kinetic theory of gases - personal.psu.edu
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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 !
1Homepage: http://math.psu.edu/nguyen2Math blog: https://sites.psu.edu/nguyen
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
Collisionless kinetic theory
VP: ∂t f + v · ∇x f −∇xφ · ∇v f = 0, −∆φ = ρ− 1.
For instance, homogenous equilibria
µ = µ(e), e :=1
2|v |2, φ0 = 0.
Arnold: Look at casimir’s invariant
A[f ] =
∫∫ [1
2|v |2f + ϕ(f )
]dvdx +
1
2
∫|∇φ|2 dx .
µ is a critical point iff ϕ′(µ) = −12 |v |
2 or ϕ′ = −µ−1. Convexity ofA[f ] requires
µe < 0.
µe < 0 implies Arnold’s nonlinear stability. But, what’s long-timedynamics?
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 11 / 20
Collisionless kinetic theory
VP: ∂t f + v · ∇x f −∇xφ · ∇v f = 0, −∆φ = ρ− 1.
For instance, homogenous equilibria
µ = µ(e), e :=1
2|v |2, φ0 = 0.
Arnold: Look at casimir’s invariant
A[f ] =
∫∫ [1
2|v |2f + ϕ(f )
]dvdx +
1
2
∫|∇φ|2 dx .
µ is a critical point iff ϕ′(µ) = −12 |v |
2 or ϕ′ = −µ−1. Convexity ofA[f ] requires
µe < 0.
µe < 0 implies Arnold’s nonlinear stability. But, what’s long-timedynamics?
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 11 / 20
Collisionless kinetic theory
VP: ∂t f + v · ∇x f −∇xφ · ∇v f = 0, −∆φ = ρ− 1.
For instance, homogenous equilibria
µ = µ(e), e :=1
2|v |2, φ0 = 0.
Arnold: Look at casimir’s invariant
A[f ] =
∫∫ [1
2|v |2f + ϕ(f )
]dvdx +
1
2
∫|∇φ|2 dx .
µ is a critical point iff ϕ′(µ) = −12 |v |
2 or ϕ′ = −µ−1. Convexity ofA[f ] requires
µe < 0.
µe < 0 implies Arnold’s nonlinear stability. But, what’s long-timedynamics?
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 11 / 20
Collisionless kinetic theory
VP: ∂t f + v · ∇x f −∇xφ · ∇v f = 0, −∆φ = ρ− 1.
For instance, homogenous equilibria
µ = µ(e), e :=1
2|v |2, φ0 = 0.
Arnold: Look at casimir’s invariant
A[f ] =
∫∫ [1
2|v |2f + ϕ(f )
]dvdx +
1
2
∫|∇φ|2 dx .
µ is a critical point iff ϕ′(µ) = −12 |v |
2 or ϕ′ = −µ−1. Convexity ofA[f ] requires
µe < 0.
µe < 0 implies Arnold’s nonlinear stability. But, what’s long-timedynamics?
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 11 / 20
Collisionless kinetic theory
v
µ(v)
vmin vvmin
µ(v)
Figure: Stable vs unstable equilibria. Beautiful Penrose’s iff criteria:
linearlyunstable iff vmin exists and
P.V.
∫R
µ′(v)dv
v − vmin> 0.
S2: Presentation on Penrose’s criteria?
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 12 / 20
Collisionless kinetic theory
v
µ(v)
vmin vvmin
µ(v)
Figure: Stable vs unstable equilibria. Beautiful Penrose’s iff criteria: linearlyunstable iff vmin exists and
P.V.
∫R
µ′(v)dv
v − vmin> 0.
S2: Presentation on Penrose’s criteria?
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 12 / 20
Collisionless kinetic theory
• Mouhot-Villani: nonlinear Landau damping of Penrose stablehomogenous 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, 2017 13 / 20
Collisionless kinetic theory
• Mouhot-Villani: nonlinear Landau damping of Penrose stablehomogenous 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, 2017 13 / 20
Collisionless kinetic theory
• Mouhot-Villani: nonlinear Landau damping of Penrose stablehomogenous 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, 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
Collisional kinetic theory
v ′∗
v vv ′
v∗ = 0 v ′∗v∗
v ′
v
Figure: Elastic collisions: momentum and energy are exchanged between particles,however conserved, after collision:
v + v∗ = v ′ + v ′∗
|v |2 + |v∗|2 = |v ′|2 + |v ′∗|2
v
v∗
v ′
v ′∗v+v∗2
Sphere Svv∗ :
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 15 / 20
Collisional kinetic theory
v ′∗
v vv ′
v∗ = 0 v ′∗v∗
v ′
v
Figure: Elastic collisions: momentum and energy are exchanged between particles,however conserved, after collision:
v + v∗ = v ′ + v ′∗
|v |2 + |v∗|2 = |v ′|2 + |v ′∗|2
v
v∗
v ′
v ′∗v+v∗2
Sphere Svv∗ :
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 15 / 20
Collisional kinetic theory
One has the Boltzmann equation:
∂t f − H, f = Q[f ]
in which collision operator Q[f ]:
Q[f ](x , v) =
∫∫∫R9
ω(v , v∗|v ′, v ′∗)︸ ︷︷ ︸Scattering
[f2(x , v ′, v ′∗)︸ ︷︷ ︸
Gain
− f2(x , v , v∗)︸ ︷︷ ︸Loss
]dv∗dv ′dv ′∗
with f2(x , v , v∗) the probability distribution of finding two particles withrespective momenta.• Molecular chaos assumption:
f2(x , v , v∗) = f (x , v)f (x , v∗)
asserting velocities are uncorrelated before the collision (unlikeHamiltonian, here it introduces a notion of time arrow!).
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 16 / 20
Collisional kinetic theory
One has the Boltzmann equation:
∂t f − H, f = Q[f ]
in which collision operator Q[f ]:
Q[f ](x , v) =
∫∫∫R9
ω(v , v∗|v ′, v ′∗)︸ ︷︷ ︸Scattering
[f2(x , v ′, v ′∗)︸ ︷︷ ︸
Gain
− f2(x , v , v∗)︸ ︷︷ ︸Loss
]dv∗dv ′dv ′∗
with f2(x , v , v∗) the probability distribution of finding two particles withrespective momenta.
• Molecular chaos assumption:
f2(x , v , v∗) = f (x , v)f (x , v∗)
asserting velocities are uncorrelated before the collision (unlikeHamiltonian, here it introduces a notion of time arrow!).
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 16 / 20
Collisional kinetic theory
One has the Boltzmann equation:
∂t f − H, f = Q[f ]
in which collision operator Q[f ]:
Q[f ](x , v) =
∫∫∫R9
ω(v , v∗|v ′, v ′∗)︸ ︷︷ ︸Scattering
[f2(x , v ′, v ′∗)︸ ︷︷ ︸
Gain
− f2(x , v , v∗)︸ ︷︷ ︸Loss
]dv∗dv ′dv ′∗
with f2(x , v , v∗) the probability distribution of finding two particles withrespective momenta.• Molecular chaos assumption:
f2(x , v , v∗) = f (x , v)f (x , v∗)
asserting velocities are uncorrelated before the collision (unlikeHamiltonian, here it introduces a notion of time arrow!).
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 16 / 20
Collisional kinetic theory
• Conservation laws: v ′, v ′∗ ∈ Svv∗ and hence
dv ′dv ′∗ = dσ(Svv∗) = |v − v∗|dσ(n), n ∈ S2.
• (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 =
∫R3
∫S2
|n · (v − v∗)|[f ′f ′∗ − ff∗
]dndv∗.
Is it solvable ? Di Perna-Lions ’89, Guo 2010. There remain manyoutstanding open problems.
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 17 / 20
Collisional kinetic theory
• Conservation laws: v ′, v ′∗ ∈ Svv∗ and hence
dv ′dv ′∗ = dσ(Svv∗) = |v − v∗|dσ(n), n ∈ S2.
• (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 =
∫R3
∫S2
|n · (v − v∗)|[f ′f ′∗ − ff∗
]dndv∗.
Is it solvable ? Di Perna-Lions ’89, Guo 2010. There remain manyoutstanding open problems.
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 17 / 20
Collisional kinetic theory
• Conservation laws: v ′, v ′∗ ∈ Svv∗ and hence
dv ′dv ′∗ = dσ(Svv∗) = |v − v∗|dσ(n), n ∈ S2.
• (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 =
∫R3
∫S2
|n · (v − v∗)|[f ′f ′∗ − ff∗
]dndv∗.
Is it solvable ? Di Perna-Lions ’89, Guo 2010. There remain manyoutstanding open problems.
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 17 / 20
Collisional kinetic theory
• Conservation laws: v ′, v ′∗ ∈ Svv∗ and hence
dv ′dv ′∗ = dσ(Svv∗) = |v − v∗|dσ(n), n ∈ S2.
• (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 =
∫R3
∫S2
|n · (v − v∗)|[f ′f ′∗ − ff∗
]dndv∗.
Is it solvable ? Di Perna-Lions ’89, Guo 2010. There remain manyoutstanding open problems.
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 17 / 20
Collisional kinetic theory
• Boltzmann’s H-Theorem: Look at the entropy:
H(t) =
∫∫R6
f log f dxdv .
• One has
d
dtH(t) =
∫R3
∫S2
|n · (v − v∗)|[f ′f ′∗ − ff∗
]log f dndxdvdv∗
=1
4
∫R3
∫S2
|n · (v − v∗)|[f ′f ′∗ − ff∗
]log
ff∗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, 2017 18 / 20
Collisional kinetic theory
• Boltzmann’s H-Theorem: Look at the entropy:
H(t) =
∫∫R6
f log f dxdv .
• One has
d
dtH(t) =
∫R3
∫S2
|n · (v − v∗)|[f ′f ′∗ − ff∗
]log f dndxdvdv∗
=1
4
∫R3
∫S2
|n · (v − v∗)|[f ′f ′∗ − ff∗
]log
ff∗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, 2017 18 / 20
Collisional kinetic theory
• Boltzmann’s H-Theorem: Look at the entropy:
H(t) =
∫∫R6
f log f dxdv .
• One has
d
dtH(t) =
∫R3
∫S2
|n · (v − v∗)|[f ′f ′∗ − ff∗
]log f dndxdvdv∗
=1
4
∫R3
∫S2
|n · (v − v∗)|[f ′f ′∗ − ff∗
]log
ff∗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, 2017 18 / 20
Collisional kinetic theory
• Boltzmann’s H-Theorem: There is more: equilibria iff
ff∗ = f ′f ′∗ , v ′, v ′∗ ∈ Svv∗
and so iff (presentation?):
Maxwellian distribution
f (x , v) = Ce−β|v−u|2
=ρ(x)
(2πθ(x))3/2e− |v−u(x)|2
2θ(x)
in which (ρ, u, θ) denotes macroscopic density, velocity, and temperatureof gases.
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
ff∗ = f ′f ′∗ , v ′, v ′∗ ∈ Svv∗
and so iff (presentation?): Maxwellian distribution
f (x , v) = Ce−β|v−u|2
=ρ(x)
(2πθ(x))3/2e− |v−u(x)|2
2θ(x)
in which (ρ, u, θ) denotes macroscopic density, velocity, and temperatureof gases.
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 19 / 20
Collisional kinetic theory
• Hydrodynamics limits (if time permits).
• Mathematical difficulty of the Cauchy problem and convergence toequilibrium.
• 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, andBose-Einstein condenstate (w/ Tran)
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 toequilibrium.
• 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, andBose-Einstein condenstate (w/ Tran)
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 toequilibrium.
• 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, andBose-Einstein condenstate (w/ Tran)
Toan T. Nguyen (Penn State) Kinetic theory of gases PSU, Jan 19th, 2017 20 / 20