Weinan E. Principles of Multiscale Modeling

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4.3. KINETIC THEORY 177

have

∂

∂t

|x|

ρ(x, t)dx =D

|x|

ρ(x, t)dx (4.2.52)

∇ · (|x|

∇ρ)dx − D

(∇|x|

) · ∇ρdx (4.2.53)

= − 2D

x · ∇ρdx (4.2.54)

=2D

(∇ · x)ρdx (4.2.55)

=6D (4.2.56)

in three dimension.

So far the discussion is limited to the macroscopic model (4.2.51). Now consider the

dynamics of a diﬀusing particle. Using y(t) =

v(t

)dt

, we have

h|y|

(t)i = h

v(t

)v(t

)dt

i (4.2.57)

hv(t

− t

)v(0)idt

(4.2.58)

h|y|

(t)i →

∞

hv(t)v(0)idt (4.2.59)

as t → ∞. Hence we have:

D =

∞

hv(t)v(0)idt (4.2.60)

This equation relates a transport coeﬃcient D to the integral of an autocorrelation

function for the microscopic system, here the velocity autocorrelation function g(t) =

hv(t)v(0)i of the diﬀusing particle. The function g(·) can be calculated using microscopic

simulations such as molecular dynamics. Similar expressions can be found for viscosity,

heat conductivity, etc. Such relationships are known as the Green-Kubo formulas [9].

4.3 Kinetic theory

Kinetic theory serves as a nice bridge between continuum and atomistic models. It can

either be viewed as an approximation to the hierarchy of equations for the many-particle

probability densities obtained from molecular dynamics, or as a microscopic model for

178 CHAPTER 4. THE HIERARCHY OF PHYSICAL MODELS

the phase-space probability distribution of a particle, from which continuum models can

be derived.

The basic object of interest in kinetic theory is the one-particle distribution function.

In the classical kinetic theory of gases, a particle is described by its coordinates in the

phase space, i.e. position and momentum. Therefore classical kinetic theory describes

the behavior of the one-particle phase space distribution function [8]. Kinetic theory

has been extended in many diﬀerent directions. One interesting extension is the kinetic

models of complex ﬂuids, such as liquid crystals made u p of rod-like molecules [13], in

which additional variables are introduced to describe the conformation of the molecules.

For example, for rod-like molecules, it is important to include the orientation of the

molecules as part of the description.

The main diﬃculty in formulating models of kinetic theory is in the mod eling of

particle-particle interaction. Indeed for this reason, kinetic th eory is mostly used when

particle-particle interaction is weak, e.g. for gases and dilute plasmas. Another case

where kinetic theory is used is when particle-particle interaction can be modeled accu-

rately by a mean ﬁeld approximation.

In this section, we will brieﬂy summarize the main features of kinetic theory for simple

systems, i.e. p articles without internal degrees of freedom.

4.3.1 The BBGKY hierarchy

At a fundamental level, kinetic equation can be viewed as an approximation to the

hierarchy of equations for the many-particle distribution functions for interacting par-

ticle systems. This hierarchy of equations is called the BBGKY hierarchy, named after

Bogoliubov, Born, Green, Kirkwood, and Yvon [36].

We will only consider a system of N identical particles interacting with a two-body

potential. In this case, the Hamiltonian is given by:

H =

j6=i

(|x

− x

|). (4.3.1)

Let z

= (p

, x

), and let ρ

, . . . , z

, t) be the probability density function of this

interacting particle system in the R

dimensional phase space at time t. Liouville’s

equation (4.2.16) can be written as [36]

∂ρ

∂t

+ H

= 0 (4.3.2)

4.3. KINETIC THEORY 179

where

i=1

· ∇

) −

i<j

. (4.3.3)

and



−(f

· ∇

) − (f

· ∇

)



, f

= −∇

(|x

− x

|) (4.3.4)

Deﬁne the 1-particle probability density as

, t) =

···

, . . . , z

, t)dz

···dz

. (4.3.5)

Similarly, we deﬁne the k-particle probability density:

, . . . , z

, t) =

···

, . . . , z

, t)dz

k+1

···dz

. (4.3.6)

Since the particles are identical, it makes sense to limit our attention to the case when

the distribution functions {ρ

}, k = 1, ··· , N, are symmetric functions. Let

i=1

· ∇

) −

i<j≤k

. (4.3.7)

Integrating over the variables z

k+1

, . . . , z

, we get from (4.3.2)

∂ρ

∂t

+ H

···

k+1

···dz

−

i=k+1

· ∇

) (4.3.8)

k+1≤i<j

i≤k,j≥k+1

, . . . , z

, t).

Assuming that ρ

decays to 0 suﬃciently fast as kz

k goes to ∞, both the ﬁrst and

second terms on the right hand side vanish after the integration. For the third term,

using the fact that the ρ

’s are symmetric functions, we have for each j ≥ k + 1,

···

k+1

···dz

, . . . , z

, t) (4.3.9)

···

k+1

···dz

i,k+1

, . . . , z

, t)

k+1

i,k+1

k+1

, . . . , z

k+1

)

180 CHAPTER 4. THE HIERARCHY OF PHYSICAL MODELS

Therefore, we have

∂ρ

∂t

+ H

=(N − k)

i≤k

k+1

i,k+1

k+1

, . . . , z

k+1

In particular, the equation for ρ

reads:

∂ρ

∂t

· ∇

)ρ

= (N − 1)

, z

, t) (4.3.10)

The equations just derived form a hierarchy of equations for all the k-particle dis-

tribution functions. Note that the equation of ρ

depends on ρ

, and the equation of

depends on ρ

. . . , and so on. This is a familiar situation that we see repeatedly in

modeling: To obtain a closed system with a small number of equations and unknowns, we

must place some assumptions on this hierarchy of distribution functions. In particular,

to obtain a closed equation for ρ

, we must make closure assumptions that express ρ

terms of ρ

only. The resulting equation for ρ

is called the kinetic equation.

4.3.2 The Boltzmann equation

Consider now a dilute system of particles in the gas phase. We will model the particle-

particle interaction by collision, i.e. particles do not interact unless two gas particles

collide, and we will model the collision process probablistically. This allows us t o derive

a closed model in terms of ρ

only. In d oing so, we have neglected the details of the

interaction process. We will present an intuitive way of deriving the kinetic equation,

the Boltzmann equation, by studying directly the collision process. Our presentation is

to a large extend inﬂuenced by the treatment in [29].

From this point on, we will write ρ

, t) = f (x, v, t), where x and v are the position

and velocity coordinates respectively. We will consider elastic collisions only, i.e. the

total energy of the particles is conserved during the collision process.

Let P

and P

be two particles with velocities v

, v

before collision and v

′

, v

′

after

collision. Since their total momentum and energy are conserved during the collision

process, we must have

+ v

= v

′

+ v

′

, (4.3.11)

+ |v

= |v

′

+ |v

′

. (4.3.12)

4.3. KINETIC THEORY 181

Deﬁne

v =

+ v

), v

′

+ v

′

), (4.3.13)

u = v

− v

, u

′

= v

′

− v

′

the conservation laws (4.3.11) and (4.3.12) imply

v = v

′

, (4.3.14)

|u| = |u

′

The second identity states that u

′

can be obtained by rotating u by some solid angle ω:

′

= R

u, (4.3.15)

Here ω ∈ S

(the unit sphere in R

) is called the scattering angle, R

is the rotation

operator by the angle ω. Knowing ω, we can relate (v

′

, v

′

) uniquely with (v

, v

′

+ u

′

) =

(v + R

u) (4.3.16)

′

− u

′

) =

(v − R

and vice versa.

The outcome of the collision process is determined uniquely by the scattering angle

ω. In kinetic theory, ω is consid ered to be a random variable, whose (unnormalized)

probability density σ(ω) is called the scattering cross section. The total scattering cross

section:

tot

σ(ω)dω (4.3.17)

which has the dimension of area, can be roughly thought of as being the area (say, normal

to v

− v

) available for the particles to collide. Outside of this area, particles pass by

each other without collision. One should note that σ(ω) depends on v

and v

. In

principle, σ(ω) = σ((v

, v

) → (v

′

, v

′

)) can be obtained from a more detailed model

for the collision process, such as a quantum mechanics model. In practice, it is often

modeled empirically.

However, independent of the details of σ(ω), it is intuitively quite clear that the

following properties hold for the collision process for simple spherical particles:

182 CHAPTER 4. THE HIERARCHY OF PHYSICAL MODELS

1. σ(ω) is invariant under rotation and reﬂection transformation of the velocity (or

momentum) space.

2. σ(ω) is reversible, i.e.

σ((v

, v

) → (v

′

, v

′

)) = σ((v

′

, v

′

) → (v

, v

)). (4.3.18)

The rate of change of f is given by:

∂f

∂t

+ (v · ∇

)f = J

− J

−

(4.3.19)

where J

−

is the loss term due to the collision of the particles having velocity v with

other particles, and J

is a gain term due to the creation of particles with velocity v as

a result of the collision process. To derive expressions for J

and J

−

, let us ﬁx v and the

center of collision x. Consider the collision of particles with velocity v and particles with

velocity w. Their relative velocity is v −w. The curr ent of such colliding particles (that

pass through the plane normal to v − w at the center of collision x) with velocity in a

neighborhood of w with size dw is equal to I = |v − w|f(x, w, t)dw. The rate at which

particles scatter with a solid angle ω after the collision process is Iσ(ω). Therefore the

loss term is equal to

−

= f(x, v, t)

tot

|v − w|f(x, w , t)dw (4.3.20)

f(x, v, t)f(x, w , t)σ(ω)|v − w|dωdw

= f(x, v, t)

σ(ω)dω

f(x, w, t)|v − w |dw

Similarly, to compute the gain term, we consider collision processes of the type:

′

, w

′

) → (v, w) (4.3.21)

Since v is ﬁxed, we can parametrize such collision processes by w and ω. Therefore we

have

f(x, v

′

, t)f(x, w

′

, t)|v

′

− w

′

|σ((v

′

, w

′

) → (v, w))dωdw (4.3.22)

f(x, v

′

, t)f(x, w

′

, t)|v

′

− w

′

|dωdw

f(x, v

′

, t)f(x, w

′

, t)σ(ω)|v − w|dωdw

4.3. KINETIC THEORY 183

Putting these together, we obtain the Boltzmann equation

∂f

∂t

+ (v · ∇

)f = B(f, f), (4.3.23)

where

B(f, f) =

σ(ω)|w − v|(f(v

′

)f(w

′

) − f (v)f(w)) dωdw. (4.3.24)

Here we have used the short-hand notation: f(v) = f(x, v, t). A more detailed derivation

can be found in [29].

4.3.3 The equilibrium states

Next we consider homogeneous equilibrium states such that f(x, v, t) = f

(v). From

(4.3.23) we see that we must have

B(f

, f

) ≡ 0. (4.3.25)

(4.3.25) is an integral constraint, stating that the net gain is equal to the net loss, summed

over all collisions. The following lemma gives a much stronger statement: The gain and

loss terms are equal for each individual pairwise collisions.

Lemma 6. A ssume that σ(ω) > 0, for all ω ∈ S

. Then for any equilibrium states f

the following holds:

′

) = f

(v)f

(w). (4.3.26)

where v

′

, w

′

and v, w are related by the conservation laws of momentum and energy, i.e.

′

, w

′

) is a possible outcome after the collision of a pair of particles with velocities v and

w respectively.

Proof. Let

H(f) =

f ln fdxdv, (4.3.27)

then

(ln f + 1)

∂f

∂t

dxdv (4.3.28)

(ln f + 1)B(f, f)dxdv

ZZZ

σ(ω)|v − w|(f

′

− f

)(1 + ln f

)dωdwdxdv.

184 CHAPTER 4. THE HIERARCHY OF PHYSICAL MODELS

Here and in what follows, we use the short-hand notation: f

= f

(v), f

= f

(w), f

′

), f

′

= f

′

). Now we will make use of the symmetry properties of our system.

First, if we exchange v and w, the result should not change, i.e.

ZZZ

σ(ω)|v − w|(f

′

− f

)(1 + ln f

)dωdwdxdv. (4.3.29)

Hence, we have

ZZZ

σ(ω)|v − w|(f

′

− f

)(2 + ln(f

))dωdwdxdv. (4.3.30)

Secondly, because of (4.3.18), if we exchange the pairs (v, w) and (v

′

, w

′

), the result

should not change either:

ZZZ

σ(ω)|v − w|(f

− f

′

)(2 + ln(f

′

))dωdwdxdv. (4.3.31)

Hence,

ZZZ

σ(ω)|v − w|(f

′

− f

) (ln(f

) − ln(f

′

)) dωdwdxdv. (4.3.32)

Using the fact that for any a, b > 0,

(b − a)(ln b − ln a) ≥ 0, (4.3.33)

we conclude that

≤ 0. (4.3.34)

In addition, if

≡ 0, we must have

′

) = f

(v)f

(w) (4.3.35)

for all ω, v and w, since the equality in (4.3.33) holds only when a = b.

(4.3.26) is referred to as the relation of detailed balance.

Equally important is the statement in (4.3.34) which holds for any solutions of the

Boltzmann equation. This statement is referred to as Boltzmann’s H-theorem. One

consequence of the H-theorem is the irreversibility of the Boltzmann equation. This is an

intriguing statement, since the collision p rocess, up on which Boltzmann’s equation was

derived, is reversible.

4.3. KINETIC THEORY 185

Using the detailed balance condition, we can derive the general form of the equi-

librium states f

(v). For this purpose, let us examine further the quantities that are

conserved during the collision process. We have known from (4.3.11) and (4.3.12) that

1, v, |v|

/2 are conserved during collision. We can show that if the collision cross section

is non-degenerate, i.e. if σ(ω) is strictly positive, then there are no additional conserved

quantities. In fact, assume to the contrary that there is such an conserved quantity, say

η, then

η(v) + η(w) = η(v

′

) + η(w

′

) (4.3.36)

for all v, w, ω. Fix (v, w), then v

′

, w

′

are uniquely determined by ω. An additional

conserved quantity also means an additional constraint on the possible values of ω, i.e. ω

can only take values on a low-dimensional subset of S

. This contradicts with the strict

positivity of σ, which guarantees that ω can take any value on S

We now write the relation of detailed balance as

ln f

′

+ ln f

′

= ln f

+ ln f

. (4.3.37)

This means that ln f is also a conserved quantity. Thus, ln f must be a linear combination

of 1, v,

|v|

, namely

f = exp



+ C

· v + C

|v|



, (4.3.38)

or equivalently

f = A exp(−C

(v − v

)

). (4.3.39)

To determine the constants A and C

, deﬁne the number density

n ≡

fdv = A exp



−C

(v − v

)



= A





3/2

, (4.3.40)

then

vfdv = nv

. (4.3.41)

In addition, it is easy to compute the internal energy of the system:

|v − v

fdv =

3ρ

. (4.3.42)

186 CHAPTER 4. THE HIERARCHY OF PHYSICAL MODELS

where m is the mass of the particles and ρ = nm. Deﬁne temperature as the thermal

energy per degree of freedom, i.e.

|v − v

fdv

T, (4.3.43)

then we obtain

. (4.3.44)

Together with (4.3.40), we obtain the well-known Maxwell-Boltzmann distribution

(v) = n



2πk



3/2

exp



−

m|v − v



. (4.3.45)

4.3.4 Macroscopic conservation laws

We are interested in conserved quantities of the form

I(x, t) =

η(v)f(t, x, v)dv. (4.3.46)

Using the Boltzmann equation (4.3.23), we have

η(v)



∂f

∂t

+ v · ∇



dv =

η(v)B(f, f )dv. (4.3.47)

If η satisﬁes

η(v)B(f, f )dv = 0, (4.3.48)

then I must be a conserved quantity. In addition, let J(x, t) =

η(v)vfdv, then we

obtain a conservation law in local form:

∂I

∂t

+ ∇

· J = 0. (4.3.49)

J is the corresponding ﬂux. In the following, we will write ∇

= ∇.

We have already seen that η(v) = 1, v,

|v|

satisfy the condition (4.3.48). They

give rise to the conservation laws of mass, momentum and energy respectively. We now

examine these conservation laws in some detail.

Deﬁne the averaging operator:

hηi =

ηfdv

fdv

(4.3.50)