introduction to kinetic equations and their applicationsamath.kaist.ac.kr/pdeschool2015/ha1.pdf ·...

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


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


Lecture Schedule

• Lecture 1: Introduction, the Boltzmann equation• Lecture 2: The Boltzmann equation• Lecture 3: The Vlasov equation, applications• Lecture 4 and 5: Applications


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


Lecture 1

A crash introduction to kinetic equations

(The Boltzmann equation and the Vlasov 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 ?• · · ·


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


• 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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xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxx

xxxx

xxxx

.(x, )ξ

ξ

ξ

x

x

F (x , ξ, t)∆x∆ξ ≈ number of particles inside ∆x∆ξ.


• 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.


Why do we need a kinetic theory ?• A thermal creep flow

(Source: Molecular gas dynamics: Theory, Techniques and Applications byYoshio Sone)


• Simulation by Y. Sone group

Thermal-creep.aviMedia File (video/avi)


Summary from thermal creep flow simulation

For a rarefied gas, we need a new theory not a hydrodynamicequations


• 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)


• 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.


• 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.


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


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 ).


• Continuum mechanics: (ρ,u,E):

∂tρ+∇x · (ρv) = 0,∂t (ρv)t +∇x · (ρv ⊗ v + pI) = 0,∂t (ρE) +∇x · (ρEv + Pv) = 0.


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))


• 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)


• 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)


Introduction to the Boltzmann equation

Ludwig Eduard Boltzmann (1844-1906), S = k lnW .


The Boltzmann equation

• Velocity Distribution function:

F = F (x , ξ, t): velocity distribution function (number densityfunction) of monatomic particles (e.g. Ar, He,· · · ).

. x

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxx

xxxx

xxxx

.(x, )ξ

ξ

ξ

x

x

F (x , ξ, t)∆x∆ξ ≈ number of particles inside ∆x∆ξ.


• 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∗.


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.


• 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.


• 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).


• 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.


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.


• 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.


• 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.


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.


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.


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

introduction to kinetic equations and their applicationsamath.kaist.ac.kr/pdeschool2015/ha1.pdf ·...

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