MacKinnon - Computational Physics (523159), страница 9
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The problem becomes one of finding the eigenvalues, * , and eigenvectors, y, ofa generalised eigenvalue problem. The eigenvectors show how the wire distorts when oscillating in eachmode and the eigenvalues give the corresponding oscillation frequencies. Low frequency modes are moreimportant than high frequency modes to the crane manufacturer.The problem can be solved most easily by using a LaPack routine which finds the eigenvalues andeigenvectors directly. However, before doing so it is necessary to eliminate the matrix M using the samemethod as discussed in problem 6.You should investigate both the computational aspects, such as the dependence of the results on , aswell as the physical ones, such as the dependence of the behaviour on the mass, .
Do your results makephysical sense? You might even compare them with a simple experiment involving a weight on the end ofa string.O3.9 Project — Phonons in a Quasicrystal3.9.1 IntroductionUntil a few years ago, it was thought that there were only two different kinds of solids: crystals, in whichthe atoms are arranged in a regular pattern with translational symmetry (there may be defects, of course);Phonons in a Quasicrystal36and amorphous solids, in which there is no long range order, although there is some correlation between thepositions of nearby atoms.
It was also known that it was impossible for a crystal to have five fold rotationalsymmetry, since this is incompatible with translational order.This was how things stood until 1984, when Shechtman et al. (Phys. Rev. Lett. 53 1951 (1984)), weremeasuring the X ray diffraction pattern of an alloy of Al and Mn and got a sharp pattern with clear fivefold symmetry. The sharpness of the pattern meant that there had to be long range order, but the five foldsymmetry meant that the solid could not be crystalline. Shechtman called the material a “quasicrystal”.One possible explanation (although this has still not been conclusively established) is that quasicrystalsare three dimensional analogues of Penrose tilings (Scientific American, January 1977 — Penrose tilingswere known as a mathematical curiosity before quasicrystals were discovered).
Penrose found that youcould put together two (or more) different shapes in certain well defined ways so that they “tiled” the planeperfectly, but with a pattern that never repeated itself. Sure enough, some of these tilings do have five foldsymmetries; and sure enough, there is perfect long range order (although no translational symmetry) sothat the diffraction pattern from a Penrose lattice would be sharp.3.9.2 The Fibonacci LatticeThe mathematical theory of Penrose tilings gets quite high brow and abstruse, but everything is very simplein one dimension.
Then the two shapes are lines of different lengths, which we shall call and , for Adultand Child (Fibonacci actually studied the dynamics of rabbit populations). Every year each adult has onechild and each child becomes an adult. Let us start with a single child(3.40)and then repeatedly apply the “generation rule,”F: F:to obtain longer and longer sequences. The first few sequences generated are,SS(3.41)[(3.42)SNoteproperty that each generation is the “sum” of the 2 previous generations: = the interestingVUWIn a one dimensional Fibonacci quasicrystal, the longs and shorts could represent the interatomic distances; or the strengths of the bonds between the atoms; or which of two different types of atom is at thatposition in the chain.3.9.3 The Model/This project is to write a program to work out the phonons (normal mode vibrations) of a Fibonacci quasicrystal with two different sorts of atom.
The “adults” and “children” are the masses,and , of thetwo kinds of atom. Since we are only interested in small vibrations, we may represent the forces betweenOOthe atoms as simple springs (spring constant ) and we will assume that all the springs are identical ( isindependent of the types of the atoms at the ends of the spring). The equation of motion of such a systemmay be written asPO(3.43)/ Q 1 0Y K 0 Q $ / Q * 0 K Q "]K QS#T $ f K Q K QeT & :KÑQ and / Q are the displacement and mass of the 8 th atom. If the chain contains whereatoms, you willthen have a set of coupled second order ordinary differential equations.The choice of boundary conditions for the atoms at either end of the chain is up to you and should notmake much difference when the chain is long enough: you could fix the atoms at either end to immovableOwalls,, or you could leave them free by removing the springs at the 2 ends,O .
(Periodic boundary conditions — when the chain of atoms is looped around and joined upin a ring — are convenient for analytic work but not so good for numerical work in this case. Why?)K K S#T á S#T á T Phonons in a Quasicrystal37*0The equations (3.43) are linear algebraic equations which may be cast as a tridiagonal matrix eigenproblem. The eigenvalues of this problem areand so give the vibrational frequencies, and the eigenvectors give the corresponding normal mode coordinates,/ .Note, however, that the presence of the massesin (3.43) means that the problem is in the generalisedform AxBx. As discussed in the notes and in problem 6 this can be transformed into the normaleigenvalue problem by making the substitutionand multiplying the th equation by.I suggest you solve the eigenproblem by using a NAG or LaPack routine for the eigenvalues only.
Thereshould be no problem in dealing with chains of several hundred atoms or more.K£Q Q/ : 8 :K Q / QegT 0 Q *03.9.4 The Physics/ Qe TR 08/ 74The idea is to investigate the spectrum (the distribution of the eigenvalues) as the ratio,, of the twomasses changes. Your program should list the eigenvalues in ascending order and then plot a graph ofeigenvalue against position in the list. When, the crystal is “perfect”, the graph is smooth, andyou should be able to work out all the eigenvalues analytically (easiest when using periodic boundaryconditions). But when andbegin to differ, the graph becomes a a “devil’s staircase” with all sorts offascinating fractal structure.
Try to understandthe behaviour 6at small(when the wavelength is long and6YXthe waves are “acoustic”) and the limits asand.6(on the axis) against the vibrational frequenAnother interesting thing to do is to plot6 values ofcies (on the axis). Choose a value of, work out all the frequencies, and put a point on the graph foreach. The graphnow has a line of points parallel to the 4axisat the given value. Do this for a number of6values ofand see how the spectrum develops aschanges.
Again, you should try to understandthe behaviour when the frequency tends to zero, and the large and small mass ratio limits.If you have time it is interesting to investigate the fractal structure by focusing in on a single peak for ashort sequence and investigating how it splits when you add another “generation”. You should find that thebehaviour is independent of the number of generations at which you start.// 7K/ / 7 F 7 / 7 aF *// 7K/ 7Chapter 4Monte Carlo Methods and Simulation4.1 Monte CarloThe term Monte–Carlo refers to a group of methods in which physical or mathematical problems are simulated by using random numbers. Sometimes this is done at a very simple level.
For example, calculationsof radiation damage in humans have been studied by simulating the firing of random particles into humantissue and randomly carrying out the various possible processes. After a lot of averaging one arrives at thelikely damage due to different forms of incident radiation. Similar methods are used to simulate the tracksleft in particle physics experiments. Here we will concentrate on 3 different types of calculations usingrandom numbers.4.1.1 Random Number GeneratorsBefore discussing the uses of random numbers it is useful to have some idea of how random numbers aregenerated on a computer.
Most methods depend on a chaotic sequence. The commonest is the multiplicative congruential method which relies on prime numbers. Consider the sequenceK1QS=Tm" [Z KÑQ& (4.1)KKwhere refers to the remainder on dividing by and and are large integers which have nocommon factors (often both are prime numbers). This process generates all integers less than in anapparently random order. After all integers have been generated the series will repeat itself.
Thus oneimportant question to ask about any random number generator is how frequently it repeats itself.K is chosenIt is worth noting that the sequence is completely deterministic: if the same initial seed,the same sequence will be generated. This property is extremely useful for debugging purposes, but can bea problem when averaging has to be done over several runs. In such cases it is usual to initialise the seedfrom the system clock.Routines exist which generate a sequence of integers, as described, or which generate floating pointnumbers in the range to . Most other distributions can be derived from these.
There are also veryefficient methods for generating a sequence of random bits (Press et al. 1992). The Numerical AlgorithmsGroup library contains routines to generate a wide range of distributions.A difficult but common case is the Gaussian distribution. One method simply averages several (say )uniform () random numbers and relies on the central limit theorem. Another method uses the factthat a distribution of complex numbers with both real and imaginary parts Gaussian distributed can also berepresented as a distribution of amplitude and phase in which the amplitude has a Poisson distribution andthe phase is uniformly distributed between and .As an example of generating another distribution we consider the Poisson case, F f\"A&il !#"%$m&\[38(4.2)Monte Carlo Methods and Simulation39B "<Ky& and \ "A& are the probability distributions of K and respectively,> ea4« ` ]_^ ª \ "A B &% B > eaª ` \ B "]K B &%1K B(4.3)K B must be the same as that for finding anymustfor all , as the probability of finding anyK B 5Kbe.
IttruefollowsthatK\ "<&i \ B "<Ky& _ w_ [(4.4)__ __KB "<KÑ&U _ , then_If is uniformly distributed between and , i.e. \1KF Lc$ b ¹ K[\ "A&; l !#"%$m&(4.5)Then, if \Hence, a Poisson distribution is generated by taking the logarithm of numbers drawn from a uniformdistribution.4.2 Monte–Carlo Integratione 0Often we are faced with integrals which cannot be done analytically. Especially in the case of multidimensional integrals the simplest methods of discretisation can become prohibitively/d expensive.
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