Quantum Chaos?

Benjamin Collar

Introduction

The term science is generally used in reference to a specific field of study, such as chemistry, biology, or physics. What one often overlooks is that all sciences have a similar purpose: to accumulate knowledge, systematize that knowledge, and formulate general truths that allow the scientist to make predictions about the world. General science is comprised of separate levels of specific studies, and the "enterprise of science involves investigating the laws at all levels, while also worki ng, from the top down and the bottom up, to build staircases between them."1 Each level of science is built upon the previous level and some correlation is necessary between each level in order to fully describe the world.

Because everything in the universe is comprised of atomic and sub-atomic particles, and quantum mechanics describes their interactions, quantum mechanics is the base level of science; it is the framework into which all other sciences must fit.2 The level directly above quantum mechanics is classical mechanics. The correspondence principle, a set of equations that translate classical systems to their quantum counterparts, provides a perfect example of a bridge between two levels of science. But there exist some problems with the correspondence principle, one of which is the existence of classically chaotic systems; chaos cannot be translated into quantum mechanics easily. Some kind of reconciliation for chaotic systems with quantum mechanics is imperative for a complete understanding of the universe.

Classically Chaotic Systems

Chaotic systems have been discovered over the entire spectrum of science, excluding quantum mechanics. All chaotic systems have three elements in common: determinism, sensitivity, and order.

The term determinism means that the system has some equations, often simple, that regulate the system's overall behavior. The Mandelbrot set, discovered in 1979 by Benoit Mandelbrot,3 offers an exquisite example of this principle (see figure A). The set is created by one iterative equation graphed on the complex plane, z --> z2 + c, where z begins at 0 and c is the complex number corresponding to the point being tested.4 The point is an element of the set if the equation converges. It is also possible to distinguish between degrees of convergence. For instance, the program graphing the set may be told to color the point blue if it converges by two or purple if it converges by three. The amazing complexity of the set, despite its blatant simplicity, is revealed when one magnifies the set. For example, if the set magnification point is at the junction between the two major spheres, the resulting graph would be one called the Julia set, reminiscent of sea-horse tails. Increasing the magnification again will reveal more Julia sets--similar in shape but never repeating. At some point, after repeated magnification, one will again find the Mandelbrot set, but it will not be identical to the initial image. It is apparent that infinitely complex syste ms can be determined by simple, compact equations.

Michael Barnsley developed a technique called the "chaos game,"5 which further elucidates the principle of determinism in chaotic systems. A set of rules that will create a graph is defined, and simple probabilities determine which rule will apply to any specific point. For example: 1. Move three inches north. 2. Move thirty percent closer to the center. After omitting the first fifty points, in order to equalize the chance distribution, a shape begins to emerge. The complexity of the resulting graph is directly proportional to the number of rules; if one increases the number of rules of the system the complexity rises accordingly. Barnsley created a fern with a set of only 28 equations with randomly selected starting points (see figure B). Furthermore, Barnsley realized that the creation process was reversible: any shape that had repetitive qualities, varying only on scale, could be represented by a set of simple equations.

The second principle of chaotic systems is sensitivity to initial conditions. Small, seemingly insignificant changes of initial parameters will lead to an enormously different and unpredictable final system. Richard Feynman elucidated this concept in the 1960's:6

"It's true classically that if we know the position and velocity of every particle in the world, or in a box of gas, we could predict exactly what would happen. And therefore the classical world is deterministic. Suppose, however, that we have a finite accuracy and do not know exactly where just one atom is, say to one part in a billion. Then as it goes along it hits another atom, and because we did not know the position better than one part in a billion, we find an even larger error in the position after the collision. And that is amplified, of course, in the next collision, so that if we start with only a tiny error it rapidly magnified to a very great uncertainty....Speaking more precisely, given an arbitrary accuracy, no matter how precise, one c an find a time long enough that we cannot make predictions valid for that long a time....The time goes, in fact, only logarithmically with the error, and it turns out that in only a very, very tiny time we lose all our information."

During the early 1960's, Edward Lorenz inadvertently provided statistical proof for Feynman's thought experiment.7 Using a Royal McBee computer he created a fairly accurate simplified weather system from twelve recursive differential equations. The computer generated two types of data: numerical and graphical. It calculated the data internally to six decimal places, but the output was only accurate to three. After running the program for some time Lorenz noticed a set of data to which he was particularly inclined and decided to enter the data by hand in order to see the system again. But the resulting system, after a short time, astonished him--it was measurably different than the original system and also completely unpredictable (see figure C). Lorenz's weather system provided the first concrete evidence of the importance of initial conditions to the final outcome of the system.

The third principle of chaos, that despite the amount of apparent disorder of any system there is some order-determining element, was realized by Heinz-Otto Peitgen and Peter H. Richter8 in their mathematical study of the magnetic transition of a system. They graphed the variations in the arrangement of the atoms in a transitional system in phase space. The area between the two types of systems--magnetic or non-magnetic--was their focus. Often areas between system types are chaotic, with order on both sides. Repeated magnification of the area revealed increasingly chaotic figures. The determining factor of the chaotic region, however, is the interesting aspect: at the greatest magnification, Peitgen and Richter found the Mandelbrot set.

The phase space graphics Peitgen and Richter used are often employed to help translate systems from one level of science to another. Lorenz offered another discovery to the science of complexity: the strange attractor. The idea of attractors when Lorenz published his findings was nothing new. Systems graphed in phase space often fall to attractors, the final state of the system. For instance, the phase space diagram of a pendulum (see figure D) spirals towards the center because of the dissipati ve force of friction. If one releases the pendulum it will rise to almost the same level on the opposite side. It returns, again not as high, and it repeats. The final state of the system is motionless--the system has arrived at the attractor.

The phase space diagrams of chaotic systems, however, do not terminate at an attractor. The phase space graph of the solution of Lorenz's weather equations has two attractors, and the lines never arrive at either, but they do approach each (see figure E). Logically it would seem that the lines at some point would intersect. But because of properties of differential equations the lines never do. If they did, the system would repeat itself although a chaotic system, such as weather, never does. This leads to the concept of infinite length within a finite area. A visual representation of this is provided by fractals. One of the first fractals was developed by Helge von Koch in 1904 (see figure F).9 The Koch curve, or Koch snowflake, is created by dividing each side of an equilateral triangle into thirds, placing a smaller equilateral triangle on the middle third of each side, and repeating the process infinitely. By tracing the perimeter of the snowflake, one will approach infinity--within a fini te area. This allows for infinite variation; a chaotic system graphed in phase space varies infinitely because it never repeats itself, but stays within a finite area because it is always self-similar.

An important property of fractals like the Koch curve and the Mandelbrot set is similarity across scale; by magnifying the system, almost-repeating images are found. Physical realization of fractals and similarity across scale was discovered by Kadanoff Libchaber in the 1977.10 Libchaber created a very small cell of liquid helium that allowed very few degrees of freedom in order to study convection currents. Two temperature probes were attached to the top of the cell and a heat source surrounded it. At only a few degrees above absolute zero, liquid helium easily changes to gas with small heat differences. The outer molecules evaporate first, causing rings of gas. Atoms are heated and rise but, upon energy loss when they reach the top, they cool and fall. Chaotic motion becomes apparent as the temperature increases. Uneven walls of the cell cause differences in the position of each helium atom. The differences in position lead to small irregularities in the structure of the rings which become more pronounced as the energy increases. At some temperature the loops will double. Increased energy will create more smaller-scaled loops and when the spectrum of the system is graphed, spikes occur, corresponding to the changes in the number and size of the rings.

The Transition to Quantum Mechanics

Bridges between classical and quantum mechanics are necessary to give an accurate description of the world. One conflict between classical and quantum mechanics that must be reconciled is that energy is a continuum in classical physics while, in most cases, energy comes in packets and is represented by spectra in quantum physics. When translating, the classical system is graphed in phase space and the length of the periodic orbits of the system correspond to the quantum spectral lines. The correspondence principle often suffices in the task of translating the systems, but problems arise as the systems become chaotic. Quantum mechanics calculates the probabilities of precise histories with no possibility of the indeterminacy present in chaotic systems:

"However, the difficulty, and a very deep one, arises form the fact that the former [quantum mechanics] is commonly accepted to be the universal theory, particularly, comprising the latter [classical mechanics] as the limiting case. Hence, the correspondence principle which requires the translation from quantum to classical mechanics in all cases including the dynamical chaos. Thus, there must exist a sort of quantum chaos!"11 Recently a new formula, the Selberg Trace formula, has been used; it "relates the lengths of periodic orbits as represented in 'phase space'...to combinations of quantum energy levels. Using the trace formula, physicists can convert a system's chaotic behavior in the classical world into predictions about the statistics of its quantum-mechanical spectrum."12

Martin Gutzwiller provides an illustration13 of the difficulties of translation between the two mechanics. Gutzwiller's example of what he terms "mild chaos" is centered on the sun-moon-earth system. Most moons in the solar system are easily described by Kepler's equations for elliptical motion. Our moon is the exception--it is strongly perturbed by the gravity of the sun and the earth. On the surface the system seems simple: it has only three degrees of freedom and one adiabatically slow perturbation.

The general equations for the motion and position of the moon have been known since ancient times, allowing for the predictions of full and new moons and eclipses. The equations of the moon in the XYZ plane is

x = a0 cos alpha0 + a1 cos alpha1 + ...

y = a0 sin alpha0 + a1 sin alpha1 + ...

z = c0 sin gamma0 + c0 sin gamma1 + ...

The coefficients ai and ci depend on the 6 initial conditions of the system. The angular variables alpha and gamma are all of the same form, alpha = omega t + theta where both the frequency omega and the phase theta are linear combinations with integer coefficients of the four basic frequencies and their associated phases.14

Unfortunately, new terms for each coordinate are being continuously discovered and it has been found that the coordinate equations do not converge, so the conditions that determine the position of the moon neither are nor ever will be fully known. This seems to suggest chaos, but Gutzwiller goes on to explain a more subtle form of complexity:

"The requirements of modern observations demand a precision of 10-10 in the coordinates of the moon. Occulations of fixed stars by the moon can be timed with the precision of the apparent star diameter, ~10-5 arc seconds. Echoes of light pulses from the reflectors on the surface of the moon give distances correct to several centimeters. But at this point, one enters the realm of slight chaos.

"First there is the problem of proliferating small terms, as discussed by Gutzwiller. The root of the sum of squares for all coefficients smaller than 10-10 turns out to be larger than 10-9. It becomes therefore necessary to compute all terms down to a precision of 10-12. Since even such small coefficients should be known with 2 decimals accuracy, one has to go all the way to 10-14 just to manage the noise from the small terms in the expansion."15 He goes on to say that, regardless of whether the moon's motion is chaotic or not, the determinism in its equations gets lost--the system is not as simple as it appeared. Gutzwiller's "mild chaos" is a prime example of the difficulty of finding equivalent equations in quantum and classical mechanics. It points out our inability as scientists to be precise enough in our measurements to make accurate predictions.

One approach to solving the chaos-quantum mechanics problem is the use of random matrices. If the system is chaotic, the chaos must be represented somehow in the Hamiltonian of the system. The classical Hamiltonian can be regarded as a matrix and the elements within the matrix will be determined by the physical models of the system. In a chaotic system the elements are exceedingly hard to determine, but mathematical methods permit some elements to be replaced with random entries, allowing one to stud y the statistical properties of the matrices.16 As a result of this theory two sets of matrices have been developed, the Gaussian orthogonal ensemble (GOE) and the Gaussian unitary ensemble (GUE), both of which only apply to chaotic systems. Non-chaotic systems have statistical properties determined by the Poisson ensemble. When a non-chaotic system becomes chaotic, there is a translation formula that relates Poisson data to GOE data.

Two classically chaotic systems that are often studied are the spectrum of actions of a Rydberg atom in a uniform field and compound nuclei spectrum. The first system is soluble only when the strength of the magnetic field is zero or infinity.17 As the field increases the phase space diagram reveals an increasing number of chaotic packets. At some field strength the entire graph becomes chaotic. The spectrum of the system is given in figure G, and one notices an increased level of repulsion as the field increases. The spectral lines fluctuate randomly. Because this in a non-integrable system, there is a mathematical transition between Poisson and GOE. The spacing and rigidity of the spectrum is compared to these in figure H.

Fluctuations in the spectrum of compound nuclei are studied by firing a neutron at the nuclei and measuring the resulting energy. In a 166Er atom the distribution of the spectral lines are smooth for about 108 levels, but at 109 the lines begin to randomly fluctuate. Both Poisson and GOE statistics are shown in figure I.

Another possible solution to the translation problem is the Riemann Zeta function:

zeta(s) = 1 + 1/2s + 1/3s + 1/4s + ... + 1/ns + ...

and equivalently:

zeta(s) = 1/(1-2-s)(1-3-s)(1-5-s)(1-7-s) ... (1-p-s) ...

where p is a prime number. Riemann discovered that this equation can also be written as a function of zeroes:

zeta(s) = f(s)(1-s/zeta(1))(1-s/zeta(2))(1-s/zeta(3)) ... (1-s/zeta(n)) ...

where zeta(n) are the zeroes of the function and f(s) is a simple fudge factor. Riemann realized that this function, after finding the zeroes, will lead to precise results of the location of prime numbers.

Physicists have discovered many potential uses for the Riemann Zeta function. Certain aspects of the equation seem to be very similar to values in quantum mechanics. First of all, the logarithm of the primes given by the function coincide with the lengths of periodic orbits of some classical systems. Second, the equivalency of the two infinite products looks suspiciously like the Selberg trace formula. Finally, the zeroes of the function correspond to the density of the spectral lines in the quantum system. With the advent of computers the zeroes of the function have become increasingly easy to compute--thus making the Riemann Zeta function an easily calculable translating scheme for chaotic systems (see figure J).18

Plotting the Zeta zeroes against the true spectrum of complex nuclei (see figure K) has increased physicist's resolve in thinking that the Riemann Zeta function provides an accurate description of the chaotic system. Similarly, there is a mathematical translation between the Riemann Zeta function and GUE (figure L), which in some systems, like the Rydberg atom in a magnetic field, can be translated to GOE (figure M).

Conclusion

The ultimate goal of science is to correctly describe the world in all of its various forms. Bridges are created with each new discovery; the search for more bridges continues. Links to concepts that are not often directly associated with the world, like mathematics, are being discovered too. The random matrix theories and the Riemann Zeta function are methods to not only bridge the gap between mathematics and physical reality, but also to reconcile the chaotic systems of the natural world with the sy stem scientists consider to be the true description of the world.

1 Gell-Mann, Murray, The Quark and the Jaguar, (New York: W.H. Freeman and Co., 1996) 112.

2 Gell-Mann, 6.

3 Gleick, James, Chaos: Making a New Science, (New York: Penguin Books, 1987) 223.

4 Gleick, 227.

5 Gleick, 237.

6 Richard Feynman, The Feynman Lectures on Physics, Vol. 3 (Massachusetts: Addison-Wesley Publishing Company, 1965) 2-9.

7 Gleick, 16.

8 Gleick, 230.

9 Gleick, 98.

10 Gleick, 202.

11 W. Dieter Heiss ed., Chaos and Quantum Chaos (Berlin: Springer-Verlag 1992) 5.

12 Barry Cipra, "Prime Formula Weds Number Theory and Quantum Mechanics," Science 274 (1996): 2014.

13 Casati, Giulio, Chaotic Behavior in Quantum Systems, (New York: Plenum Press 1983) 153.

14 Casati, 155.

15 Casati, 156.

16 Heiss, 229.

17 Seligman, T.H. and Nishioka, H., eds., Quantum Chaos and Statistical Nuclear Physics, (New York: Springer-Verlag 1986) 34.

18 Cipra, 2015.


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