Kinetic Theory of Gases

Toan T. Nguyen1,2 Penn State University

Graduate Student seminar, PSU Jan 19th, 2017

Fall 2017, I teach a graduate topics course: same topics !

1Homepage: http://math.psu.edu/nguyen 2Math blog: https://sites.psu.edu/nguyen Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 1 / 20 • James Clerk Maxwell, in 1859, gave birth to kinetic theory: use the statistical approach to describe the dynamics of a (rareﬁed) gas.

Figure: 4 states of matter, plus Bose-Einstein condensate!

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 2 / 20 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, 2017 2 / 20 Gas molecules are identical. d d One-particle phase space: position x ∈ Ω ⊂ R , momentum p ∈ R 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.

(x, p)

Figure: Kinetic theory of gases: phase space!

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 3 / 20 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.

(x, p)

Figure: Kinetic theory of gases: phase space!

Gas molecules are identical. d d One-particle phase space: position x ∈ Ω ⊂ R , momentum p ∈ R

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 3 / 20 (x, p)

Figure: Kinetic theory of gases: phase space!

Gas molecules are identical. d d One-particle phase space: position x ∈ Ω ⊂ R , momentum p ∈ R 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, 2017 3 / 20 Collisionless kinetic theory

I. Collisionless Kinetic Theory

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 4 / 20 Examples: Free particles: 1 H(x, p) = |p|2 2m Particles in a potential: 1 H(x, p) = |p|2 + V (x) 2m pot

Collisionless kinetic theory

Hamilton 1830’s classical mechanics3: one-particle Hamiltonian H(x, p) (induced from Lagrangian):

x˙ = ∇pH, p˙ = −∇x H

3In 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, 2017 5 / 20 Particles in a potential: 1 H(x, p) = |p|2 + V (x) 2m pot

Collisionless kinetic theory

Hamilton 1830’s classical mechanics3: one-particle Hamiltonian H(x, p) (induced from Lagrangian):

x˙ = ∇pH, p˙ = −∇x H

Examples: Free particles: 1 H(x, p) = |p|2 2m

Hamilton 1830’s classical mechanics3: one-particle Hamiltonian H(x, p) (induced from Lagrangian):

x˙ = ∇pH, p˙ = −∇x H

Examples: Free particles: 1 H(x, p) = |p|2 2m Particles in a potential: 1 H(x, p) = |p|2 + V (x) 2m pot

3In 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, 2017 5 / 20 • Force acting on particles: q F = −∇ H = E + (p × B) x m

which is the Lorentz force, with E = −∇x φ and B = ∇x × A.

Collisionless kinetic theory

• Also for plasmas: charged particles in electromagnetic ﬁelds 1 H(x, p) = |p − qA(x)|2 + φ(x) 2m with electric and magnetic potentials φ, A.

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 6 / 20 Collisionless kinetic theory

• Also for plasmas: charged particles in electromagnetic ﬁelds 1 H(x, p) = |p − qA(x)|2 + φ(x) 2m with electric and magnetic potentials φ, A.

• Force acting on particles: q F = −∇ H = E + (p × B) x 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, 2017 6 / 20 This yields the kinetic equation: ∂t f = {H, f } (Poisson bracket), explicitly d 0 = f (t, x(t), p(t)) = ∂ f + ∇ H · ∇ f − ∇ H · ∇ f dt t p x x p = ∂t f + v · ∇x f + Fforce · ∇v f

Collisionless kinetic theory

(xt , pt ) t > 0

(x0, p0)

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, 2017 7 / 20 Collisionless kinetic theory

(xt , pt ) t > 0

(x0, p0)

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 d 0 = f (t, x(t), p(t)) = ∂ f + ∇ H · ∇ f − ∇ H · ∇ f dt t p x x p = ∂t f + v · ∇x f + Fforce · ∇v f

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 7 / 20 Vlasov dynamics (transport in phase space):

∂t f + v · ∇x f + F · ∇v f = 0

Vlasov-Poisson: Z F = −∇x φ, −∆x φ = σ f (t, x, v) dv, R3 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).

Collisionless kinetic theory

Three examples of kinetic equations (remain very active):

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 8 / 20 Vlasov-Poisson: Z F = −∇x φ, −∆x φ = σ f (t, x, v) dv, R3 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).

Collisionless kinetic theory

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, 2017 8 / 20 Vlasov-Maxwell (plasma): Lorentz force F = E + v × B with the electromagnetic ﬁelds E, B solving Maxwell (generated by charge and current density).

Collisionless kinetic theory

Three examples of kinetic equations (remain very active): Vlasov dynamics (transport in phase space):

∂t f + v · ∇x f + F · ∇v f = 0

Vlasov-Poisson: Z F = −∇x φ, −∆x φ = σ f (t, x, v) dv, R3 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, 2017 8 / 20 Collisionless kinetic theory

Three examples of kinetic equations (remain very active): Vlasov dynamics (transport in phase space):

∂t f + v · ∇x f + F · ∇v f = 0

Vlasov-Poisson: Z F = −∇x φ, −∆x φ = σ f (t, x, v) dv, R3 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, 2017 8 / 20 In fact, it’s simply ODEs:

x˙ = v, v˙ = F (t, x, v)

Then, Vlasov f (t, x, v) = f0(x−t , v−t ).

• One needs Lipschitz bounds on F (t, x, v). R • Vlasov-Poisson: F = −∇x φ and −∆φ = ± f (t, x, v) dv. Roughly 2 −1 speaking: ∇x F = ∇x (−∆x ) 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.

Collisionless kinetic theory

• Can you solve Vlasov, the Cauchy problem?

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 9 / 20 • One needs Lipschitz bounds on F (t, x, v). R • Vlasov-Poisson: F = −∇x φ and −∆φ = ± f (t, x, v) dv. Roughly 2 −1 speaking: ∇x F = ∇x (−∆x ) 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.

Collisionless kinetic theory

• Can you solve Vlasov, the Cauchy problem? In fact, it’s simply ODEs:

x˙ = v, v˙ = F (t, x, v)

Then, Vlasov f (t, x, v) = f0(x−t , v−t ).

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 9 / 20 R • Vlasov-Poisson: F = −∇x φ and −∆φ = ± f (t, x, v) dv. Roughly 2 −1 speaking: ∇x F = ∇x (−∆x ) 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.

Collisionless kinetic theory

• Can you solve Vlasov, the Cauchy problem? In fact, it’s simply ODEs:

x˙ = v, v˙ = F (t, x, v)

Then, Vlasov f (t, x, v) = f0(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, 2017 9 / 20 • Global unique solutions (1991): Pfaﬀelmoser, Lions-Perthame, Schaeﬀer, Glassey’s book.

• Vlasov-Maxwell: Outstanding open problem! Glassey Strauss’s theorem.

Collisionless kinetic theory

• Can you solve Vlasov, the Cauchy problem? In fact, it’s simply ODEs:

x˙ = v, v˙ = F (t, x, v)

Then, Vlasov f (t, x, v) = f0(x−t , v−t ).

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 9 / 20 • Vlasov-Maxwell: Outstanding open problem! Glassey Strauss’s theorem.

Collisionless kinetic theory

• Can you solve Vlasov, the Cauchy problem? In fact, it’s simply ODEs:

x˙ = v, v˙ = F (t, x, v)

Then, Vlasov f (t, x, v) = f0(x−t , v−t ).

• Global unique solutions (1991): Pfaﬀelmoser, Lions-Perthame, Schaeﬀer, Glassey’s book.

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 9 / 20 Collisionless kinetic theory

• Can you solve Vlasov, the Cauchy problem? In fact, it’s simply ODEs:

x˙ = v, v˙ = F (t, x, v)

Then, Vlasov f (t, x, v) = f0(x−t , v−t ).

• 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, 2017 9 / 20 Inﬁnitely many conserved quantities, and inﬁnitely many equilibria: D ϕ(f ) = 0, f = ϕ(H(x, p)) dt ∗ 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!

Collisionless kinetic theory

Several mathematical issues: ∂t f = {H, f } (focus on previous examples).

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 10 / 20 Hence, large time dynamics and stability of equilibria are very delicate!

Collisionless kinetic theory

Several mathematical issues: ∂t f = {H, f } (focus on previous examples). Inﬁnitely many conserved quantities, and inﬁnitely many equilibria: D ϕ(f ) = 0, f = ϕ(H(x, p)) dt ∗ 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, 2017 10 / 20 Collisionless kinetic theory

Hence, large time dynamics and stability of equilibria are very delicate!

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 10 / 20 µe < 0 implies Arnold’s nonlinear stability. But, what’s long-time dynamics?

Arnold: Look at casimir’s invariant ZZ h1 i 1 Z A[f ] = |v|2f + ϕ(f ) dvdx + |∇φ|2 dx. 2 2

0 1 2 0 −1 µ is a critical point iﬀ ϕ (µ) = − 2 |v| or ϕ = −µ . Convexity of A[f ] requires µe < 0.

Collisionless kinetic theory

VP: ∂t f + v · ∇x f − ∇x φ · ∇v f = 0, −∆φ = ρ − 1.

For instance, homogenous equilibria 1 µ = µ(e), e := |v|2, φ = 0. 2 0

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 11 / 20 µe < 0 implies Arnold’s nonlinear stability. But, what’s long-time dynamics?

0 1 2 0 −1 µ is a critical point iﬀ ϕ (µ) = − 2 |v| or ϕ = −µ . Convexity of A[f ] requires µe < 0.

Collisionless kinetic theory

VP: ∂t f + v · ∇x f − ∇x φ · ∇v f = 0, −∆φ = ρ − 1.

For instance, homogenous equilibria 1 µ = µ(e), e := |v|2, φ = 0. 2 0 Arnold: Look at casimir’s invariant ZZ h1 i 1 Z A[f ] = |v|2f + ϕ(f ) dvdx + |∇φ|2 dx. 2 2

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 11 / 20 µe < 0 implies Arnold’s nonlinear stability. But, what’s long-time dynamics?

Collisionless kinetic theory

VP: ∂t f + v · ∇x f − ∇x φ · ∇v f = 0, −∆φ = ρ − 1.

For instance, homogenous equilibria 1 µ = µ(e), e := |v|2, φ = 0. 2 0 Arnold: Look at casimir’s invariant ZZ h1 i 1 Z A[f ] = |v|2f + ϕ(f ) dvdx + |∇φ|2 dx. 2 2

0 1 2 0 −1 µ is a critical point iﬀ ϕ (µ) = − 2 |v| or ϕ = −µ . Convexity of A[f ] requires µe < 0.

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 11 / 20 Collisionless kinetic theory

VP: ∂t f + v · ∇x f − ∇x φ · ∇v f = 0, −∆φ = ρ − 1.

For instance, homogenous equilibria 1 µ = µ(e), e := |v|2, φ = 0. 2 0 Arnold: Look at casimir’s invariant ZZ h1 i 1 Z A[f ] = |v|2f + ϕ(f ) dvdx + |∇φ|2 dx. 2 2

0 1 2 0 −1 µ is a critical point iﬀ ϕ (µ) = − 2 |v| or ϕ = −µ . Convexity of A[f ] requires µe < 0.

µe < 0 implies Arnold’s nonlinear stability. But, what’s long-time dynamics?

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 11 / 20 linearly unstable iﬀ vmin exists and Z µ0(v)dv P.V. > 0. R v − vmin S2: Presentation on Penrose’s criteria?

Collisionless kinetic theory

µ(v) µ(v)

vmin v min vv

Figure: Stable vs unstable equilibria. Beautiful Penrose’s iﬀ criteria:

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 12 / 20 Collisionless kinetic theory

µ(v) µ(v)

vmin v min vv

Figure: Stable vs unstable equilibria. Beautiful Penrose’s iﬀ criteria: linearly unstable iﬀ vmin exists and Z µ0(v)dv P.V. > 0. R v − vmin S2: Presentation on Penrose’s criteria?

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 12 / 20 • 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).

Collisionless kinetic theory

• 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, 2017 13 / 20 • 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).

Collisionless kinetic theory

• 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, 2017 13 / 20 Collisionless kinetic theory

• 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, 2017 13 / 20 Collisional kinetic theory

II. Collisional Kinetic Theory

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 14 / 20 v∗

0 0 0 v + v∗ = v + v∗ v∗ 2 2 0 2 0 2 v+v∗ |v| + |v∗| = |v | + |v∗| 2

0 Sphere Svv∗ : v v

Collisional kinetic theory

v 0

0 v v v v

v∗ v 0 v 0 v∗ = 0 ∗ ∗

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, 2017 15 / 20 Collisional kinetic theory

v 0

0 v v v v

v∗ v 0 v 0 v∗ = 0 ∗ ∗

Figure: Elastic collisions: momentum and energy are exchanged between particles, however conserved, after collision: v∗

0 0 0 v + v∗ = v + v∗ v∗ 2 2 0 2 0 2 v+v∗ |v| + |v∗| = |v | + |v∗| 2

0 Sphere Svv∗ : v v

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 15 / 20 ZZZ 0 0 h 0 0 i 0 0 Q[f ](x, v) = ω(v, v∗|v , v∗) f2(x, v , v∗) − f2(x, v, v∗) dv∗dv dv∗ R9 | {z } | {z } | {z } Scattering Gain Loss

with f2(x, v, v∗) the probability distribution of ﬁnding two particles with respective momenta. • Molecular chaos assumption:

f2(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!).

Collisional kinetic theory

One has the Boltzmann equation:

∂t f − {H, f } = Q[f ] in which collision operator Q[f ]:

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 16 / 20 • Molecular chaos assumption:

f2(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!).

Collisional kinetic theory

One has the Boltzmann equation:

∂t f − {H, f } = Q[f ] in which collision operator Q[f ]: ZZZ 0 0 h 0 0 i 0 0 Q[f ](x, v) = ω(v, v∗|v , v∗) f2(x, v , v∗) − f2(x, v, v∗) dv∗dv dv∗ R9 | {z } | {z } | {z } Scattering Gain Loss with f2(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, 2017 16 / 20 Collisional kinetic theory

One has the Boltzmann equation:

f2(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, 2017 16 / 20 Is it solvable ? Di Perna-Lions ’89, Guo 2010. There remain many outstanding open problems.

• (symmetric) Scattering: Elastic hard-sphere collisions:

0 0 |n · (v − v∗)| ω(v, v∗|v , v∗) = (angle of collision) |v − v∗|

• The Boltzmann equation (1872) for hard-sphere gases: Z Z h 0 0 i ∂t f − {H, f } = |n · (v − v∗)| f f∗ − ﬀ∗ dndv∗. 3 R S2

Collisional kinetic theory

0 0 • Conservation laws: v , v∗ ∈ Svv∗ and hence

0 0 dv dv∗ = dσ(Svv∗ ) = |v − v∗|dσ(n), n ∈ S2.

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 17 / 20 Is it solvable ? Di Perna-Lions ’89, Guo 2010. There remain many outstanding open problems.

• The Boltzmann equation (1872) for hard-sphere gases: Z Z h 0 0 i ∂t f − {H, f } = |n · (v − v∗)| f f∗ − ﬀ∗ dndv∗. 3 R S2

Collisional kinetic theory

0 0 • Conservation laws: v , v∗ ∈ Svv∗ and hence

0 0 dv dv∗ = dσ(Svv∗ ) = |v − v∗|dσ(n), n ∈ S2.

• (symmetric) Scattering: Elastic hard-sphere collisions:

0 0 |n · (v − v∗)| ω(v, v∗|v , v∗) = (angle of collision) |v − v∗|

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 17 / 20 Is it solvable ? Di Perna-Lions ’89, Guo 2010. There remain many outstanding open problems.

Collisional kinetic theory

0 0 • Conservation laws: v , v∗ ∈ Svv∗ and hence

0 0 dv dv∗ = dσ(Svv∗ ) = |v − v∗|dσ(n), n ∈ S2.

• (symmetric) Scattering: Elastic hard-sphere collisions:

0 0 |n · (v − v∗)| ω(v, v∗|v , v∗) = (angle of collision) |v − v∗|

• The Boltzmann equation (1872) for hard-sphere gases: Z Z h 0 0 i ∂t f − {H, f } = |n · (v − v∗)| f f∗ − ﬀ∗ dndv∗. 3 R S2

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 17 / 20 Collisional kinetic theory

0 0 • Conservation laws: v , v∗ ∈ Svv∗ and hence

0 0 dv dv∗ = dσ(Svv∗ ) = |v − v∗|dσ(n), n ∈ S2.

• (symmetric) Scattering: Elastic hard-sphere collisions:

0 0 |n · (v − v∗)| ω(v, v∗|v , v∗) = (angle of collision) |v − v∗|

• The Boltzmann equation (1872) for hard-sphere gases: Z Z h 0 0 i ∂t f − {H, f } = |n · (v − v∗)| f f∗ − ﬀ∗ dndv∗. 3 R S2 Is it solvable ? Di Perna-Lions ’89, Guo 2010. There remain many outstanding open problems.

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 17 / 20 • One has Z Z d h 0 0 i H(t) = |n · (v − v∗)| f f∗ − ff∗ log f dndxdvdv∗ dt 3 R S2 Z Z 1 h 0 0 i ff∗ = |n · (v − v∗)| f f∗ − ff∗ log 0 0 dndxdvdv∗ 4 3 f f R S2 ∗ ≤ 0.

• Entropy is decreasing ! (unlike Hamiltonian, there is a time arrow!).

Collisional kinetic theory

• Boltzmann’s H-Theorem: Look at the entropy: ZZ H(t) = f log f dxdv. R6

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 18 / 20 • Entropy is decreasing ! (unlike Hamiltonian, there is a time arrow!).

Collisional kinetic theory

• Boltzmann’s H-Theorem: Look at the entropy: ZZ H(t) = f log f dxdv. R6 • One has Z Z d h 0 0 i H(t) = |n · (v − v∗)| f f∗ − ff∗ log f dndxdvdv∗ dt 3 R S2 Z Z 1 h 0 0 i ff∗ = |n · (v − v∗)| f f∗ − ff∗ log 0 0 dndxdvdv∗ 4 3 f f R S2 ∗ ≤ 0.

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 18 / 20 Collisional kinetic theory

• Entropy is decreasing ! (unlike Hamiltonian, there is a time arrow!).

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 18 / 20 Maxwellian distribution

2 2 ρ(x) − |v−u(x)| f (x, v) = Ce−β|v−u| = e 2θ(x) (2πθ(x))3/2

in which (ρ, u, θ) denotes macroscopic density, velocity, and temperature of gases.

Collisional kinetic theory

• Boltzmann’s H-Theorem: There is more: equilibria iﬀ

0 0 0 0 ﬀ∗ = f f∗, v , v∗ ∈ Svv∗ and so iﬀ (presentation?):

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 19 / 20 Collisional kinetic theory

• Boltzmann’s H-Theorem: There is more: equilibria iﬀ

0 0 0 0 ﬀ∗ = f f∗, v , v∗ ∈ Svv∗ and so iﬀ (presentation?): Maxwellian distribution

2 2 ρ(x) − |v−u(x)| f (x, v) = Ce−β|v−u| = 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, 2017 19 / 20 • 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)

Collisional kinetic theory

• Hydrodynamics limits (if time permits).

Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 20 / 20 • 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)

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, 2017 20 / 20 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, 2017 20 / 20