Department of Physics, Chemistry and Biology
Master’s Thesis
Computer simulations of open acoustic Sinai
billiards
Lina F¨
alth
LiTH-IFM-EX–05/1518–SEDepartment of Physics, Chemistry and Biology Link¨opings universitet, SE-581 83 Link¨oping, Sweden
Master’s Thesis LiTH-IFM-EX–05/1518–SE
Computer simulations of open acoustic Sinai
billiards
Lina F¨
alth
Adviser: Irina Yakimenko
Theoretical Physics
Karl-Fredrik Berggren
Theoretical Physics
Examiner: Irina Yakimenko
Theoretical Physics
Avdelning, Institution Division, Department
Theoretical Physics
Department of Physics, Chemistry and Biology Link¨opings universitet, SE-581 83 Link¨oping, Sweden
Datum Date 2005-10-26 Spr˚ak Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats ¨Ovrig rapport
URL f¨or elektronisk version
ISBN
ISRN
Serietitel och serienummer Title of series, numbering
ISSN
Titel Title
Emulering av ¨oppna akustiska Sinai biljarder Computer simulations of open acoustic Sinai billiards
F¨orfattare Author
Lina F¨alth
Sammanfattning Abstract
In this work we have studied energy flow in acoustic billiards, focusing on irregular billiards with and without current effects. The open systems were modeled with an imaginary potential as a source and drain. We have used the finite difference method to model the billiards. General features of the systems are reported and effects of the measuring probe on the wave function are discussed.
Nyckelord Keywords
Acoustic, Sinai billiard
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-4774
—
LiTH-IFM-EX–05/1518–SE
Abstract
In this work we have studied energy flow in acoustic billiards, focusing on irregular billiards with and without current effects. The open systems were modeled with an imaginary potential as a source and drain. We have used the finite difference method to model the billiards. General features of the systems are reported and effects of the measuring probe on the wave function are discussed.
Acknowledgements
I would like to thank my supervisors Irina Yakimenko and Karl-Fredrik Berggren for giving me an interesting diploma work, and for all encouraging help they gave me during the work. I also would like to thank Dragan Adamovic for support and valuable discussions. I am also very grateful to Johan Larsson and Jani Hakanen for all their computational help.
For useful comments on my report I thank my opponent Vivianne Deniz.
Contents
1 Introduction 1
2 Theory 3
2.1 Billiards in general . . . 3
2.1.1 Classical chaos and billiards . . . 3
2.2 Acoustics . . . 4
2.2.1 Basic acoustics . . . 4
2.2.2 Imaginary potential and energy flow . . . 6
2.3 Microwave billiards . . . 6
2.4 Statistics in acoustics . . . 7
2.4.1 Distribution of eigenvalues . . . 7
2.4.2 Wave functions and amplitudes . . . 8
3 Modelling 11 3.1 Discretization . . . 11
3.1.1 Finite difference method . . . 11
3.2 Boundary conditions . . . 12
3.2.1 Neumann . . . 13
3.3 Solving the matrix . . . 13
3.4 Visualisation . . . 13
3.4.1 Wave intensity . . . 14
3.4.2 Energy flow arrows . . . 14
3.4.3 Energy flow densities . . . 14
4 Results 15 4.1 General results . . . 15 4.1.1 Resolution . . . 15 4.2 Closed billiards . . . 17 4.2.1 Rectangular billiard . . . 17 4.2.2 Sinai billiards . . . 17 4.3 Open billiards . . . 19
4.3.1 Source and drain . . . 19
4.3.2 Antenna added . . . 22
4.4 Statistics extracted from our models . . . 25
4.4.1 Spacings between energy levels . . . 25 ix
x Contents 4.4.2 Wave function statistics . . . 26
5 Conclusions and future work 31
5.1 Concluding comments . . . 31 5.2 Future work . . . 31
A Derivation of the energy flow 33
Chapter 1
Introduction
In the past few years, interest has grown regarding a certain type of systems, which are called billiards. A vast amount of numerical studies of quantum mechanical and microwave billiards [1]-[3] and also simulations for electrical resonance circuits [4] have been performed with similar results.
The history of billiard experiments goes as far back as the end of the eighteenth century when Ernst Florens Chladni (1756-1827) made an experiment which can be considered as precursor to todays research. He had dust randomly distributed on glass or metal plates, which he set in vibration by means of a violin bow. He then studied how the dust arranged itself in characteristic figures on the plates. What he found there was the nodal lines of the standing wave which was created in the plates, i.e. where the amplitude of the wave function equals zero. He had “made the sound visible”.
Acoustic research plays a big part in the engineering area. However, within this area of acoustics, experiments and simulations [5, 6] are more rare but experiments regarding acoustic chaos were quite recently studied by Schaadt [7, 8].
The aim of this report is to examine the energy flow in acoustic, irregular systems which includes imaginary potentials mimiking source and drain and an antenna. The antenna represents a measuring probe, and it is interesting to see how the acoustic wave is affected when a measurement is made on it. We will also study the behavior of the statistics for eigenvalues and -functions.
The disposition of this thesis begins with a chapter dealing with the theory con-cerning the acoustics and statistics used in the report. The next part of the report is a chapter regarding the model used in the calculations, boundary conditions and visualisation. At last but not at least we present our results.
Chapter 2
Theory
2.1
Billiards in general
In this chapter we will present the foundations of billiards and chaos.
2.1.1
Classical chaos and billiards
This work will focus on three types of two-dimensional billiards. Theoretically a billiard is a perfect billiard table on which a ball bounces from wall to wall without experiencing any friction or other energy loss. In this work we study acoustic waves instead of particles, but the equations are still the same. Analytical expressions for the wave functions and eigenvalues may be derived if the billiard is, for example, rectangular. Then the solutions can be described by two independent equations, one describing the horizontal- and one describing the vertical movement.
a)
b)
c)
Figure 2.1. Three different types of billiards; (a) rectangular billiard (b) Sinai billiard
with broken corner (c) Sinai billiard with internal hole.
Figure 2.1 shows the different billiards we will use in this work. The rectangular billiard is regular since we can separate the different degrees of freedom.1 This
1
The movements in x- and y-direction.
4 Theory billiard is mostly interesting because of the possibility to compare the numerical results with analytical calculations. The rectangle with a piece of one corner gone and the rectangle with a circular hole inside are called Sinai billiards. The Sinai billiards are asymmetric and hence the variables can not be seperable in their degrees of freedom, they are chaotic.
There is an extreme oversensitivity for initial conditions (i.c.) in chaotic sys-tems. Since, in reality, it is impossible to start experiments in entirely the same way, different solutions will be obtained for each run. In an entirely symmectric geometry a tiny change in the i.c. would not affect the result very much, but for a chaotic system even small changes would create a completely different path. There are two types of chaos, i.e. if we have a system with no energy loss, we speak of deterministic chaos, for example, mecanichal systems which obey the laws of classical physics, such as moons and asterioids in our solar system, and in the case of energy loss we have dissipative chaos [9], for example turbulence in gases and fluids.
Time reversal symmetry (TRS)
Time-reversed acoustic process is possible with the aid of an array of microphones and loudspeakers. This array can re-create sound and send it back to its source as if time had been reversed in a practical sense. Any sound that goes into the trans-ducers comes back, but played in reverse. The difference between time-reversed acoustics and the simple procedure as playing a tape backward is that in time-reversed acoustics the sound is projected back exactly toward its source.
In an experiment where the time-reversal process was studied through a medium analogous to a chaotic pinball machine, interesting results have been obtained. The time-reversal process is surprisingly stable when it comes to acoustics. A parti-cle follows a well-defined trajectory and a small change in the initial velocity or position might make the particle to miss an obstacle and then totally change its trajectory. On the other hand a wave travels along all possible trajectories, and that makes it much more steady because the wave amplitude results from the in-terference of all those paths. Thus, in chaotic environments, wave physics is more robust than particle physics [10].
2.2
Acoustics
The basic concepts of acoustics are presented here. For convenience we only sum-marize the aspects needed for this work.
2.2.1
Basic acoustics
The time independent Helmholz equation describes the properties of a wave func-tion in a classical mechanical system. It may, for example, describe elastic waves in solids including vibrating strings, bars and membranes. It also appears in acoustics [11], where the time independent Helmholz equation is of the form
2.2 Acoustics 5 where ∇2= ∂ 2 ∂x2 + ∂2 ∂y2 (2.2)
is the two-dimensional Nabla operator, ψmn(x, y) is the amplitude function of the
mnth resonance and kmn is the associated wave number
kmn2 =
ω2 mn
v2 (2.3)
for the mnth eigenmode, v is the phase velocity of waves and ωmnthe angular
fre-quency for this eigenmode. In this thesis the time independent Helmholz equation is used since we are only interested in studying standing waves.
Analytical expressions
In this thesis we will compare our results with analytical results for different eigen-modes. Here we will give the analytical expressions for the wave function and frequency of the wave in the rectangular billiard. We will not go through the derivations of these expressions, which can be found in [12]. For a gas in a 2D box with a width Lxand lenght Ly, the expression for the wave function ψmn(x, y) is
ψmn(x, y) = A · cos mπ Lx x cos nπ Ly y , (2.4)
where A is a normalization constant. The expression for the nodal frequency ωmn
is ω2 mn v2 = mπ Lx 2 + nπ Ly 2 (2.5) where m and n are resonance modes of the system (m,n=1,2,3,...). We seek the frequencies ωmn as ωmn= s mπ Lx + nπ Ly v. (2.6)
From equation 2.4 we may easily plot the analytical solutions. The nodal structure of three low modes are presented in figure 2.2.
6 Theory
2.2.2
Imaginary potential and energy flow
In this work we want to look at the energy flow if we disturb the wave somehow, we will use an imaginary potential to simulate transmitters and an antenna. An interesting discussion about the conservation of the position probability density when a well is present were made by Schiff [13]. The imaginary potential helps us to model the flow, since, as in other areas of physics,2 the imaginary part of
the potential denotes an emission or absorption of particles. This was studied by Feshbach et al. in connection with inelastic nuclear scattering as far back as the fifties [14].
Energy flow
The probability energy flow of the system is given by the Poynting vector [2]
S= Imψ∗(r)∇ψ(r). (2.7)
For a derivation of equation 2.7, see Appendix A.
2.3
Microwave billiards
Figure 2.3. A
two-lead cavity.
Microwave cavities are the most common system described as billiards that billiard experiments have been performed on. Experiments on microwave cavities have been performed by Barth [3], and Kim [15]. A billiard shaped as in figure 2.3 was also studied recently using the finite difference method which we also will use [1].3 The equation used in that work
was the Schr¨odinger equation which is the quantum mechan-ical counterpart to the Helmholz equation. Please, note the similiarities between the Helmholz and Schr¨odinger equa-tions. The Helmholz equation for the perpendicular electric field E⊥, in a planar microwave resonator is [16]
h ∇2+ k2+ (nπ d ) 2iE ⊥= 0. (2.8) For frequencies υ < c
2d only modes with n=0 are possible. 4
So, for small d equation 2.8 reduces to
(∇2+ k2)E⊥= 0. (2.9)
Equation 2.9 has a similar form as the Schr¨odinger equation for a particle in a box, with infinite walls
∇2+2mE ~2
!
ψ = 0, (2.10)
2
For example in quantum mechanics.
3
This method will be further explored in next chapter.
4
Corresponding to wave numbers k < π
2.4 Statistics in acoustics 7 if k in equation 2.9 corresponds to q2mE
~2 in equation 2.10 and E⊥ corresponds
to ψ. Thus, in this particular case it is clear that the equations 2.9 and 2.10 are closely related.
2.4
Statistics in acoustics
Statistics are used to study a systems properties. Objects interesting to study might be if a system or a particular state is chaotic or if TRS is present, in this context it means that the wave function is complex or real [16, 17].
2.4.1
Distribution of eigenvalues
In this chapter we will study the statistics of spacings between eigenenergies. This is done after solving equation 2.1 in the case of a closed billiard, i.e. the case with no source and drain. What we do is line up all the eigenenergies, Ei, in
order of increasing energy. First we want to normalize the mean distance between the energies to one, which can be done by calculating the positive and normal-ized spacings, si = (Ei+1− Ei)/∆E, between neighbouring energies. In order to
visualize this in a histogram an array of slots is created, each slot representing intervals of spacings. We place each of the spacings in their correct slot and count the amount of spacings in each slot. When this is done we have a histogram, P(s), which we scale to have a total probability of one.
By studying these histograms we can extract important information about our system. For an integrable, i.e. non-chaotic, system with TRS the eigenfrequencies are distributed according to the Poisson distribution
P (s) = e−s (2.11)
and when the system undergo chaos effects, but still with TRS present,5 we find
the Wigner-Dyson distribution [9]
P (s) = sπ 2 e
−πs2
4 . (2.12)
The Poisson distribution has its maximun at zero. This means that there exist many nearby states, i.e. states with nearly the same or the same frequency. The Wigner-Dyson distribution on the other hand shows no degeneracies at all. This can be understood quite easily, by considering a wave in a square box, where the time-independent Helmholz equation can be separated in both directions, one realizes that many states are similar. For a chaotic system, on the other hand, the states come more randomly, the more chaotic the system is, the less amount of degeneracies present. This is known as level repulsion, i.e. the frequency levels repel each other as the degree of chaos increases [9, 16].
5
TRS is present because of the fact that we look at the squared absolute value of the eigen-functions.
8 Theory
2.4.2
Wave functions and amplitudes
We have two different possibilities for how the statistics for the spacings si can
follow. It all depends on whether TRS is present or not. If the system has TRS, Random Matrix Theory governs that the statistics follow the Gaussian Orthogonal Ensemble (GOE), and when not, the Gaussian Unitary Ensemble (GUE). The form of the wave and the type of statistics linked to this are connected, and the two cases can be distinguished just by computing statistics for the amplitude of the wave in each point.6
We want to study the amplitude for the wave ψ, and in order to do so we have to normalize ψ according to
Z
S
|ψ|2dS = 1 (2.13)
where S is the area or volume where the wave propagates. We can from this information obtain a plot P (ρ), where ρ = S|ψ|2. This gives the probability to
find a certain amplitude. The plot is made by measuring the amplitudes, and then counting how many amplitudes one has in certain intervals. According to RMT P (ρ) follows the Porter-Thomas distribution given by
P (ρ) =√1 2πρe
−ρ
2 (2.14)
for GOE statistics, and
P (ρ) = e−ρ (2.15)
which is the Rayleigh distribution for GUE statistics [9]. If we divide the complex wave function in two parts,
ψ = u + iv (2.16)
where u is the real and v the imaginary part, treated as Gaussian fields, we can also extract a Gaussian distribution given by
P (s) = 1 σ√2πe
−(s−µ)2σ2 (2.17)
for u and v, σ is the standard deviation and µ the mean value of the desired curve [18].
If the Porter-Thomas distribution is found in a system,7 the real part can be
made dominant over the imaginary by multiplying the wave function by eiα. We
let h...i indicate the mean value, α can be written as α = 1
2arctan
2huvi
hu2i − hv2i (2.18)
6
Since both GOE and GUE are related to chaotic modes, we can not expect fully reliable statistics studying them.
7
2.4 Statistics in acoustics 9 giving a rotation of all points by the angle α in the complex plane [17, 19]. This rotation is in fact derived from the expression huvi = 0, thus also ensuring that the fields are independent. Note that it is also possible to make the imaginary part dominant by adding one more rotation with an odd multiple of π/2.
If, on the other hand, TRS is not present and we have an intermediate dis-tribution or the Rayleigh disdis-tribution, the real and imaginary parts of the wave function are of equal magnitude. Hence, neither of the two fields can be made dominant over the other, and the rotations only make the fields independent [19]. Thus, if we have GOE statistics two phases are dominant, and if we have GUE there exists no such dominance.
Chapter 3
Modelling
In this chapter we present the billiards that we modeled and we also show how they may be implemented. MatLab was used both in the calculations and for the visualisation.
Since analytical expressions only exist for the rectangular billiard, all the cal-culations will be performed numerically.
It takes four steps to do this. First we must transform our continuous system to a discrete one, then apply the boundary conditions to this discrete system. When this is done we have achieved a matrix from which we will extract our frequencies (eigenvalues) and the corresponding wave-functions (eigenvectors). Finally the results are visualised.
3.1
Discretization
The continuous system has to be discretized in order to make our numerical calcu-lations. The two most common methods to do this are the finite difference method (FDM) and the finite element method (FEM). The FDM is very easy to under-stand and to implement, while FEM is harder. Why one would like to use FEM is because it gives faster calculations, although for this work we have chosen FDM because of its simplicity and numerical robustness.
3.1.1
Finite difference method
As the name of the method implies, the FDM uses finite differences to produce a grid of equidistant points.1
The derivatives in FDM can be calculated in several ways but the most common ones are the five points- and the nine points approximation. In this work the five point approximation is used which means that the derivative depends on the value at a given point (fi,j) and its four nearest neighbours.
1
Methods that uses non-equidistant points do exist but that will not be further discussed in this thesis.
12 Modelling a a 1 3 3 3 3 2 2 2 2
Figure 3.1. Grid showing the discretization of the system with FDM. The point 1 is
dependent on its nearest neighbours; 2. 3 are points on the boundary, thus with Neumann conditions, and a is the distance between the grid points.
The first two derivatives in the five point approximation are given by, with respect to x ∂f (x, y) ∂x → fi+1,j− fi−1,j 2a (3.1) ∂2f (x, y) ∂x2 →
fi+1,j− 2fi,j+ fi−1,j
a2 (3.2)
and with respect to y
∂f (x, y) ∂y → fi,j+1− fi,j−1 2a (3.3) ∂2f (x, y) ∂y2 →
fi,j+1− 2fi,j+ fi,j−1
a2 (3.4)
Thus for ∇2we have
∇2f (x, y) → fi+1,j+ fi,j+1− 4fai,j2 + fi−1,j+ fi,j−1 (3.5)
where a is the distance between the grid points.
3.2
Boundary conditions
Boundary conditions (b.c.) are necessary in order to model our systems. Assume a surface S with a boundary ∂S, and a value ψ(x,y) at the point (x,y) on S.
There exist two kinds of b.c. we may use, one is the Dirichlet b.c. which tells that the function should have a certain value at the boundary, most often zero. In this work we will use another type of b.c., the Neumann b.c., which instead tells that the normal derivative of the function should be zero at the boundary.
3.3 Solving the matrix 13
3.2.1
Neumann
The Neumann b.c. means that the normal derivative of the function should be zero at the boundary [11], i.e.
ψ0(x
b, yb) = 0; (xb, yb) ∈ ∂S (3.6)
Assume a point on the boundary at i=M, then the second derivative becomes ∇2f (x, y) →fM,j+1− 4fM,j+ fa2M,j−1+ 2fM −1,j (3.7)
The Neumann b.c. is neutral to the wave function, i.e., not repulsive nor attractive. This property gives that the Neumann b.c. is good to use when the boundary is open to the outside, for example for systems with source and drain. For further discussion about the Neumann b.c., see Fletcher et al. [12].
3.3
Solving the matrix
The FDM together with the Neumann b.c. gives a penta-diagonal matrix, see figure 3.2. The matrix represents the operator ∇2, and can be of three different
types depending on which of of the Sinai billiards that is managed. Solving the eigenvalue problem ∇2ψ
mn = −k2mnψmn gives the solution where we can extract
the frequency for the system for the eigenstate mn from k2
mn, and ψmnis the wave
function for that particular eigentstate. Because of its convenience in handling matrices, MatLab was a natural choice to solve this eigenvalue problem.
When the system is solved, the eigenfunctions have to be ordered according to some criterion. Since we are interested in the frequency, we choose to sort the states according to the increasing frequency. In some of our solutions we get a complex part of the frequency, however, we choose to sort only according to the real part. 5 10 15 20 25 30 35 5 10 15 20 25 30 35 −5 0 5 10 15 x 105
Figure 3.2. 6x6 point-system matrix with Neumann boundary conditions representing
the rectangular billiard.
3.4
Visualisation
There are several interesting methods of visualisation we want to examine. This section presents the different kinds we are interested in.
14 Modelling
3.4.1
Wave intensity
One thing we might be interested in is looking at the wave distribution, i.e. |ψ|2of
the system. The wave density plots show this and gives the probability of finding a certain amplitude. The color in the plots represents the wave density, with white color representing the highest density.
3.4.2
Energy flow arrows
The plots showing the flow arrows help us a great deal in understanding how the energy flows in the system, although they only show relative values, not the absolute magnitude of the flow. The plots were done with the MatLab function quiver, which given four matrices (x, y, dx, dy) plots the gradient of the matrices. The arrows show the direction and the relative magnitude of the flow.
Figure 3.3 presents to us where the potentials representing the transducers and the antenna are situated.
x y Ly Lx
-V’
-V’’
+V
Figure 3.3. The potentials used in this work. The positive V represents a transmitter
and the negative V0 a reciever, i.e. the energy flow in the x-direction. The negative,
small V00represents the effects of an antenna. The sum of the potentials equals zero, i.e.
V − V0− V00= 0.
3.4.3
Energy flow densities
In order to easily see where the highest energy flow densities are we use plots of the absolute value of the flow in each point. To do this we calculate |S| =qS2
x+ Sy2.
This is then plotted the same way as the wave intensity, i.e. the lighter the color, the higher currrent density.
Chapter 4
Results
In this chapter we present the results of the calculations for the three billiards: a) a rectangular billiard; b) a Sinai billiard with broken corner; c) a Sinai billiard with an internal hole. The first section contains general results about the model itself.
4.1
General results
In this section we present some general results for the model itself, i.e. tests with the rectangular billiard, for which there are analytical expressions with which we can compare our results.
4.1.1
Resolution
In order to decide which resolution to use, we performed some calculations on the system for which there exists analytical expressions for the wave functions and the energies, i.e. the rectangular billiard. In figure 4.1 we compare the frequencies of our calculations with the analytical expressions for the frequencies. It is clear that a grid with 17x25 points is not able to give the right frequencies except for the first few frequencies. A grid consistent of 37x55 points gives fairly good results but we have used the 59x87 point grid because it is even closer to the analytical values and the percentual error is smaller.
16 Results 0 20 40 60 80 100 120 140 160 180 200 0 1000 2000 3000 4000 5000 6000 7000 Eigenvalue number
Angular frequency [rad/s]
Analytical 17x25 point grid 37x55 point grid 59x87 point grid (a) Frequency. 0 20 40 60 80 100 120 140 160 180 200 0 2 4 6 8 10 12 14 16 Eigenvalue number
Relative error in frequency [%]
17x25 point grid 37x55 point grid 59x87 point grid
(b) Error in frequency.
Figure 4.1. Plots of frequencies for a standing wave, as a function of the eigenvalue
4.2 Closed billiards 17
4.2
Closed billiards
First of all we want to examine our system in its normal state without any distur-bance to be able to compare the changes when we add our potentials.
4.2.1
Rectangular billiard
In order to verify the numerical model used herein, we compare three example modes, figure 4.2, with corresponding analytical solutions, figure 2.2, and conclude that they are in excellent agreement
(a) (b) (c)
Figure 4.2. Three low modes of the rectangular billiard, plotted as |ψ|2
.
4.2.2
Sinai billiards
In all the calculations throughout this work the 96th eigenmode is used and we will not state this in the figures. In this section we present the standing waves in figure 4.3 (a), (c) and (e) and the intensity in (b), (d) and(f).
An interesting aspect to look at is the shape of the wave function. We want to study the nodal lines, where the amplitude is equal to zero, ψmn(x, y) = 0. For
the rectangular shaped billiard, these lines form a perfect symmetric net. If we instead consider a skew billiard, as the Sinai billiards, the nodal lines turn into an asymmetric complex pattern. For a complete asymmetric system, no nodal line cross another. Note that in the intensity plots one can clearly see the nodal lines, i.e. where the wave function equals zero.
18 Results −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5 x [m] y [m] −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 (a) w = 4.5413 · 103 rad/s −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5 x [m] y [m] 0.005 0.01 0.015 0.02 0.025 0.03 0.035 (b) −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5 x [m] y [m] −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 (c) w = 4.7549 · 103 rad/s −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5 x [m] y [m] 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 (d) −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5 x [m] y [m] −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 (e) w = 4.7146 · 103 rad/s −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5 x [m] y [m] 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 (f)
Figure 4.3. (a),(c) and (e) shows the real part of our wave functions at the 96th
eigenmode, (b),(e) and (f) maps |ψ|2
4.3 Open billiards 19
4.3
Open billiards
In order to create an energy flow through the system we open up our billiards and study two cases, when there is one in- and one outflow and when there also exists a well, representing an antenna. Experiments have shown that the presence of antennas broaden and shifts the resonances somewhat, but this does not change the distributions in chaotic systems [3].
In both cases the total in- and outflow are equal in size. For a better veiw how the energy flow through the system see figure 3.3 where the potentials simulating the transducers are shown. Please note that in the first case we have not included an antenna and thus V00= 0.
4.3.1
Source and drain
In this part of our work we have added two potentials representing transducers, one positive, V , and one negative, V0, to model a transmitter and reciever. This
gives rise to an energy flow in our system. In this case, when there only exist one inflow and one outflow, we have that V = −V0.
The in- and outflow potentials are located a bit below the center of the side of the billiard in order to reduce the possibility to be placed on a node. In figure 4.4 (a) and 4.5 (a) it is nice to see how the energy flow through the billiard. One can clearly see the higher intensity between the potentials and how it weakens the further along the y-axis and from the source and drain we go. Figures 4.4 (b)-(c) and 4.5 (b)-(c) does not show the same symmetry although it is possible to distinguish a relation between how the energy flows and the intensity of the wave from figure 4.3.
20 Results −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5
x [m]
y [m]
(a) w = 4.5413 · 103 rad/s. −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5x [m]
y [m]
(b) w = 4.7549 · 103 rad/s. −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5x [m]
y [m]
(c) w = 4.7146 · 103 rad/s.Figure 4.4.Current created by the in- and outflow, represented by the complex
4.3 Open billiards 21 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5
x [m]
y [m]
1 2 3 4 5 6 x 10−14 (a) −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5x [m]
y [m]
0.5 1 1.5 2 2.5 3 x 10−14 (b) −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5x [m]
y [m]
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 x 10−14 (c)22 Results
4.3.2
Antenna added
It is interesting to see how much we disturb the system when we perform measure-ments with a probe (antenna). Experimeasure-ments performed on systems with antennas experience problems with determination of flow distributions. The probe antenna gives rise to a leakage flow spoiling the statistical properties of the energy flow distribution.
We simulate the effect of an antenna by adding the imaginary potential V00,
and V = −(V0+ V00). By comparison between figure 4.6 and figure 4.7 it is clearly
visible that the energy flow changes when the antenna is added. The effect of the antenna is most noticable in our rectangular system, where it seems like the current is pushed away from the antenna. In the two chaotic Sinai billiards the visible effect is smaller although it is still present.
The influence from the antenna can be made smaller by using a considerable larger flow through the system [3]. This was also confirmed during the calculations. The potential representing the antenna is fairly large compared with the potentials representing the transducers, around 18% of the size of the positive transducer and around 22% of the size of the negative transducer. This is because we actually want it to affect the energy flow.
Different locations for the antenna were tested, and depending on where the antenna was located the effects of it varied. If the antenna was placed on a nodal line no effect on the current was noted. The antenna in the current plots in this thesis is situated where it will give an effect on the energy flow.
4.3 Open billiards 23 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5
x [m]
y [m]
(a) w = 4.5413 · 103 rad/s. −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5x [m]
y [m]
(b) w = 4.7549 · 103 rad/s. −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5x [m]
y [m]
(c) w = 4.7146 · 103 rad/s.24 Results −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5
x [m]
y [m]
1 2 3 4 5 6 7 8 x 10−14 (a) −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5x [m]
y [m]
0.5 1 1.5 2 2.5 x 10−14 (b) −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5x [m]
y [m]
0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10−14 (c)4.4 Statistics extracted from our models 25
4.4
Statistics extracted from our models
Acoustic resonators were studied by Schaadt [7, 8] where the same statistical properties as in this work were investigated. The statistics are only valid for modes in a certain class of symmetry. The procedure to isolate such a class is explored in [4]. Since this work mostly focuses on chaos effects we have chosen not to look at the symmetric billiard, but to only examine the two Sinai billiards. Thus this chapter will only present results for our Sinai billiards.
4.4.1
Spacings between energy levels
It is probable that the correspondence betwen angular frequencies ω and the en-ergy E in the quantum mechanical case is given by equations 2.9 and 2.10. For microwave resonators, it is stated that k2 = ω2/c2 corresponds to the energy E
in quantum mechanics. If this is the case, we should find the eigenmodes corre-sponding to the angular frequencies, square the angular frequencies and compute the spacings, and proceed as outlined above to produce a histogram.
The Wigner-Dyson distribution was discovered when analyzing the energy lev-els for atomic cores. Later on experiments and analytical simulations confirmed that the energy levels in all examined quantum mechanical system which are clas-sical chaotic follow the Wigner-Dyson distribution law. Thus, what we have here is a classical system which has been quantisized and as a consequence of this the energy levels of that system follow the Wigner-Dyson distribution law.
0 0.5 1 1.5 2 2.5 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 P(s) s (a) 0 0.5 1 1.5 2 2.5 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 P(s) s (b)
Figure 4.8.Statistics for the angular frequency spacings, corresponding to the spacings
between eigenenergies in quantum mechanics. (a) represents energies extracted from the first Sinai billiard and (b) is achieved from the second Sinai billiard and thus have a Wigner-Dyson distribution (equation 2.12). The histograms are generated of 4789 respectively 4755 spacings.
26 Results spacings would be distributed if the system would be symmetrical. Considering the shapes of the first and second Sinai billiards one can presume that the second is of a more chaotic nature than the first. If this is so, the spacings from the first Sinai billiard does not necessarily have to follow the Wigner-Dyson curve as strictly as the spacings from the second. One can actually see in figure 4.8 (a) that the histogram is pushed to the left, that is, the histogram is under the influence of both the Poisson and Wigner-Dyson distributions. By comparison with figure 4.8 (b) one can see that the histogram consisting of data from the second and more chaotic Sinai billiard is mostly controlled by the Wigner-Dyson distribution.
4.4.2
Wave function statistics
The procedure to create a histogram that show the probability of finding a certain amplitude is as follows. At first a mode is specified, giving the eigenfunction for that particular mode. The function values in each grid point are lined up, and normalized according to equation 2.13. When this is done, one just places the normalized function values in discrete trays. The result is yet another histogram which indicates how common the amplitudes of that single wave function are.
A nice way of showing the difference between states following either GOE statistics och GUE statistics is to study the phase plots. When GOE is present, there are only two phases dominant, that means that a phase plot are dominated by two colors only, and the difference in color corresponds to π. In our systems we find only distributions which follow GOE statistics, see figure 4.9 and 4.10.
4.4 Statistics extracted from our models 27 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5
x [m]
y [m]
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 (a) w = 4.7549 · 103 rad/s. −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5x [m]
y [m]
−3 −2 −1 0 1 2 3(b) The phase in each point.
0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
P(
ρ
)
ρ
(c) Wave function statistics. The amplitudes of the wave function are plotted along the x-axis and the probability of finding a certain amplitude is plotted along the y-axis.
Figure 4.9. Statistics regarding the first Sinai billiard. Figure (a) maps the intensity of
the wave function. (b) Shows the phase of the wave function in each grid point. (c) The curve is the Porter-Thomas distribution given by equation 2.14.
28 Results −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5
x [m]
y [m]
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 (a) w = 4.7146 · 103 rad/s. −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5x [m]
y [m]
−3 −2 −1 0 1 2 3(b) The phase in each point.
0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
P(
ρ
)
ρ
(c) Wave function statistics. The amplitudes of the wave function are plotted along the x-axis and the probability of finding a certain amplitude is plotted along the y-axis.
Figure 4.10.Statistics regarding the second Sinai billiard. Figure (a) maps the intensity
of the wave function. (b) Shows the phase of the wave function in each grid point. (c) The curve is the Porter-Thomas distribution given by equation 2.14.
The statistics for the real and imaginary parts of the amplitudes follow a Gaus-sian distribution, this has been validated for different modes. Figures 4.11 and 4.12 show distributions for the real and imaginary parts of the amplitudes for the two Sinai billiards, for a high mode. The plots show the distribution both before and
4.4 Statistics extracted from our models 29 −30 −2 −1 0 1 2 3 0.05 0.1 0.15 0.2 0.25 P(Re( ψ )) Re(ψ)
(a) Real part, before the rotation. σ = 0.9, and µ = −0.9. −3 −2 −1 0 1 2 3 0 0.05 0.1 0.15 0.2 0.25 P(Im( ψ )) Im(ψ)
(b) Imaginary part, before the rotation. σ = 0.9, and µ = −0.9. −30 −2 −1 0 1 2 3 0.05 0.1 0.15 0.2 0.25 P(Re( ψ )) Re(ψ)
(c) Real part, after the rotation. σ = 0.9, and µ = −0.9. −3 −2 −1 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P(Im( ψ )) Im(ψ)
(d) Imaginary part, after the rotation. σ = 0.045, and µ = 0.00001.
Figure 4.11. This figure presents statistics for the first Sinai billiard, with w = 4.7549 ·
103
rad/s. Figures (a)-(d) show the distributions of the real and imaginary part of our Gaussian fields, before and after the rotation by the angle α. The curves are Gaussian,
given by equation 2.17. The real part of the function spans over an energy spectra 1010
times larger than the imaginary part does.
after the rotation given by equation 2.18. The histograms representing the imagi-nary part of the wave function do look equal in size to the histograms representing the real part. This is because the plots are not scaled with respect to each other. The imaginary part is so much smaller than the real one so it would not be possible to see any noticable change after the rotation. As the plots show, the imaginary part becomes very small after the rotation. Although, since the real party is so much larger than the imaginary one, the rotation is pretty much unnecessary in practice.
30 Results −3 −2 −1 0 1 2 3 0 0.05 0.1 0.15 0.2 0.25 P(Re( ψ )) Re(ψ)
(a) Real part, before the rotation. σ = 0.9, and µ = −0.9. −3 −2 −1 0 1 2 3 0 0.05 0.1 0.15 0.2 0.25 P(Im( ψ )) Im(ψ)
(b) Imaginary part, before the rotation. σ = 0.79, and µ = −0.796. −3 −2 −1 0 1 2 3 0 0.05 0.1 0.15 0.2 0.25 P(Re( ψ )) Re(ψ)
(c) Real part, after the rotation. σ = 0.86, and µ = −0.9. −3 −2 −1 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P(Im( ψ )) Im(ψ)
(d) Imaginary part, after the rotation. σ = 0.045, and µ = 0.00001.
Figure 4.12. This figure presents statistics for the second Sinai billiard, with w =
4.7146 · 103
rad/s. Figures (a)-(d) show the distributions of the real and imaginary part of our Gaussian fields, before and after the rotation by the angle α. The curves are Gaussian, given by equation 2.17. The real part of the function spans over an energy
spectra 1010
Chapter 5
Conclusions and future work
5.1
Concluding comments
The objective of this work was to examine systems representing different kinds of billiards which includes imaginary potentials mimiking source, drain and an antenna. What we noted here was that the size of the antenna had to be fairly large, around 20% of the source and drain potentials, in order to give a visible disturbance on the energy flow. The tool used in these studies was based on finite difference approximations.
The electron cavity (see figure 2.3) studied by Hakanen [1] was modelled in a similar fashion to our systems. It also had a source and drain, and the equa-tion used was the the quantum mechanical correspondence to Helmholz equaequa-tion (equation 2.1), the Schr¨odinger equation (equation 2.10).
Statistics show that the first Sinai billiard is of less chaotic nature than the second one. Histograms showing the spacings between eigenenergies imply that the distribution for the first Sinai billiard is affected by both chaotic and non-chaotic distribution while the second Sinai billiard is mostly governed by a non-chaotic distribution.
One conclusion of this chapter is that our Sinai billiards have GOE statistics. Experiments have already shown that the time-reversal process is stable and thus our result was expected.
5.2
Future work
There are still interesting work to be done using the present model. One should improve the speed of the calculations so larger systems can be used. This would in turn allow higher frequencies to be included. This could be done by using a better eigenvalue solver for large systems.
In order to make comparison between our acoustic results and quantum theory more straight forward one could do the same analysis but in a system equivalent to figure 2.3. This system has already been studied using both electrical circuits
32 Conclusions and future work [4] and using quantum theory [1]. It would be interesting do study this both with and without an antenna.
One last thing that would be interesting is to apply the Schr¨odinger equation to the billiards in this work, yet another way to be able to perform analog studies between classical and quantum mechanical systems.
Appendix A
Derivation of the energy flow
The time-independent Helmholz equation can be considered as a complex ampli-tude ψ in a scalar single-frequencied wave field. Presume that the medium in which it propagates is isotropic. The field is expressed by the complex analytic signal1
φ(r, t) = ψ(r)e−iωt, (A.1)
where ω is the angular frequency and r = (x, y, z). In a source-free region, the complex amplitude satisfies
∂2ψ ∂x2 + ∂2ψ ∂y2 + ∂2ψ ∂z2 + ω2 v2ψ = 0, (A.2)
which we recognize as the three dimensional version of equation 2.1. The analytic signal φ may represent the complex time-varying sound pressure amplitude P , with which we can extract energetic quantities of the field. The sound potential energy density time averaged over one period is given by [20]
wpot=1
2κ|P |
2, (A.3)
where κ denotes the compressibility. The kinetic time-averaged energy density of the wave is given by
wkin =1 2ρV · V
∗, (A.4)
where V is the sound velocity vector and ρ is the density of the medium. Newton’s relation iωρV = ∇P (A.5) where ∇ = ˆx∂ ∂x+ ˆy ∂ ∂y + ˆz ∂
∂z, together with A.4 give
wkin= 1
2ρω2 · |∇P |
2. (A.6)
1
This derivation has earlier been done by Ebeling [5].
34 Derivation of the energy flow Now, we seek the time-averaged sound energy flux density, S. From Morse and Ingard [20] and a simple dimension analysiswe know that we can express S as P V . But P , V ∈ C so S= 1 2P ∗V +1 2P V ∗. (A.7)
We know from equation A.5 that
V = 1 iωρ∇P (A.8) and V∗= 1 −iωρ∇P ∗. (A.9)
So we have from equations A.7, A.8 and A.9 that
S= i
2ωρ(P ∇P
∗− P∗∇P ). (A.10)
We can express the energy and energy flux densities in terms of the analytic signal φ or the complex amplitude ψ and their gradients. For convenience, we suppress unimportant proportionality constants and define our energy densities with the aid of the complex amplitude. The potential energy density is given by
wpot= |ψ|2, (A.11)
and the kinetic energy density by
wkin = |∇ψ|2. (A.12)
and thus the energy flux from A.3, A.6, A.11, and A.12 by
S= i(ψ∇ψ∗− ψ∗∇ψ)/2. (A.13)
We can simplify this equation by using ψ = a + ib and ψ∗ = a − ib, which gives
that
S= i(a + ib)∇(a − ib) − (a − ib)∇(a + ib)/2. (A.14) We know that z = Re(z)+ iIm(z) where z in our case equals the expression within the brackets in equation A.14 and the real and imaginary parts are:
Re(z) = (a∇a + b∇b) − (a∇a + b∇b) = 0 (A.15)
Im(z) = (−a∇b + b∇a) + (−a∇b + b∇a) = 2(b∇a − a∇b) (A.16) which gives that equation A.14 can be written as
S= (a∇b − b∇a) (A.17)
which is the same as
S = Im(a − ib)∇(a + ib) (A.18)
i.e. equation A.13 can be written as
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