Weinan E. Principles of Multiscale Modeling

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2.6. RENORMALIZATION GROUP ANALYSIS 77

Figure 2.6: Typical spin conﬁgurations for the two-dimensional Ising model at low, critical

and high temperatures (produced by Hao Shen).

78 CHAPTER 2. ANALYTICAL METHODS

3. ν can be found via linearizing R at its ﬁxed point. For |K − K

∗

| ≪ 1,

= R(K) = K

∗

+ λ(K − K

∗

) + ···

where λ =

∗

), therefore

ξ(K

)

ξ(K)

− T

∗

−ν

|T − T

∗

−ν

∗

− K

−ν

∗

− K|

−ν

= λ

−ν

This gives a prediction on the value of ν.

2.6.2 An illustration of the renormalization transformation

It is useful to ﬁrst illustrate the basic ideas of renormalization group analysis using

the one-dimensional Ising model. Consider a one-dimensional lattice with N sites. At

the jth site, we assign a spin variable s

, which takes two possible values +1, −1. The

Hamiltonian of this model is given by

H = −J

|i−j|=1

. (2.6.5)

J > 0 so that neighboring spins tend to align with each other to lower the value of the

Hamiltonian.

Renormalization group analysis proceeds as follows. First, let us deﬁne a coarse-

graining operator, the renormalization group ﬂow. For example, starting with a system

with N sites (N is assumed to be even), we integrate out the spin variables on the even

sites and take only the odd ones {s

, s

, . . . , s

N−1

} as the variables of our system.

Next we deﬁne the renormalization group ﬂow. The transformation

, s

, . . . , s

} → {s

, s

, . . . , s

N−1

}

induces a transformation in the space of the models (the Hamiltonians). After summing

over s

, s

, etc, the partition function becomes

Z(N, K) =

}

exp(K(s

+ s

+ . . .))

...

(exp(K(s

+ s

)) + exp(−K(s

+ s

)))

× (exp(K(s

+ s

)) + exp(−K(s

+ s

))) ···

2.6. RENORMALIZATION GROUP ANALYSIS 79

We view this new form of the partition function as the partition function of a new

Hamiltonian on a lattice with N/2 sites. This means that we would like to ﬁnd a constant

(L = 2) such that

Z(N, K) =

...

exp(K

+ s

+ . . .))f(K)

N/2

. (2.6.6)

Here the last term is a scaling factor that has to be introduced to compensate for the

reduction of the conﬁguration space. For (2.6.6) to hold, it suﬃces to require

exp(K(s

+ s

)) + exp(−K(s

+ s

)) = f (K) exp(K

) (2.6.7)

for all choices of s

and s

. The solution to these equations is given by

= ln cosh(2K)/2, f(K) = 2 cosh

1/2

(2K) (2.6.8)

The ﬁrst equation deﬁnes the renormalization group ﬂow in the space of the model

parameters, here the value of K. We then have

Z(N, K) = f(K)

N/2

Z(N/2, K

). (2.6.9)

The renormalized Hamiltonian is

= −J

|i−j|=2

, (2.6.10)

where J

= k

T K

. From (2.6.9), we obtain

−ζ(K) =

ln f(K) −

ζ(K

). (2.6.11)

From (2.6.8), if we know the value of ζ at K, we can obtain the value of ζ at K

To analyze the dynamics of the renormalization group ﬂow and the change of the free

energy along th e ﬂow, we may proceed as follows. When K = 0, the partition function

is simply Z(N, 0) = 2

. Therefore for some small value of K, say 0.01, we may set

Z(N, 0.01) ≈ 2

. We then use the renormalization transformation (2.6.8) iteratively to

ﬁnd the free energy at other values of K. For the 1D Ising model, th is approximation is

indeed quite good [?]. See Table 2.6.2 (from [?]).

Is a permission required in this case?

80 CHAPTER 2. ANALYTICAL METHODS

ζ(K)

K Renormalization group Exact

0.01 ln 2 0.693 197

0.100 334 0.698 147 0.698 172

0.327 447 0.745 814 0.745 827

0.636 247 0.883 204 0.883 210

0.972 710 1.106 299 1.106 302

1.316 710 1.386 078 1.386 080

1.662 637 1.697 968 1.697 968

2.009 049 2.026 876 2.026 877

2.355 582 2.364 536 2.364 537

2.702 146 2.706 633 2.706 634

In this example, the renormalized Hamiltonian (2.6.10) is in exactly the same form

as the Hamiltonian we began with, i.e. we were able to ﬁnd values of f(K) and K

such that (2.6.7) is satiﬁed for all values of s

and s

and no approximations are needed.

This is rarely the case in general. Instead, one has to make approximations in order to

maintain some feasibility for the renormalization group analysis, as the next example

shows.

2.6.3 RG analysis of the two-dimensional Ising model

We now look at the Ising model on two-dimensional triangular lattice with lattice

constant a [?]. To coarse-grain the model, we divide the lattice into a triangular lattice

of triangular blocks indexed by I with three vertices inside each block, and deﬁne the

block spin S

= sgn(s

+ s

), where s

, s

are the spins on the vertices inside the

I-th block, ordered in some particular way. After coarse-graining, the lattice constant

changes from a to

√

3a. When S

= 1, we have four possible spin conﬁgurations for s

and s

that we will denote by σ

= 1, 2, 3, 4: + + −, + −+, −+ + and + + +, where

we use + for spin up and − for spin down. These are the internal variables that we will

integrate over. To renormalize, note that we can write H as a function of the variables

, σ

}: H = H(K, {S

, σ

}). Integrating out the variables {σ

}, we can write Z(K, N)

2.6. RENORMALIZATION GROUP ANALYSIS 81

in the form of

exp(−H(K

, {S

})) by taking

H(K

, {S

}) = −log

{σ

}

exp(−H(K, {S

, σ

})). (2.6.12)

It would be nice if H(K

, {S

}) has the same form as the original Hamiltonian, say

H = −J

hI,Ji

for some new parameter J

. However, this is impossible unless we

make some approximations.

Figure 2.7: Coarse-graining th e Ising model on triangular lattice.

Figure 2.8: Interaction between two block spins.

82 CHAPTER 2. ANALYTICAL METHODS

The starting point is

P {S

} =

}

P {s

} (2.6.13)

where

}

means summing over all conﬁgurations {s

} that give rise to {S

} after

coarse graining. For later use, we mention the obvious relation:

}

(2.6.14)

and the elementary but useful factorization:

···

) ···f

) = (

)) ···(

)) (2.6.15)

Rewriting (2.6.13), we have

}

(2.6.16)

In the last equation, K

|i−j|=1

consists of two parts: the part that corresponds to

the internal block, including all terms for which s

and s

are in the same block, and the

inter-block part that includes all terms for which s

and s

are in two diﬀerent blocks.

We denote them by H

and H

respectively.

Claim: For (2.6.16) to hold, it is suﬃcient that

}

−H

/(k

T )

(2.6.17)

Proof. Note that

}

−H

/(k

T )

I=1

K(S

)

= (e

+ 3e

−K

)

(2.6.18)

where M is the number of blocks. Note also that the right hand side of (2.6.18) does not

depend on {S

}. Summing over {S

} on both sides of (2.6.17), we get

}

−H

/(k

T )

where we used (2.6.14). The ratio of (2.6.17) and the last equation gives (2.6.16).

2.6. RENORMALIZATION GROUP ANALYSIS 83

Next deﬁne the weighted average:

hAi =

}

−H

/(k

T )

}

−H

/(k

T )

then (2.6.17) can be written as

= he

−H

/(k

T )

i = e

−hH

i/(k

T )−

(hH

i−hH

)/(k

T )

+ ··· (2.6.19)

We will make the approximation:

−H

/(k

T )

i ≈ e

−hH

i/(k

T )

(2.6.20)

This is a drastic approximation. Now compare the exponents:

−K

|I−J|=1

= hH

i/(k

T ) = −

|I−J|=1



i + hs



where s

, s

are shown in Figure 2.6.3.

Using (2.6.15) again, we have

i =

}

−H

/(k

T )

}

−H

/(k

T )

K(s

)

K(s

)

K(s

)

K(s

)

Similar to (2.6.18), we get

K(s

)

= S

+ e

−K

)

Therefore, we have

|I−J|=1

2K(

+ e

−K

+ 3e

−K

)

namely,

= R(K) = 2K(

+ e

−K

+ 3e

−K

)

This is the renormalization group ﬂow under the approximation (2.6.20).

The nontrivial ﬁxed point of this map is K

∗

ln(1 + 2

√

2) ≈ 0.3356, from which

we also obtain ν = ln L/ ln R

′

∗

) ≈ 1.134. As a comparison, note that Onsager’s exact

solution for the 2D Ising model on triangular lattice gives K

∗

ln 3 ≈ 0.2747 and

ν = 1.

In principle, we may try to improve upon the approximation (2.6.20). In practice,

this is not easy since the computation quickly becomes very involved.

84 CHAPTER 2. ANALYTICAL METHODS

2.6.4 A PDE example

Consider the example







∂

= ∂

− u

∂

= D∂

+ u

(2.6.21)

with initial data that decays suﬃciently fast at inﬁnity. This example was analyzed in

[?]. Our interest here is to see how simple-minded renormalization group methods can

be used for this problem.

First let us deﬁne the renormalization transformation. Let L > 1 be ﬁxed, and let

, u

) be the solution of (2.6.21). Deﬁne







˜u

(x, t) = L

(Lx, L

˜u

(x, t) = L

(Lx, L

(2.6.22)

Next, we deﬁne the renormalization group ﬂow, now in the space of nonlinear opera-

tors. It is easy to see that (˜u

, ˜u

) satisﬁes







∂

˜u

= ∂

˜u

− L

2(1−α

)

˜u

∂

˜u

= D∂

˜u

+ L

2−(α

+α

)

˜u

(2.6.23)

Therefore, under the renormalization transformation (2.6.22), the operator in the PDE

changes to



∂

− u

D∂

+ u



−→



∂

− L

2(1−α

)

D∂

+ L

2−(α

+α

)



(2.6.24)

This suggests that we should consider the class of operators with the form







∂

+ c

D∂

+ c



(2.6.25)

Indeed, under the same renormalization transformation, such an operator changes to







∂

+ c

2(1−α

)

D∂

+ c

2−(α

+α

)



(2.6.26)

which is still in the same class of operators. Therefore our renormalization group ﬂow is

deﬁned by:

→ c

2(1−α

)

, c

→ c

2−(α

+α

)

(2.6.27)

2.7. THE MORI-ZWANZIG FORMALISM 85

We are interested in the ﬁxed points of the map (2.6.22) and (2.6.27). Fixed points

of (2.6.27) should satisfy

∗

= c

∗

2(1−α

)

, c

∗

= c

∗

2−(α

+α

)

(2.6.28)

It is beyond our scope to carry out a complete analysis of all the ﬁxed points. But one

interesting set of ﬁxed points identiﬁed by Bricmont et al. is given by [?]

∗

= −1, c

∗

= 0, α

= 1, α

> 1 (2.6.29)

This ﬁxed point corresponds to the PDE







∂

= ∂

− u

∂

= D∂

(2.6.30)

The ﬁnal step is to study the linearization of the renormalization group ﬂow (2.6.27)

around this ﬁxed point. We refer to [?] for some details. Let us note that this ﬁxed point

PDE has solutions that are ﬁxed point of the map (2.6.22). The value of α

can be found

by looking for solutions of (2.6.30) which are also ﬁxed points of the renormalization

transformation (2.6.22). Such solutions must take the form

(x, t) = t

−



√



, u

(x, t) = t

−



√



It is easy to see that

(ξ) = Ae

−

and u

satisﬁes

−

dξ

−

dξ

−

+ u

= 0

For small values of A, α

can be calculated perturbatively, and it has the form [?]

= 1 +

4π

√

2D + 1

+ O(A

2.7 The Mori-Zwanzig formalism

So far we have discussed two special situations in which simpliﬁed eﬀective models

can be obtained using asymptotic analysis. The ﬁrst is the case when the system has

86 CHAPTER 2. ANALYTICAL METHODS

disparate scales. The second is when the system has self-similar behavior at diﬀerent

scales. A natural question is: What happens in more general situations? To address this

question, we turn to the Mori-Zwanzig formalism.

The Mori-Zwanzig formalism is a strategy for integrating out a subset of variables

in a problem, which is precisely what we would like to do when performing coarse-

graining [?, ?]. What is remarkable about the Mori-Zwanzig formalism is its generality

– it applies to general nonlinear systems. However, at some level, it is also a tautology,

since what it does is simply to represent the solution for a subset of the variables (the

ones to be eliminated) as functionals of the other variables (the ones to be kept) and

then substitute back into the (projected form of the) original equation. In this way, some

variables are formally eliminated, but the remaining m odel is often far from being explicit.

Nevertheless, it is still a very good starting point for model reduction or coarse-graining

for the following reasons:

1. It is a general strategy.

2. It suggests the form of the reduced model. For example, if we start with the

equations in molecular dynamics, then the model obtained after elimination of

some variables can be viewed as a generalized Langevin equation, with noise and

memory terms.

3. It provides the starting point for making further approximations. From a practical

viewpoint, this is the most imp ort ant application of the Mori-Zwanzig formalism.

For example, the abstract generalized Langevin equation that arises from the appli-

cation of the Mori-Zwanzig formalism to molecular dynamics models can sometimes

be approximated by a much simpler Langevin equation.

We start with a simple example to illustrate the basic idea. Consider a linear system

= A

p + A

q, (2.7.1)

= A

p + A

q. (2.7.2)

Our objective is to obtain a closed model for p, which is the quantity that we are really

interested in. To this end, we solve the second equation for q, viewing p as a parameter:

q(t) = e

q(0) +

(t−τ)

p(τ) dτ.