lions 89, Guo There remain many outstanding open problems. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
47 Collisional kinetic theory Boltzmann s H-Theorem: Look at the entropy: H(t) = f log f dxdv. R 6 Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
48 Collisional kinetic theory Boltzmann s H-Theorem: Look at the entropy: H(t) = f log f dxdv. R 6 One has d [ ] dt H(t) = n (v v ) f f ff log f dndxdvdv R 3 S 2 = 1 [ ] n (v v ) f f ff log ff 4 R 3 S 2 f f dndxdvdv 0. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
49 Collisional kinetic theory Boltzmann s H-Theorem: Look at the entropy: H(t) = f log f dxdv. R 6 One has d [ ] dt H(t) = n (v v ) f f ff log f dndxdvdv R 3 S 2 = 1 [ ] n (v v ) f f ff log ff 4 R 3 S 2 f f dndxdvdv 0. Entropy is decreasing! (unlike Hamiltonian, there is a time arrow!). Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
50 Collisional kinetic theory Boltzmann s H-Theorem: There is more: equilibria iff and so iff (presentation?): ff = f f, v, v S vv Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
51 Collisional kinetic theory Boltzmann s H-Theorem: There is more: equilibria iff ff = f f, v, v S vv and so iff (presentation?): Maxwellian distribution f (x, v) = Ce β v u 2 = ρ(x) v u(x) 2 e 2θ(x) (2πθ(x)) 3/2 in which (ρ, u, θ) denotes macroscopic density, velocity, and temperature of gases. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
52 Collisional kinetic theory Hydrodynamics limits (if time permits). Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
53 Collisional kinetic theory Hydrodynamics limits (if time permits). Mathematical difficulty of the Cauchy problem and convergence to equilibrium. Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
54 Collisional kinetic theory Hydrodynamics limits (if time permits). Mathematical difficulty of the Cauchy problem and convergence to equilibrium. Some of my works (available on my homepage): On justification of Maxwell-Boltzmann distribution (w/ Bardos, Golse, Sentis) on Quantum Boltzmann, interaction between fermions, bosons, and Bose-Einstein condenstate (w/ Tran) Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, / 20
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