Weinan E. Principles of Multiscale Modeling

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4.3. KINETIC THEORY 187

First we let η(v) = 1. Then I = n = ρ/m, J = nu where u = hvi is the mean velocity.

Mass conservation becomes:

∂ρ

∂t

+ ∇ · (ρu) = 0. (4.3.51)

For what follows, it is convenient to split the velocity as the sum of its mean and

ﬂuctuation:

v = u + v

′

, (4.3.52)

where v

′

has mean 0. Let η(v) = v, then

I =

vfdv = nu =

ρu, (4.3.53)

and the momentum ﬂux becomes

J =

v ⊗ vfdv =

(u + v

′

) ⊗ (u + v

′

)fdv

′

(4.3.54)

= nu ⊗ u +

′

⊗ v

′

fdv

′

Let us deﬁne the tensor τ as

τ =

′

⊗ v

′

fdv

′

, (4.3.55)

then conservation of momentum reads

∂(ρu)

∂t

+ ∇(ρu ⊗u) + ∇ · τ = 0. (4.3.56)

Finally, let η(v) =

|v|

, then

I =

|v|

fdv =

|u + v

′

fdv

′

(4.3.57)

|u|

n +

′

fdv

′

Now deﬁne the internal energy density and temperature using the following relation

e =

T = h

′

i, (4.3.58)

we have

I =



|u|

+ ρe



E (4.3.59)

188 CHAPTER 4. THE HIERARCHY OF PHYSICAL MODELS

The energy ﬂux is

J =

|v|

vfdv (4.3.60)



|u|

u + τ · u + eρu



′

fdv

′

Let

q =

′

fdv

′

, (4.3.61)

we have

∂

E + ∇ · (Eu + τ · u) + ∇ · q = 0. (4.3.62)

4.3.5 The hydrodynamic regime

The Knudsen number

The most important p arameter in kinetic theory is the Knudsen number, deﬁned as

the ratio between the mean free path of the particles λ and the system size L:

Kn =

Here by “the system size”, we mean the scale over which averaged quantities such as

density, momentum and energy change. In typical dense gas, λ is of the order 10

−7

−

−8

m, while the system size is, say O(1m). Hence the Knudsen number is very small. As

we will see below, in th is case, hydrodynamic approximations are very accurate. However,

there are exceptions, for example:

1. at shocks, the relevant system size becomes the width of the shock, which can be

very small;

2. at the boundary, the relevant system size becomes the boundary layer width;

3. when modeling high frequency waves, in which case the time scale of interest is

speciﬁed by the frequency of the waves and the Knudsen numb er is the ratio between

the frequency of the waves and the collision frequency;

4. when the gas is dilute, then the mean free path becomes large.

4.3. KINETIC THEORY 189

In these cases, the full Boltzmann equation is needed to model the dynamics of the gas.

If we nondimensionalize the Boltzmann equation, using x = Lx

′

, f = f

′

/λ

, v =

′

, w = vw

′

, t = t

′

L/v, σ = λ

′

and omitting the primes, we obtain

∂f

∂t

+ v · ∇

f =

B(f, f), (4.3.63)

where ε = Kn is the Knudsen number.

The local equilibrium approximation

We now turn to the asymptotic analysis of (4.3.63). When ε ≪ 1, we have, to leading

order:

B(f, f) = 0

Therefore, as discussed above, f must be of the form:

f(x, v, t) =

ρ(x, t)

(2πT (x, t))

3/2

exp



−

m|v − u(x, t)|

T (x, t)



. (4.3.64)

This is called the local equilibrium approximation.

It is straightforward to calculate that under the local equilibrium approximation, we

have

τ = pI, q = 0

where

p = ρk

Together with the conservation laws, this gives the Euler’s equation for ideal gas:

∂

ρ + ∇ · (ρu) = 0 (4.3.65)

∂

(ρu) + ∇(ρu ⊗ u) + ∇p = 0 (4.3.66)

∂

E + ∇ · ((E + p)u) = 0 (4.3.67)

The Chapman-Enskog expansion

The leading order result does not in clude any dissipative eﬀects such as the viscous

eﬀects or thermal conduction. To see these, we have to go to the next order. The stand ard

way of doing this is the Chapman-Enskog expansion. Here we give a brief summary of

this procedure. More details can be found, for example, in [29].

190 CHAPTER 4. THE HIERARCHY OF PHYSICAL MODELS

We look for approximate solutions of the Boltzmann equation in the form:

f = f

(1 + εϕ

+ ε

+ ···). (4.3.68)

To leading order, we have

B(f

, f

) = 0, (4.3.69)

∂f

∂t

+ v · ∇f

= B(f

, f

) + B(f

, f

) ≡ −

J(ϕ

). (4.3.70)

The last equality also serves as the deﬁnition of the operator J. Equation (4.3.69) shows

that f

actually follows the Maxwell-Boltzmann distribution. For simplicity, we write

ϕ = ϕ

. The left hand side of (4.3.70) can be calculated using the local equilibrium

approximation and the Euler’s equation for ideal gas, and this gives:

J(ϕ) = −

ρk

ρT



|v − u|

−



(v − u) · ∇T (4.3.71)

−

ρk



(v − u) ⊗ (v − u) −

|v − u|



: D, (4.3.72)

where D ≡



∇u + (∇u)



is the strain rate.

Using linearity, we can write ϕ as a linear combination of

∂T

∂x

and D

ϕ = −

ρk

∂T

∂x

−

ρk

(4.3.73)

where {a

}, {b

} satisfy

J(a

) = ˜a



|v − u|

−



− u

), (4.3.74)

J(b

) =

= (v

− u

)(v

− u

) −

|v − u|

It can be concluded by symmetry considerations that the solutions to these equations

should be of the form:

= α(T, |v − u|

)˜a

, (4.3.75)

= β(T, |v − u|

)

4.3. KINETIC THEORY 191

for some functions α and β. From this we get

q = −κ∇T, τ = ρk

T I + 2µD (4.3.76)

where I is the identity tensor,

κ =

2ρk

′

α(T, |v

′

)



′

−



′

(4.3.77)

µ = −

2ρk

β(T, |v

′

)(v

′

⊗ v

′

) : (v

′

⊗ v

′

−

′

I)dv

′

(4.3.78)

Remark: Note that the viscous and heat conduction terms are of O(ε) compared

with the terms obtained under the local equilibrium approximation.

There are two general approaches for obtaining more accurate approximations:

1. The ﬁrst is to include higher order moments. Grad’s 13-moment model is a pri-

marily example of this kind [20, 38].

2. The second is to include higher order derivatives. The Burnett equation is a typical

example of this class of models [17].

4.3.6 Other kinetic models

A much simpliﬁed model in kinetic theory is the BGK model, which approximates the

collision term by a simple relaxation form [2]:

∂f

∂t

+ (v · ∇

)f = M

− f, (4.3.79)

where M

is local Maxwell-Boltzmann distribution

(x, v, t) = n(x, t)



2πk

T (x, t)



3/2

exp



−

m|v − u(x, t)|

T (x, t)



, (4.3.80)

n, u, T are obtained from f through

n(x, t) =

fdv, n(x, t)u(x, t) =

vf(x, v, t)dv, (4.3.81)

T (x, t)n(x, t) =

|v − u(x, t)|

f(x, v, t)dv (4.3.82)

Note that this model is still quite nonlocal and nonlinear, due to the term M

. However, it

is certainly much simpler than the original Boltzmann equation. For some problems, this

simple model gives fairly accurate results compared to the original Boltzmann equation.

192 CHAPTER 4. THE HIERARCHY OF PHYSICAL MODELS

4.4 Electronic structure models

Strictly speaking, none of the models we have discussed so far can be regarded as

being the true ﬁrst principle. They all contain aspects of empirical modeling. Models in

molecular dynamics use empirical inter-atomic potentials. Kinetic theory models rely on

the empirical scattering cross section σ. In continuum theory, the constitutive relations

are empirical in nature. For the true ﬁrst principle, we need quantum mechanics and we

should consider the quantum many-body problem. As we discuss below, this quantum

many-body problem contains no empirical parameters. The diﬃculty, however, lies in

the complexity of its mathematics. For this reason, approximate models have been

developed which describe the electronic structure of matter at various levels of detail.

These approximate models include:

1. The Hartree-Fock model, in which the many-body electronic wave function is lim-

ited to the form of a single Slater determinant [34].

2. Kohn-Sham density functional theory, a form of the density functional theory

(DFT) that uses the ﬁctitious orbitals. DFT can in principle be made exact by

using the right functionals. But in practice, approximations have to be made for

the exchange-correlation functional [34].

3. Orbital-free density fun ctional theory. Thomas-Fermi model is the simplest ex-

ample of this type. These models do not involve any wave functions. They use

the electron density as the only variable. They are in general less accurate than

Kohn-Sham DFT, but are much easier to handle [7].

4. Tight-binding models. These are much simpliﬁed electronic structure models in

which the electronic wave functions are expressed as linear combinations of a mini-

mal set of atomic orbitals. Electron-electron interaction is usually neglected except

that Pauli exclusion principle is imposed.

In the following sections, we will ﬁrst discuss the quantum many-body problem. We

will then discuss the approximate models. We will focus on the zero temperature case

and neglect the consideration of the spins.

4.4. ELECTRONIC STRUCTURE MODELS 193

4.4.1 The quantum many-body problem

Quantum mechanics is built upon two fundamental principles:

1. The form of the Hamiltonian operator and the Schr¨odinger equation.

2. The Pauli exclusion principle.

Consider a system consists of N

nuc

nuclei and N electrons. It is described by a

Hamiltonian which, in atomic units, is given by

H =

nuc

I=1

i=1

+ V (R

, . . . , R

nuc

, x

, . . . , x

). (4.4.1)

Here M

is the mass of the I-th nucleus – the mass of an electron is 1 in the atomic units;

is the position coordinate for the I-th nucleus, ( x

, ··· , x

) are the spatial coordinates

for the N-electron system; P

, p

are the corresponding momentum operators: p

−i∇

, P

= −i∇

. V is the interaction potential between the nuclei as well as the

electrons, given by

V (R

, . . . , R

nuc

, x

, . . . , x

) =

I6=J

− R

i6=j

− x

−

i,I

− R

(4.4.2)

where Z

is the charge that the I-th nucleus carries. We can write the Hamiltonian as

H = −

nuc

I=1

∆

−

i=1

∆

+ V (R

, . . . , R

nuc

, x

, . . . , x

), (4.4.3)

The ground state of a system of electrons and nuclei can be found by solving the

Schr¨odinger equation

HΨ = EΨ, Ψ = Ψ(R

, . . . , R

nuc

; x

, . . . , x

) (4.4.4)

subject to the symmetry constraints determined by the statistics of the particles (nuclei or

electrons). For example, since the electrons are fermians, Ψ is an antisymmetric function

in (x

, ··· , x

), i.e., Ψ changes sign if any pair of coordinates x

and x

are interchanged.

Ψ(R

, . . . , R

nuc

; x

, . . . , x

)

= −Ψ(R

, . . . , R

nuc

; x

, . . . , x

), 1 ≤ i < j ≤ N. (4.4.5)

194 CHAPTER 4. THE HIERARCHY OF PHYSICAL MODELS

This quantum many-body problem contains no empirical parameters. One only has to

enter the mass {M

} and charge {Z

} of the participating nuclei, then the description is

complete.

Born-Oppenheimer approximation. The time scales of the nuclei and electronic

degrees of freedom are determined by their respective masses. Since M

is usually on

the scale of 10

or larger, the electrons relax at a much faster pace than the nuclei. This

suggests making the approximation that the electronic structure is always at the ground

state given by the st ate of the nuclei. Under this approximation, the electronic state

is slaved by the state of the nuclei. This is the Born-Oppenheimer approximation, also

referred to as the adiabatic approximation. In addition, in many cases, it is accurate

enough to treat the nuclei classically.

The Born-Oppenheimer approximation allows us to solve the coupled nuclei-electron

problem in two steps:

1. Given the positions of the nuclei, we can ﬁnd the ground state of the electrons by

solving

= E

, (4.4.6)

where Ψ

is the antisymmetric wave function for the electronic degrees of freedom,

= −

∆

+ V (R

, . . . , R

nuc

, x

, . . . , x

) (4.4.7)

2. Find the state of the nuclei using the classical Hamiltonian:

H =

+ E

, . . . , R

nuc

) (4.4.8)

where E

= hΨ

|Ψ

i is the eﬀective potential due to the electrons.

We often write the Hamiltonian H

as:

= −

∆

ext

) + V

, . . . , x

) + V

, . . . , R

nuc

) (4.4.9)

where

ext

(x) = −

|x − R

, V

, . . . , x

) =

i6=j

− x

, . . . , R

nuc

) =

I6=J

− R

(4.4.10)

4.4. ELECTRONIC STRUCTURE MODELS 195

The term V

is an important contribution in E

. However, in the Schr¨odinger equation

(4.4.6), the positions of nuclei are ﬁxed, V

is just a constant p otential. Therefore, we

often use

= −

∆

ext

) + V

, . . . , x

), (4.4.11)

with a slight abuse of notation. It is understood that the V

term should be added back

in the Bohn-Oppenheimer surface energy E

It is well-known that the Schr¨odinger equation admits an equivalent variational for-

mulation. For example, corresponding to (4.4.6), we can deﬁne the functional

I(Ψ

) = hΨ

|Ψ

i =

i=1



|∇

+ V |Ψ



··· dx

(4.4.12)

We then have

= inf

I(Ψ

) (4.4.13)

subject to the normalization and antisymmetry constraints.

First principle-based inter-atomic potential. (4.4.8) gives the eﬀective inter-

atomic potential E

, . . . , R

nuc

) derived from the ﬁrst principle. It gives a rigorous

starting point for thinking about inter-atomic potentials. However, it is very hard to use

this in practice. Th e wave function is a function of many variables. Even for a nano-sized

system, the wave function depends on so many variables that it is practically impossible

to solve for such wave functions. Therefore further approximations have to be made. We

now proceed to discuss some of these approximations.

4.4.2 Hartree and Hartree-Fock approximation

The simplest approximation is the Hartree approximation, which assumes that the

N-electron wave function takes the form of a product of single-electron wave functions:

, . . . , x

) = ψ

) . . . ψ

), (4.4.14)

with the normalization constraints

|ψ

(x)|

dx = 1, j = 1, 2, ··· , N. (4.4.15)

196 CHAPTER 4. THE HIERARCHY OF PHYSICAL MODELS

Putting this ansatz into the variational principle, we get

({ψ

}) =

i=1

|∇ψ

+ V

ext

i=1

|ψ

dx +

×R

i6=j

|ψ

(x)|

|ψ

(y)|

|x − y|

dx dy.

(4.4.16)

The corresponding Euler-Lagrange equations become

−

∆ + V

ext

(x) +

j6=i

|ψ

(y)|

|x − y|

(x) = ε

(x), , i = 1, ··· , N (4.4.17)

where ε

is the Lagrange multiplier corresponding to the normalization constraint.

Hartree’s approximation has a simple interpretation, namely that each electron feels

a mean-ﬁeld potential given by the other electrons in the system, via the Coulomb inter-

action.

Hartree’s approximation violates the Pauli exclusion principle. This problem is ﬁxed

in the Hartree-Fock approximation. In the Hartree-Fock approximation, the wavefunction

is assumed to be in the form of a single Slater determinant:

, . . . , x

) =

√

det







) . . . ψ

)

) . . . ψ

)







. (4.4.18)

Here {ψ

, . . . , ψ

} is a set of orthonormal (one-particle) wave functions:

∗

(x)ψ

(x) dx = δ

. (4.4.19)

The Hartree-Fock functional is obtained from the functional (4.4.12) using the Hartree-

Fock approximation (4.4.18):

({ψ

}) =

i=1

|∇ψ

(x)|

+ V

ext

(x)|ψ

(x)|

i,j=1

∗

(x)ψ

∗

(y)ψ

(y)

|x − y|

dx dy

−

i,j=1

∗

(x)ψ

(y)ψ

∗

(y)ψ

(x)

|x − y|

dx dy

(4.4.20)