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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Introduction to kinetic equations and theirapplications
Seung Yeal Ha
Department of Mathematical SciencesSeoul National University
Feb. 9th , 2015
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Outline
What is a kinetic equation ?
Why do we need a kinetic theory ?
When do we use kinetic equations ?
Hilbert’s 6th problem
The Boltzmann equation
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Lecture Schedule
• Lecture 1: Introduction, the Boltzmann equation• Lecture 2:
The Boltzmann equation• Lecture 3: The Vlasov equation,
applications• Lecture 4 and 5: Applications
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Goal of my lectures is two-fold:
1. To give a general idea and basics for kinetic equations
2. To prepare other invited talks for kinetic equations
• Yong-Kum Cho: Infinite energy solutions of the
Boltzmannequation: Recent developments
• Sunho Choi: Scattering and modified scattering of
theVlasov-Poisson system
• Ho Lee: Global properties of solutions to
theEinstein-Boltzmann system with Bianchi 1 symmetry
• Seok-Bae Yun: An introductio to BGK models
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Lecture 1
A crash introduction to kinetic equations
(The Boltzmann equation and the Vlasov equation)
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
If you hear of kinetic equation, you might ask the
followingquestions:
• QA 1: What is kinetic equation ?• QA 2: Why do we need kinetic
equations ?• QA 3: When do we use kinetic equations ?• QA 4: What
are the relations with other fluid equations ?• · · ·
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
What is kinetic equation ?
• Configuration and phase spaces
Consider a mechanical system consisting of N point particles ina
region of 3-space
We set
xi ,pi : position and momentum of i-th particle
• Configuration (physical) space K:
K := {(x1, · · · , xN)} ⊂ R3N .
• Phase space P:
P := {(x1, · · · , xN ,p1, · · · ,pN)} ⊂ R6N .
cf. N-particle phase(state) space
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• One-particle distribution function: (number density function,
velocitydistribution function)
(Probability) density function F = F (x , ξ, t) in
one-particlephase space R3x × R3ξ :
. x
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xxxx
xxxx
.(x, )ξ
ξ
ξ
x
x
F (x , ξ, t)∆x∆ξ ≈ number of particles inside ∆x∆ξ.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• Defintion: (Kinetic equation)
A kinetic equation is a partial differential equation
describingthe phase spatial-temporal evolution of the kinetic
density
F = F (x , ξ, t) on the one-particle phase space R6.
• Examples
1. The Boltzmann equation for a dilute gas
∂tF + ξ · ∇xF = Q(F ,F ).
2. The Vlasov equation for plasma and galaxy of stars.
∂tF + ξ · ∇xF + E(x , t) · ∇ξF = 0.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Why do we need a kinetic theory ?• A thermal creep flow
(Source: Molecular gas dynamics: Theory, Techniques and
Applications byYoshio Sone)
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• Simulation by Y. Sone group
Thermal-creep.aviMedia File (video/avi)
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Summary from thermal creep flow simulation
For a rarefied gas, we need a new theory not a
hydrodynamicequations
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• Hueristic explanation using a kinetic theory
(James Clerk Maxwell: 13 June 1831 - 5 November 1879,Scottish
Physicist)
On stresses in rarefied gases arising from inequalities of
temperature, Philos.Trans. R. Soc. 170 (1879), 231-256.
cf. Kinetic theory of gases by Kennard E. H. (1938)
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• A rigorous formulation using the Boltzmann equation
1. Y. Sone: Thermal creep in rarefied gas. J. Phys. Soc. Jpn.21
(1966), 1836-1837.
2. Y. Sone: A note on thermal creep in rarefied gas. J.
Phys.Soc. Jpn. 29 (1970), 1655.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• A rigorous analysis by mathematician
1. C.-C. Chen, I.-K. Chen, T.-P. Liu and Y. Sone:
Thermaltranspiration for the linearized Boltzmann equation. CPAM60
(2007), 147- 163.
2. Y.-J. Kim, M. Lee and M. Slemrod: Thermal creep of ararefied
gas on the basis of non-linear Korteweg-theory.ARMA 215 (2015),
353-379.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
When do we use kinetic equations ?• Different view of
scales:
Source: Cross-Scale Numerical Simulations using discrete
particle modelsby Vcislo R. et al, Molecular simulation (1999),
397-418
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Relations between physical models
• Quantum mechanics: ψ = ψ(x , t)
i~∂tψ =[− ~
2
2m∆x + V (x , t)
]ψ.
• Classical mechanics: (xi , vi)
dxidt
= vi , midvidt
= Fi .
• Kinetic theory:
∂tF + ξ · ∇xF = Q(F ,F ).
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• Continuum mechanics: (ρ,u,E):
∂tρ+∇x · (ρv) = 0,∂t (ρv)t +∇x · (ρv ⊗ v + pI) = 0,∂t (ρE) +∇x ·
(ρEv + Pv) = 0.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Hilbert’s 6th Problem
In 1900, David Hilbert presented 10 of his problems in
Parisconference of 2nd ICM (full list was published in
GottingerNachrichten, 1900, pp. 253-297, and Archiv der Mathematik
und Physik,3dser., vol. 1 (1901), pp. 44-63, and 1902 Bulletin of
AMS, 213-237 (1902))
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• Hilbert’s 6th problem
Mathematical treatment of the axioms of Physics
"To treat in the same manner, by means of axioms, thosephysical
sciences in which mathematics plays an importantpart; in the first
rank are the theory of probabilities andmechanics"
cf. A. Kolmogorov (Foundations of the Theory of Probability,
1933)
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• Hilbert’s sixth problem is to provide rigorous proof for
Newton’s equations =⇒ Boltzmann equation =⇒Hydrodynamic
equations.
cf. Oscar Lanford III (1975), F. Golse-L. Saint-Raymond
(2004)
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Introduction to the Boltzmann equation
Ludwig Eduard Boltzmann (1844-1906), S = k lnW .
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
The Boltzmann equation
• Velocity Distribution function:
F = F (x , ξ, t): velocity distribution function (number
densityfunction) of monatomic particles (e.g. Ar, He,· · · ).
. x
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xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxx
xxxx
xxxx
.(x, )ξ
ξ
ξ
x
x
F (x , ξ, t)∆x∆ξ ≈ number of particles inside ∆x∆ξ.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• The Boltzmann equation (1872)
In the absence of external forces,
∂tF + ξ · ∇xF︸ ︷︷ ︸Rate of change in F along particle
trajectory
=1
KnQ(F ,F ),
Q(F ,F )(x , ξ, t) =∫
R3×S2q(ξ − ξ∗, ω)(F ′F ′∗ − FF∗)dωdξ∗.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
The Kudsen number Kn
• Excluded volume, perfect gas
For a monatomic gas at room temperature and atmosphericpressure,
about 1020 gas molecules with a radius 10−8 cm areto be found in
any volume of 1cm3.
Excluded volume: Total volume occupied by a gas particles ifthey
are tightly packed
1020 × 4π3× (10−8)3 ≈ 5× 10−4cm3 � 1cm3
This is called an ideal (perfect) gas condition.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• Perfect gas (ideal gas)
p = ρRθ,
where R : gas constant.
• Mean free path (1858, Rudolf Clausius)
Roughly speaking, the average distance between twosuccessive
collisions for any given molecule in the gas.
` ≈ 1N × A
,
where N is the number of particles and A is the
cross-sectionalarea of each particle.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• Fluid (hydrodynamic) domain
For the same monatomic gas as before (at room temperatureand
atmospheric pressure),
N = 1020 molecules/cm3, A = π × (10−8)2 ≈ 3× 10−16cm2.
` ≈ 13× 10−4cm� 1cm.
• Kinetic domain
While keeping the same temperature, lower the pressure at10−4
atm.
N = 1016molecules/cm3 and ` ≈ 13
cm,
which is comparable to the size of the 1cm3
container(characteristic length of the physical domain).
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• Defintion: (Knudsen number Kn)
Martin Hans Christian Knudsen, Danish physicist 1(871 Ð
1949)
• Kundsen number: (measures Degree of rarefaction)
Kn =`
L.
where L is the macroscopic length.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Fluid and Kinetic regimes
• Fluid regimes are characterized by Kn� 1 ; the gas is inlocal
thermodynamic equilibrium: its state can be described by
ρ = ρ(x , t) : density u = u(x , t) : bulk velocityθ = θ(x , t)
: temparature
• Kinetic regimes are characterized by Kn = O(1) ; since thegas
is more rarefied, there are not enough collisions per unit oftime
for a local thermodynamic equilibrium to be reached, andthe state
of the gas is adequately described by
F = F (x , ξ, t) : one-particle distribution function.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• Macroscopic observables
Macroscopic quantities (observables) are computed byaveraging
the corresponding quantity for a single particle withrespect to the
measure F (x , ξ, t)dxdξ:
∫1Fdxdξ : total mass,∫ξFdxdξ : total momentum,∫|ξ|2
2Fdxdξ : total energy.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
• Macroscopic densities:
ρ(x , t) :=∫
1Fdξ : local mass density,
(ρu)(x , t) :=∫ξFdξ : local momentum density,
ρE = ρ( |u|2
2+ e)
=
∫|ξ|2
2Fdξ : local energy density.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
References
1. C. Cercignani, R. Illner and M. Pulvirenti: Themathematical
theory of dilute gases. Applied Math.Sciences 106,
Springer-Verlag.
2. R. T. Glassey: The Cauchy problem in kinetic theory.SIAM.
3. Harold Grad: On the kinetic theory of rarefied gases.4.
Yoshio Sone: Molecular gas dynamics: Theory,
Techniques and Applications.5. S. Ukai and T. Yang: Mathematical
theory of Boltzmann
equation. Lecture notes series No. 8. Liu Bie Ju Center forMath.
Sci. 2006.
6. C. Villani: A review of mathematical topics in
collisionalkinetic theory.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Summary of Lecture 1
Today, we have studied• There are three scales in a classical
regime:
Micro-scale =⇒ Meso-scale =⇒ Macro-scale.
• There exist a phenomenon (thermal creep flow) thatcannot be
described by the classical fluid equations
• Kinetic equation is a PDE for a kinetic density
(one-particledistribution function)
• The Boltzmann equation describes a state of dilute gases.
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WHAT IS A KINETIC EQUATION ? WHY DO WE NEED A KINETIC THEORY ?
WHEN DO WE USE KINETIC EQUATIONS ? HILBERT’S 6TH PROBLEM THE
BOLTZMANN EQUATION
Tomorrow, I will continue the study of basic properties of
theBoltzmann collision operator, conservation laws, symmetryrelated
to the Boltzmann equation.
What is a kinetic equation ?Why do we need a kinetic theory
?When do we use kinetic equations ?Hilbert's 6th problemThe
Boltzmann equation