Anderson Localization

8/10/2019 Anderson Localization

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Mech 506

Exploration of Anderson localizationusing n-pendula model

Author:

Miayan YeremiStudent #:

18213074

December 5, 2014


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pendulum is i= g

L(1+Li). In addition, lets label the frequency domain analogue of displacement

as ci andV = km

. From this we get:

2ci= (2V +i)ci+V ci+1+V ci1

Lastly, if we label Ui = 2 2V i. We get the exact equation presented in Hodgess paper.

V ci+1+Uici+V ci1 = 0 (1)

Lets turn our attention to the system of masses on a string. We have to discuss the analogiesbetween the two systems, and in addition discuss what it means to have coupling in the stringsystem. So, lets start out making a couple of observations. If the masses on the string wereinfinitely massive then the pieces of string in between masses, which I will refer to as bays, will notfeel the effect of vibrations of the neighbouring bays. This is like having no coupling in the system.So heavy masses is equivalent to weak coupling between bays. It is also worth mentioning thatthe bays are the analogues of the pendula. Finally, we can guess the mode shapes of the masseson a string system. We can say the masses vibration amplitude is small compared to the stringamplitude vibration. The masses can be thought of as being displaced from each other by straight

lines, and the bays can be thought of as having sinusoidal mode shapes. So, if we say the massesvibration amplitude is bi and the bays vibration amplitude is ai, where bi


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Figure 1: Two pendula example

3 Two pendula example

First I will start by deriving the equations of motion (EOM) for the simple two pendula example.After, this derivation I will generalize the equation to n number of pendula with appropriateboundary conditions.

Start by writing out the kinetic and potential energies of the system.

T = m

2(L1)

2 +m

2(L2)

2

U= k

2(L1 L(1 + L)2)

2 +mgL(1 cos(1) + +mgL(1 + L)(1 cos(2)

Now from the kinetic and potential energies, I applied Lagranges equations to arrive at thefollowing EOM (note, I assumed that the angles are small in order to linearise the equations).

mL21+kL(L1 L(1 + L)2) +mgL1 = 0

mL2(1 + L)22+kL(1 + L)(L1 L(1 + L)2) +mgL(1 + L)2 = 0

Next, I write the EOM in matrix form.

d2

dt2 1

2

= ( k

m

+ g

l

) k

m

(1 + L)km

1(1+L)

( km

+ gl

1(1+L) )

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From the above equation it is trivial to write down the characteristic equation used to find theeigenvalues of the system.

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41,2+

2k

m +

g

l +

g

l

1

(1 + L)

21,2+

gk

Lm+

gk

(1 + L)m+

g2

L2(1 + L)= 0

It is convenient to non-dimensionalize above equation by dividing above expression by g2

L2 and

definingR2 = kLmg

. After, non-dimensionalization 21,2 becomes 21,2 =

21,2g

L

.Next, it is easy to show

using the quadratic equation that,

221,2 = 2R2 + 1 +

1

(1 + L)

4R4 + 1

2

(1 + L)+

1

(1 + L)2. (4)

At this point, I would like to introduce some terminology. I am studying the effects of weakdisorder, which means small values of L, and both cases of strong/weak coupling, which can beviewed in the non-dimensional form as large/small values ofR2.

The two regimes I am interested in, lend themselves well to Taylor expansions of equation 4.Lets Taylor expand the strong coupling case, which gives:

21,2 = 1 +R

2

L

2 +

(L)2

2 R2

1 +

(L)2

8R4

+O(L3

).

This approximation works well for strong coupling, where R2 is large, but it is worth noticingthat in the weak coupling regime this approximation breaks down. Therefore, the moral of thisstory is that the smallness ofR2 has to be considered as well in the weak coupling case. In the nextsections, when I generalize these ideas to n pendula I will treat the perturbation methods in bothregimes differently to make sure I do not run into divergent solutions when switching between thestrong and weak coupling cases.

4 Exact Equations of Motion for n-pendula

The first thing I will present is an extension of the equations of motions (EOM) arrived at in Section3. Note, I am assuming small angular values through out entire discussion.

To start out, I will write the EOM for a pendulum in the interior of n-pendula. An interiorpendulum has two pendula on either side of it, so the equations presented in Section 3 can begeneralized to:

mL2(1 + Li)2i+kL(1 + Li)(L(1 + Li)i L(1 + Li+1)i+1) +

+kL(1 + Li)(L(1 + Li)i L(1 + Li1)i1+mL(1 + L)gi= 0

From here, I non-dimensionlize (i.e. same method used in Section 3) above equation and writeit in matrix form:

d2

dt2= A,

A= B

R2(1 + Li1)

1 + Li; 2R2 +

1

1 + Li;

R2(1 + Li+1)

1 + Li

.

B is a tridiagonal banded matrix. Lastly, I assumed fixed-fixed boundary conditions.

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5 Strong coupling weak disorder n-Pendula Perturbation

In order to develop a strong coupling perturbation we take the equations of motion (EOM) providedin Section 4 and Taylor expand using Li as a small parameter. The two terms that need to be

expanded are,1

1 + Li= 1 Li+ (Li)

2 + H.O.T.

1 + Lj1 + Li

= 1 + (Lj Li) + (L2i LiLj) + H.O.T.

H.O.T stands for higher order terms. Let Li= Li, where Li is order one and


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The only way this equation can be solved is ifx1iis orthogonal tox0i, which is what we implicitlyimposed when solving for 1i. So to solve for x1i we can just find its projections on to the rest ofthe eigenvectors of the unperturbed problem. If we do this we get the following,

x1i=j

1(0i 0j)

x

0jA1x0ix0jx0j

x0j i=j.

Finally, we proceed to the second order perturbation problem.

O(2) : (A0 0i)x2i= A1x1i+1ix1i+2ix0i A2x0i

Again, multiply by x0i and solve for 2i, which gives,

2i = x0i(A1x1i+A2x0i)

x0ix0i.

By a very similar process as described for the first order perturbation we get the following forthe second order correction for the eigenvector,

x2i =j

1

(0i 0j)

x0j(A1x1i+A2x0i)

x0jx0j

x0j i=j.

There is one slight issue I have been ignoring here. I have been implicitly assuming there areno multiple eigenvectors with the same eigenvalues. If there is a degeneracy in the eigenvalues thanthe equation above have to be modified.

So how well is this perturbation method actually working? Lets try an example with Matlab.Like I mentioned previously, I calculated the correction to the ground state of a system with

strong coupling and smaller disorder. In this example I wanted to demonstrate that the perturbationmethod performs well even if the disorder is to weak. Now, because the equations I am workingwith are non-dimensionalized I can compare the number I used for the order of the disorder and theorder of the coupling. I set R2 = 1 andepsilon= 0.1 so the disorder is only one order of magnitude

smaller than the coupling parameter. In addition, I randomly and uniformly set the changes inpendula lengths using a uniform distribution. The eigenvalues I got were as follows:

exact= 1.1779

0i= 1.0015 0th order perturbation

1i= 1.1533 1st order perturbation

2i= 1.1750 2nd order perturbation

Now, lets see how the method performed in terms of guessing the eigenvector correction, seeFigure 2.

The second order perturbation approximated the exact solution very closely, even for a systemwith fairly large disorder. Another interesting observation is that even though the system is fairlydisordered there is still transmission through the system at the zero harmonic (i.e. the mode spansthrough all the pendula and is not localized). This as we shall see in the next section is not true forsystem with weak coupling. This difference in behaviour was hinted at in Section 1, when it became

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0 10 20 30 40 50 60 70 80

0

0.05

0.1

0.15

0.2

0.25Strong coupling weak disorder (n=80)

0th order perturbation

exact solution



Figure 2: Perturbation eigen vectors vs. Exact solution (strong coupling)

clear that the same approximation methods could not be used for both weak and strong coupling.In the weak coupling weak disorder regime we actually get Anderson localization, as opposed toglobal mode transmission.

6 Weak coupling weak disorder n-pendula

For the weak coupling weak disorder system, we have to rethink the perturbation scheme. What

this amounts to is a different Taylor approximation of the exact equations presented in Section 4.We still have the same structure for the perturbation scheme,

A0+A1+

2A2+

x= x

Except the A0, A1, and A2 matrices are different. In addition, I am relabelling the coupling

parameter to R2w, and R2w = R

2, where R2 is order one, and I am assuming R2wLi

is order one. Iam also assuming all the Li are unique.

A0 = diag (1 + Li) .

A1 = B

R2 ; 2R2 ; R2

.

A2 = B R2(Li1 Li) ; L2i ; R2(Li+1 Li) .x= x0+x1+

2x2+

= 0+1+22+

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0 2 4 6 8 10 12 14 16 18 20

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1Weak coupling weak disorder (n=20)


exact solution

1st order perturbation

2nd order perturbation

Figure 3: Perturbation eigen vectors vs. Exact solution (weak coupling)

The same exact theoretical frame work, as presented in Section 5, goes into solving the pertur-bation problem in this section.

For this example, R2w, and Li were on the order of 0.01, I found that anything bigger thanthis order seemed to worsen the performance of the perturbation; in addition, the exact solution forbigger values ofR2w, and Li was less localised. This lead me to conclude that 0.01 was the orderwere I could assume weak coupling and weak localization. I used 20 pendula and assigned theirdeviations in length randomly, using a uniform distribution. The eigenvalues I got were as follows:

exact= 1.0443

0i= 1.0238 0th order perturbation

1i= 1.0438 1st order perturbation

2i= 1.0441 2nd order perturbation

Figure 6 displays the results of the eigenvector perturbation corrections. After running this cal-culation for several random length deviation distribution I noticed that the first order perturbationworks quite well in the weak coupling weak disorder parameter regime. I believe that this is do tothe localised nature of this problem.

Next, I will briefly discuss the affects of having two of the length are equal.

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7 Weak coupling weak disorder n-pendula with degenerate

length

An interesting case arises if we allow m (m < n) of the pendula to have equal lengths. In order to

understand what happens when m of the pendula have equal length, it is most convenient to startwith the first order perturbation matrix.

A0 = diag (1 + Li) .

If two of the pendula length are the same than this matrix has degenerate eigenvalues of multi-plicity m. The problem with the ordinary perturbation method is that when we go to solve for theeigenvectors we have to divide by the difference of the unperturbed eigenvalues. This is of courseproblematic if two, or more of the eigenvalues are the same.

What we have to do is approach the perturbation problem a little differently. Lets just look atthe first order problem.

(A0+A1+ ) x= x,

where this time we will write x a little differently (note, I am performing an expansion for theeigenvectors that are associated with the degenerate eigenvalue).

x=j

jx0j+ x1+ , where j runs over the degenerate eigenvectors.

= 0+1+

When we proceed to solve this problem we actually have introduced multiple unknowns (i.e.

js), but we can use the multiple x0js to come up with multiple equations which allow to solve for

the js and the new multiple eigenvalue corrections. In the process of solving for the js and firstorder eigenvalue corrections we get the following new eigenvalue problem.

x01A1x01

x01x01

x01A1x02

x01x01

x01A1x0m

x01x01

x0mA1x01

x0mx0m

x0mA1x02

x0mx0m

x

0mA1x0m

x0mx0m

= 1

Now that we have the m , ands we can proceed with the perturbation to higher orders. Thefirst order corrections will usually lift the degeneracy in the unperturbed problem. The lifting of de-generacy can be seen in many fields in physics. For example, degeneracy lifting due to perturbationis seen in fine structure splitting, Zeeman effect, Stark effect, etc.

In the pendula systems I am considering there is an interesting effect taking place due to thestructure of the perturbing matrices, more specifically the perturbing matrices are tridiagonal.

Because the perturbing matrices are tridiagonal the machinery develop above only has to be

used for certain orders of perturbation corrections depending on the locations of the equal lengthpendula. Essentially if the spacing between the two equal pendula is k pendula then the degenerate

perturbation method only needs to be employed at the kth order perturbation. This suggest a scal-ing for the amplitude transmission between two degenerate pendula. The amplitude transmission

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0 5 10 151

0.5

0

0.5

1nearest neighbour degeneracy (deg.)

pendula location

eigenvectoramplitude

0 5 10 151

0.5

0

0.5

1next (n.) nearest neighbour deg.

pendula location

eigenvectorlocation

0 5 10 15

1

0.5

0

0.5n. n. nearest neighbour deg.

pendula location


0 5 10 15

1

0.5

0

0.5n. n. n. nearest neighbour deg.

pendula location


Figure 4: Degenerate pendula at different distances from each other

might take on an order that is proportional to k. To be more precise, I am defining amplitudeof transmission as magnitude of vibration felt by one pendulum do to an excitation in a differentpendulum.

I did not pursue this idea any further do to time constraints, but I did use the exact solution tosee if the amplitude of transmission between two degenerate pendula decreases as the degeneratependula are moved further apart from each other (this is not very surprising physically speaking).Figure 4 shows the results of this experimentation. I positioned the first pendulum at the beginning

(i.e. location 1)of the pendula chain and then move the degenerate pendulum further and furtheraway from location 1. The results do suggest some amplitude of transmission scaling with distancebetween degenerate pendula, that is proportional to .

8 Future Work

I think the most interesting extension of this project would be to set the experimental portionpresented in Hodges paper and observe Anderson localization in the lab.

I would also like to further investigate the scaling law for the transmission of two, or moredegenerate pendula with the parameter

Lastly, if I could work the previous part out, I think it would be interesting to come up with

an experimental setup validating the scaling proportionality.

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References

[C. Pierre, E. H. Dowell, 1986] C. Pierre, E. H. Dowell (1986) Localization of vibration by structural irregularity.Journal of sound and vibration.

[C. H. Hodges, J. Woodhouse, 1983] C. H. Hodges, J. Woodhouse (1983) Vibration isolation from irregularities in

a nearly periodic structure: Theory and measurements. Journal of acoustics.[E. J. Hinch ] E. J. Hinch Perturbation Methods.

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Anderson Localization