Kinetic theory of gases Toan T. Nguyen 1, 2 Penn State University Graduate Student seminar, PSU Jan 19th, 2017 Fall 2017, I teach a graduate topics course: same topics ! 1 Homepage: http://math.psu.edu/nguyen 2 Math blog: https://sites.psu.edu/nguyen Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 1 / 20
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Kinetic theory of gases
Toan T. Nguyen1,2
Penn State University
Graduate Student seminar, PSUJan 19th, 2017
Fall 2017, I teach a graduate topics course: same topics !
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 1 / 20
Figure: 4 states of matter, plus Bose-Einstein condensate!
• James Clerk Maxwell, in 1859, gave birth to kinetic theory: use thestatistical approach to describe the dynamics of a (rarefied) gas.
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 thestatistical approach to describe the dynamics of a (rarefied) gas.
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 2 / 20
(x , p)
Figure: Kinetic theory of gases: phase space!
Gas molecules are identical.
One-particle phase space: position x ∈ Ω ⊂ Rd , momentum p ∈ Rd
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
(x , p)
Figure: Kinetic theory of gases: phase space!
Gas molecules are identical.
One-particle phase space: position x ∈ Ω ⊂ Rd , momentum p ∈ Rd
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
(x , p)
Figure: Kinetic theory of gases: phase space!
Gas molecules are identical.
One-particle phase space: position x ∈ Ω ⊂ Rd , momentum p ∈ Rd
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
Collisionless kinetic theory
Hamilton 1830’s classical mechanics3: one-particle Hamiltonian H(x , p)(induced from Lagrangian):
x = ∇pH, p = −∇xH
Examples:
Free particles:
H(x , p) =1
2m|p|2
Particles in a potential:
H(x , p) =1
2m|p|2 + Vpot(x)
3In truth, it should be an N-particle Hamiltonian dynamics, then mean field limit orBBGKY 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
Collisionless kinetic theory
Hamilton 1830’s classical mechanics3: one-particle Hamiltonian H(x , p)(induced from Lagrangian):
x = ∇pH, p = −∇xH
Examples:
Free particles:
H(x , p) =1
2m|p|2
Particles in a potential:
H(x , p) =1
2m|p|2 + Vpot(x)
3In truth, it should be an N-particle Hamiltonian dynamics, then mean field limit orBBGKY 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
Collisionless kinetic theory
Hamilton 1830’s classical mechanics3: one-particle Hamiltonian H(x , p)(induced from Lagrangian):
x = ∇pH, p = −∇xH
Examples:
Free particles:
H(x , p) =1
2m|p|2
Particles in a potential:
H(x , p) =1
2m|p|2 + Vpot(x)
3In truth, it should be an N-particle Hamiltonian dynamics, then mean field limit orBBGKY 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
Collisionless kinetic theory
• 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 = −∇xH = E +q
m(p × B)
which is the Lorentz force, with E = −∇xφ and B = ∇x × 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
H(x , p) =1
2m|p − qA(x)|2 + φ(x)
with electric and magnetic potentials φ,A.
• Force acting on particles:
F = −∇xH = E +q
m(p × B)
which is the Lorentz force, with E = −∇xφ and B = ∇x × A.
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 6 / 20
Collisionless kinetic theory
(x0, p0)
(xt , pt)t > 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
dtf (t, x(t), p(t)) = ∂t f +∇pH · ∇x f −∇xH · ∇pf
= ∂t f + v · ∇x f + Fforce · ∇v f
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 7 / 20
Collisionless kinetic theory
(x0, p0)
(xt , pt)t > 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
dtf (t, x(t), p(t)) = ∂t f +∇pH · ∇x f −∇xH · ∇pf
= ∂t f + v · ∇x f + Fforce · ∇v f
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 7 / 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:
F = −∇xφ, −∆xφ = σ
∫R3
f (t, x , v) dv ,
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 theelectromagnetic fields E ,B solving Maxwell (generated by charge andcurrent density).
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:
F = −∇xφ, −∆xφ = σ
∫R3
f (t, x , v) dv ,
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 theelectromagnetic fields E ,B solving Maxwell (generated by charge andcurrent density).
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:
F = −∇xφ, −∆xφ = σ
∫R3
f (t, x , v) dv ,
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 theelectromagnetic fields E ,B solving Maxwell (generated by charge andcurrent density).
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:
F = −∇xφ, −∆xφ = σ
∫R3
f (t, x , v) dv ,
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 theelectromagnetic fields E ,B solving Maxwell (generated by charge andcurrent density).
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 8 / 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).
• One needs Lipschitz bounds on F (t, x , v).
• Vlasov-Poisson: F = −∇xφ and −∆φ = ±∫
f (t, x , v) dv . Roughlyspeaking: ∇xF = ∇2
x(−∆x)−1ρ ≈ 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, 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).
• One needs Lipschitz bounds on F (t, x , v).
• Vlasov-Poisson: F = −∇xφ and −∆φ = ±∫
f (t, x , v) dv . Roughlyspeaking: ∇xF = ∇2
x(−∆x)−1ρ ≈ 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, 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).
• One needs Lipschitz bounds on F (t, x , v).
• Vlasov-Poisson: F = −∇xφ and −∆φ = ±∫
f (t, x , v) dv . Roughlyspeaking: ∇xF = ∇2
x(−∆x)−1ρ ≈ 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, 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).
• One needs Lipschitz bounds on F (t, x , v).
• Vlasov-Poisson: F = −∇xφ and −∆φ = ±∫
f (t, x , v) dv . Roughlyspeaking: ∇xF = ∇2
x(−∆x)−1ρ ≈ 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, 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).
• One needs Lipschitz bounds on F (t, x , v).
• Vlasov-Poisson: F = −∇xφ and −∆φ = ±∫
f (t, x , v) dv . Roughlyspeaking: ∇xF = ∇2
x(−∆x)−1ρ ≈ 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, 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).
• One needs Lipschitz bounds on F (t, x , v).
• Vlasov-Poisson: F = −∇xφ and −∆φ = ±∫
f (t, x , v) dv . Roughlyspeaking: ∇xF = ∇2
x(−∆x)−1ρ ≈ 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, 2017 9 / 20
Collisionless kinetic theory
Several mathematical issues: ∂t f = H, f (focus on previous examples).
Infinitely many conserved quantities, and infinitely many equilibria:
D
dtϕ(f ) = 0, f∗ = ϕ(H(x , p))
for arbitrary ϕ(·). Which one is dynamically stable ? and shouldn’t itbe just Gaussian ? Maxwell-Boltzmann distribution (next section) ?
Hence, large time dynamics and stability of equilibria are verydelicate!
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 10 / 20
Collisionless kinetic theory
Several mathematical issues: ∂t f = H, f (focus on previous examples).
Infinitely many conserved quantities, and infinitely many equilibria:
D
dtϕ(f ) = 0, f∗ = ϕ(H(x , p))
for arbitrary ϕ(·). Which one is dynamically stable ? and shouldn’t itbe just Gaussian ? Maxwell-Boltzmann distribution (next section) ?
Hence, large time dynamics and stability of equilibria are verydelicate!
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 10 / 20
Collisionless kinetic theory
Several mathematical issues: ∂t f = H, f (focus on previous examples).
Infinitely many conserved quantities, and infinitely many equilibria:
D
dtϕ(f ) = 0, f∗ = ϕ(H(x , p))
for arbitrary ϕ(·). Which one is dynamically stable ? and shouldn’t itbe just Gaussian ? Maxwell-Boltzmann distribution (next section) ?
Hence, large time dynamics and stability of equilibria are verydelicate!
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 10 / 20