Chaos In Physics

New developments in science that come under the general heading of "chaos" have been described as revolutionary and the basis for a genuinely new paradigm. It has brought about the realization that instead of understanding most of nature in principle, science has really addressed only the very restricted subset of phenomena that can be analyzed by simple methods. Chaos has had a broad impact in diverse disciplines, influencing mechanics, astronomy, solid-state physics, ecology, meteorology, and biology, as a few examples. For those interested in a popular book on the subject, Chaos: Making a New Science (Gleick 1987) is probably the best, with Fractals in Your Future (Lewis 1991) being particularly appropriate for teaching high-school level students. At a slightly more advanced level is Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (Schroeder 1991).

Photos of the Mandelbrot Set (FRACTINT. PROGRAM), a classic example of chaos.

To understand the revolutionary nature of chaos requires some history. In the 1600s Newton developed classical mechanics to describe the motions of bodies in our solar system. In the 1800s Laplace extended the ideas of classical mechanics to suggest determinism — the motion of every body in the universe is completely determined. As a result, "Universal, immutable laws, absolute precision, and strict predictability were ideas that were habitually believed to characterize science." Newtonian physics assumed "that the initial data determine the future, unambiguously, uniquely, and forever ... [and] that the qualitative aspects of the motion are not too sensitive to the precise initial data." Newton's laws could be solved exactly for 2 bodies, such as the sun and the earth. For a system containing more than 2 bodies, such as the earth-moon-sun system, accurate approximations were used. More involved problems were — so was the common belief — only technically different from the special examples. Newtonian physics was a dramatic success in predicting the return of Halley's comet in 1757 after its observation in 1682, and in predicting a new planet (Neptune) based on the observed irregularities in the orbit of Uranus. "These ideas were so successful and suggestive that they stimulated the search for similar laws in all of physics (and most of science)" (Dresden 1992).
Unfortunately, "the general attitude toward classical physics was based on an uncritical, unanalyzed acceptance of the ideology of Newtonian physics." "It is not clear whether physicists and astronomers were aware of the tenuous mathematical basis of their (often unspoken) beliefs, but it is pretty clear that they didn't worry too much about it" (Dresden 1992). At the turn of the last century, Poincare in his The New Methods of Celestial Mechanics wondered about the stability of solar systems. He "discovered that with even the very smallest perturbation, some orbits behaved in an erratic, even chaotic way. This was true even for a closed system, that should be particularly amenable to analysis by classical mechanics. Quantum mechanics and the Heisenberg uncertainty principle yielded indeterminacy in another area a few years later, and Poincare's ideas were forgotten for a time. "Small wonder, since even Poincare had abandoned the ideas, saying, 'These things are so bizarre that I cannot bear to contemplate them'" (Briggs & Peat 1989). Poincare's ideas have only come to the forefront again in the last 15 years or so.
Now classical mechanics is being reevaluated. It has been found that "properties were inferred from the detailed examination of very few examples — maybe five or six altogether." "... all the general features attributed to classical mechanics are in general wrong. The exactly soluble examples are not generic; they are in fact quite atypical." "... chaotic behavior, contrary to earlier beliefs, was a rather general property and not a pathological feature of some contrived system," and even relatively simple systems can exhibit frightfully complex behavior (Dresden 1992).
For chaotic systems a slight imprecision results in indeterminism. This slight imprecision may result from uncertainty in the initial data, or from approximate calculations based on perturbations of an exact solution. For such systems, predictive errors develop exponentially with time and an initial, small imprecision eventually results in total loss of predictability (Davies 1990).
Examples of chaotic behavior are numerous. The famous Butterfly Effect suggests "that a butterfly stirring the air today in Peking can transform storm systems next month in New York" (Gleick 1987). Population dynamics of rabbits can be affected in unpredictable ways by small changes in food supply. Earthquakes, snow avalanches, and dinosaur extinctions have been studied using chaos methods. One simple example involved adding sand to a sand pile on a 4 cm plate, one grain at a time. Sand avalanches would occur after as few as one or as many as several thousand grains were added to the pile. The addition of a sand grain would cause an avalanche in unpredictable ways (Bak & Chen 1991).
Chaos theory brings attention to the fact that errors in describing the future based on present approximate data in classical mechanics develop exponentially with time. It is possible that errors in extrapolation from present earth-history data into the past also develop exponentially with time and therefore, less is known about the past than previously realized.

Benjamin L. Clausen

Selected References

• Bak, P. and K. Chen. 1991. Self-organized criticality. Scientific American 264(1):46-53.
• Briggs, J. and F. D. Peat 1989. Turbulent mirror. Harper & Row, New York, pp. 26-29.
• Davies, P. 1990. Chaos frees the universe. New Scientist 128(1737):48-51.
• Dresden, M. 1992. Chaos: a new scientific paradigm — or science by public relations? The Physics Teacher 30(1):10-14.
• Gleick, J. 1987. Chaos: making a new science. Viking Penguin. New York. 352 pp.
• Lewis, R.S. 1991. Fractals in your future. Media Magic, Nicasio, California. 265 pp.
• Schroeder, M. 1991. Fractals, chaos, power laws: minutes from an infinite paradise. Freeman, New York. 429 pp.

New Law of Physics Could Explain Quantum Mysteries

Since the early days of quantum mechanics, scientists have been trying to understand the many strange implications of the theory: superpositions, wave-particle duality, and the observer’s role in measurements, to name a few. Now, a new proposed law of physics that describes the geometry of physical reality on the cosmological scale might help answer some of these questions. Plus, the new law could give some clues about the role of gravity in quantum physics, possibly pointing the way to a unified theory of physics.

Tim Palmer, a weather and climate researcher at the European Centre for Medium-Range Weather Forecasts in Reading, UK, has been interested in the idea of a new geometric framework for quantum theory for a long time. Palmer’s doctoral thesis was in general relativity theory at Oxford University in the late 1970s. His studies convinced him that a successful quantum theory of gravity requires some geometric generalization of quantum theory, but at the time he was unsure what specific form this generalization should take. Over the years, Palmer’s professional research moved away from this area of theoretical physics, and he is now one of the world’s experts on the predictability of climate, a subject which has considerable input from nonlinear dynamical systems theory. In a return to his original quest for a realistic geometric quantum theory, Palmer has applied geometric thinking inspired by such dynamical systems theory to propose the new law, called the Invariant Set Postulate, described in a recent issue of the Proceedings of the Royal Society A.

As Palmer explained to PhysOrg.com, the Invariant Set Postulate is proposed as a new geometric framework for understanding the basic foundations of quantum physics. "Crucially, the framework allows a differentiation between states of physical reality and physical 'unreality,'" he said.

The theory suggests the existence of a state space (the set of all possible states of the universe), within which a smaller (fractal) subset of state space is embedded. This subset is dynamically invariant in the sense that states which belong on this subset will always belong to it, and have always belonged to it. States of physical reality are those, and only those, which belong to this invariant subset of state space; all other points in state space are considered “unreal.” Such points of unreality might correspond to states of the universe in which counterfactual measurements are performed in order to answer questions such as “what would the spin of the electron have been, had my measuring apparatus been oriented this way, instead of that way?” Because of the Invariant Set Postulate, such questions have no definite answer, consistent with the earlier and rather mysterious notion of “complementarity” introduced by Niels Bohr.

According to Palmer, quantum mechanics is not itself sufficiently complete to determine whether a point in state space lies on the invariant set, and indeed neither is any algorithmic extension to quantum theory. As Palmer explains, in quantum theory, states associated with these points of unreality can only be described by abstract mathematical expressions which have the algebraic form of probability but without any underlying sample space. It is this which gives quantum theory its rather abstract mathematical form.

As well as being able to provide an understanding of the notion of complementarity, the two-fold ontological nature of state space can also be used to explain one of the long-standing mysteries of quantum theory: superpositions. According to the Invariant Set Postulate, the reason that Schrodinger’s cat seems to be both alive and dead simultaneously is not because it is, in reality, in two states at once, but rather because quantum mechanics is ignorant of the intricate structure of the invariant set which determines the notion of reality. Whichever point (alive or dead) lies on the invariant set, that one is real. The notion of quantum coherence, which is reflected in the concept of superposition, is, rather, carried by the self-similar geometry of the invariant set.

With superposition seemingly resolved from the perspective of the Invariant Set Postulate, other aspects of quantum mechanics can also be explained. For instance, if states are not in superpositions, then making a measurement on the quantum system does not “collapse the state” of the system. By contrast, in Palmer’s framework, a measurement merely describes a specific quasi-stationary aspect of the geometry of the invariant set, which in turn also informs us humans about the invariant set.

The Invariant Set Postulate appears to reconcile Einstein’s view that quantum mechanics is incomplete, with the Copenhagen interpretation that the observer plays a vital role in defining the very concept of reality. Hence, consistent with Einstein’s view, quantum theory is incomplete since it is blind to the intricate structure of the invariant set. Yet consistent with the Copenhagen interpretation, the invariant set is in part characterized by the experiments that humans perform on it, which is to say that experimenters do indeed play a key role in defining states of physical reality.

Yet another quantum mechanical concept that the Invariant Set Postulate may resolve is wave-particle duality. In the two-slit experiment, a world where particles travel to areas of destructive interference simply does not lie on the invariant set, and therefore does not correspond to a state of physical reality.

Among the remaining mysteries of quantum mechanics that the Invariant Set Postulate might help explain is the role of gravity in quantum physics. As Palmer notes, gravity has sometimes been considered as an objective mechanism for the collapse of a superposed state. However, since the Invariant Set Postulate does not require superposed states, it does not require a collapse mechanism. Rather, Palmer suggests that gravity plays a key role in defining the state space geometry of the invariant set. This idea fits with Einstein’s view that gravity is a manifestation of geometry. As such, Palmer suggests, unifying the concepts of non-Euclidean causal space-time geometry and the fractal atemporal geometry of state space could lead to the long-sought theory of “quantum gravity.” Such a theory would be very different from previous approaches, which attempt to quantize gravity within the framework of standard quantum theory.

Palmer’s paper is an exploratory analysis of this Invariant Set Postulate, and he now hopes to develop his ideas into a rigorous physical theory. Just as global space-time geometric methods transformed our understanding of classical gravitational physics in the 1960s, Palmer hopes that the introduction of global state space geometric methods could give scientists a deeper understanding of quantum gravitational physics. And, as suggested above, combining these two types of geometry might help lead to the long-sought unified theory of physics.

More information: T.N. Palmer. “The Invariant Set Postulate: a new geometric framework for the foundations of quantum theory and the role played by gravity.” Proceedings of the Royal Society A. doi:10.1098/rspa.2009.0080

PhysOrg

Scientists propose lab-grade black holes

One day, scientists may create the ultimate tempest in a teapot — an artificial black hole in a millimeter-long gadget. Such laboratory-grade black holes may illuminate enigmatic physical properties of their wild galactic counterparts, all from the safety of a lab bench, a study to appear in Physical Review Letters suggests.

“For black holes, we just don’t understand the physics at all,” says physicist William Unruh of the University of British Columbia in Vancouver, Canada, who was not involved in the new study. The prospect of conducting actual experiments on systems resembling black holes is exciting, he says. “Belief is not the same as doing an experiment.”

Mysterious black holes were originally thought to gobble up everything around them, including light (hence the name). But in the 1970s, British physicist Stephen Hawking predicted that because of quantum effects, these voracious monsters should emit photons. Right on the brink of the black hole, these photons “are so energetic that they go beyond what we understand,” says study coauthor Miles Blencowe of Dartmouth College in Hanover, N.H. Such emitted photons, known as Hawking radiation, have not yet been caught in the wild, nor have they been simulated in an experiment, leaving knowledge of their basic properties — and existence — in limbo.

In the new study, the researchers propose using a series of tiny, cold superconducting devices called SQUIDs in a linear, train-track–shaped arrangement to create a black hole analog. “But unlike a black hole out in space, we know the physics of this system,” says study coauthor Paul Nation, also of Dartmouth College.

Particles inside a black hole’s boundary, called the horizon, get sucked into the depths of the black hole, while particles outside the horizon can escape. Blencowe likens the horizon to a steep waterfall, where a fish above the drop can swim at normal speeds, but once a fish hits the fast-flowing water in the waterfall, it is swept down into the water below.

Similarly, the proposed system also creates a horizon, in the form of an electromagnetic wave that moves across the device in response to a magnetic pulse. Photons behind this horizon are trapped, while photons ahead of it move normally. By detecting and studying the photons that emerge from the device, researchers hope to have a better idea of what happens to particles near the edge of a black hole, both those that escape and those that are pulled in.

Changing the strength of the horizon-creating magnetic pulse may create conditions that fluctuate, making a system that simulates “shaking spacetime,” Nation says. Watching how photons behave in such a quantum system may answer some basic questions about the quantum nature of gravity, he says.

Building the new system has many challenges. “All of these experiments have a long way to go before they’ll deliver their promise,” comments Unruh, who has proposed a black hole analog that relies on sound waves.

Nation says that stringing together the 4,000 or so SQUIDs needed to create the artificial black hole would be a difficult endeavor. The largest string built so far is only 400 units long. Another hurdle to creating this system is designing a detector sensitive enough to catch single photons that would have a frequency much lower than that of visible light. “People are close to making a detector, but technically, it hasn’t been done,” says Nation.

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