Wednesday, August 29, 2012

Thoughts on P. W. Anderson's "More is Different"

Philip Anderson wrote a well-known article for Science in 1972 entitled "More is Different" whose goal was to refute the "constructionist hypothesis," i.e. the idea that all phenomena can be explained by a small set of fundamental laws. Presumably, these were the laws that govern elementary particle interactions. The constructionist hypothesis states that everything, from cellular biophysics to human thought processes, can be understood in terms of these laws so long as one is sufficiently clever in applying them. This hypothesis also leads many scientists to consider other fields as applied subsets of the fundamentals, such as biology existing as a form of applied chemistry, which would be just applied many-body physics and so-on down the line until particle physics is reached again.

Anderson claimed that, contrary to the constructionist hypothesis, new and "fundamental" science is performed at each level of the logical hierarchy of scientific fields and that this is because of the emergence of unexpected behavior at each level of complexity. His primary arguments lay with many-body physics and the idea of broken symmetry. As a system becomes more complex (that is, it takes on more components or the interactions between components become more intricate), it seeks to minimize the interaction energy between its components, which leads to a reduction in the symmetries of the components and an entirely different behavior of the system.

One example of emergent behavior in many-body physics is a crystal lattice, whereby translational and rotational symmetry is reduced by the ordered arrangement of atoms. Instead of a continuous translation or rotation, space must be shifted by an integer amount before the lattice looks the same again, and so these symmetries are reduced. The behavior that emerges from this is rigidity. If certain regions of the crystal experience a force, then the entire crystal moves as a result.

Another example—which demonstrates the unpredictability of emergent behavior—from many-body physics is the ammonia molecule. The nitrogen atom in ammonia undergoes inversion at a rate of roughly 30 billion times per second, which means that the nitrogen atom flips between its location above and below the plane containing the hydrogen atoms. Quantum mechanically, the stationary state of the molecule is a superposition of the two states representing the location of the nitrogen atom. This stationary state is symmetrical and represents what is actually measurable about the molecule. However, the understanding of inversion as a superposition of two unsymmetrical and unmeasurable states required intellectual machinery that was independent of the fundamental rules of atoms. Anderson's argument here suggests that human intuition led to the understanding of inversion, not the laws of physics, which at the fundamental level deal with symmetries and their consequences.

On a minor level, Anderson notes that scale and complexity are what lead to faults with the constructionist hypothesis. He also cautions that the nature of emergence at one level of complexity may not be the same at other levels.

My only question from this article is exactly what does fundamental mean? He seems to assume that fundamental science is always good science, so with his arguments chemists, biologists, and even psychologists can use the word to describe their work and win back their prestige from the particle physicists. However, it also might suggest that any scientific work is fundamental, thereby reducing the word's value and meaning.

Tuesday, August 28, 2012

Notes from the Chaos Cookbook, Chapter 15

I've skipped ahead to this short chapter in the Chaos Cookbook since I wanted to incorporate some of its ideas into my dissertation proposal. This chapter is entitled "An overview of complexity" and provides a brief and limited definition of what complexity is and several examples to broaden this definition.

Complexity is the study of emergent behavior from systems operating on the verge between stability and chaos. However, chaos is considered a subset of complexity. Complex systems also involve interactions between their individual components. The behavior that emerges from these interactions is often unexpected since the rules of the components don't necessarily predict this behavior.

Examples of complex systems in this book include traffic, autocatalytic systems, sand piles, and economies.

The bunching of cars and subsequent spreading out on highways is an emergent phenomenon that can depend on factors such as driver reaction times, car speeds, and the distances that drivers feel comfortable with when following other cars. I think that the variability in these individual factors leads to the random bunching of cars on the road.

The angle of repose of a sand pile is the angle that the pile makes with the horizontal plane that the pile is on. This angle emerges as the pile grows and may depend on how the pile is formed (dumping, pouring, etc.). Any changes to this angle caused by the addition of more sand leads to small avalanches that "correct" the perturbation so that the angle of repose is maintained. This is known as a self-organized critical state.

Not all sets of system behaviors can give rise to complex behavior.

Economies represent adaptive systems. In these systems, each agent adapts their behavior to the rules of the system to maximize their profits/utility. There is not one best strategy for this; rather, each agent must adapt their strategy according to what the whole system is doing to succeed.

Wednesday, August 22, 2012

Notes from the Chaos Cookbook, Chapter 2, pp. 37-42

Today I have more notes and insights from the Chaos Cookbook, Chapter 2. I'm trying to understand what characteristics are common to all chaotic systems and what it means for these systems to be chaotic. I'm also trying to find what links the ideas chaos, fractals, emergence, and complexity together. Chapter 2 deals with a specific type of chaotic entity: an iterated function.

"For a function to exhibit chaotic behavior, the function used has to be nonlinear." However, a nonlinear equation does not necessarily display chaotic behavior (example: y = x**2).

An iterated function is one whose value depends on the prior iteration. This is a form of mathematical feedback. Feedback is common in all chaotic systems.

There are two important parameters that describe an iterated function: the initial value and the number of iterations. Chaotic functions may be very sensitive to the initial value, meaning that very small changes to it will produce very drastic differences in the outcome of the iteration process.

Despite the sensitivity of the values of an iterated function to the initial value used, each graph may display common features.

The sequence of numbers obtained by iterating over a function is known as an orbit. Orbits may be stable, unstable, or chaotic.

Tuesday, August 21, 2012

Harnessing Light 2

The US National Academy of Sciences has released the second iteration of Harnessing Light, which analyzes and recommends action for maintaining or increasing US competitiveness in global photonics markets.

I just watched the OSA webinar of the roundtable discussion on this document that occurred today at Stanford. Of interest was the committee's strong recommendation to increase US manufacturing capabilities in both optics and other areas that utilize optics for their manufacturing processes. They also addressed the stigma of manufacturing being a blue-collar field and made note that manufacturing engineers address very challenging and technical problems. One member also mentioned that the personal satisfaction from manufacturing jobs is often very great because of the tangible reward of seeing a product that one has designed come to market.

Tom Baer said that industry is better-suited for multidisciplinary research because the historical barriers across fields do not exist there.

Also of interest was the notion that the US has been a good innovator for ideas and technologies but has increasingly lost its ability to capitalize on these ideas to other countries.

Wednesday, August 15, 2012

Notes from The Chaos Cookbook, Chapter 1

I'm reading through a used copy of Joe Pritchard's "The Chaos Cookbook" that I recently obtained from Amazon. This post contains my notes from Chapter 1.

On page 26 he notes that Newton's laws of motion cannot predict the general result of three solid balls colliding simultaneously. At first I thought that this was not true since I remember solving problems like this in general physics. However, after further thought I realized that we can predict which direction a ball will be traveling after the collision only if we already know what happened to the other balls. In other words, we cannot predict the outcome of the collision entirely; we must know something about what happened.

The above example may be considered as a subsequent loss of information that occurs following the collision of all three balls.

Historically speaking, nonlinear systems subjected to analysis were either 1) treated only to first order where higher order effects were negligible (e.g. a simple pendulum displaced slightly from its resting position), or 2) too difficult to analyze thoroughly.

A system may appear periodic in a reduced-dimensional phase space (see Fig. 1.3). A corkscrew trajectory along one axis will appear circular when looking down this axis.

Period doubling leads to splitting in a system's power spectrum. As the number of period doublings occur, does the spectrum fill with peaks and appear as white noise (uniform power spectrum)? Does this form a link to diffusive processes? (The Facebook link to white noise contains some block diagrams. Perhaps these can be shown to be equivalent to some chaotic systems.)

Chaos can emerge both in iterated function systems and from sets of differential equations.

Chaotic systems may eventually settle into seemingly stable states and vice versa.

Monday, August 13, 2012

Sequestration: yes or no?

Members of scientific organizations, like myself, are being urged to sign petitions asking that congress resume talks on methods to avoid resorting to sequestration. In case you haven't heard, we've run out of fancy terms to describe globally important ideas. As a result, sequestration is no longer a group of techniques for removing carbon from the atmosphere but now refers to budget cuts to U.S. spending that give no regard to which government programs receive the cuts. More precisely, funds that exceed the budget set forth by congress are sequestered by the treasury and not made available to congress for appropriations. This is generally viewed as a bad thing, and through this post I'm trying to determine why it is bad and whether or not the arguments make sense.

The argument of we scientists and engineers appears to be this: science and engineering help the U.S. remain competitive in an increasingly global technology market while contributing more to economic growth than other government-funded programs. Therefore, science and technology should receive less of the burden imposed by the budget cuts. Sequestration is seen as unfair and harmful since other organizations that contribute less to the economy receive more-or-less the same cuts to their funding. I have not seen arguments from other federally-funded sectors that may be hurt by sequestration, but I can imagine that similar appeals are being made.

Two possible requests are made, so far as I can tell, in the petition above. The first is that we are asking that sequestration not occur. The second is that so long as budget cuts are made, fewer cuts should be applied towards science and technology than other sectors. If this is indeed a zero sum game, then that would seem to indicate that other organizations will suffer more cuts than they would under sequestration. Perhaps some government-sponsored organizations are viewing sequestration as very appealing in this light.

I have two concerns about the value of the arguments put forth by APS and the many other scientific professional organizations. The first is that science and engineering is a very big field that employs many people. If every bit of it is contributing equally to the growth of the economy, then I agree that sequestration is unfair. But we also know very well that there are too many people, at least in academia, and that this is driving down the quality of scientific research in favor of publishing to gain a competitive edge. I suspect that only a certain percentage of science and engineering is actually of value to our economy, so perhaps it would be more appropriate to suffer sequestration and let the ineffective parts choke from the lack of funding. Harsh, I know, but this is the reality of cutting a budget.

My second concern is based on an intuition I have about complex systems, particularly the stock market. Over the long run (20 years or more), index mutual funds will always outperform actively traded funds, and they do this by matching the mean growth of the market rather than reacting to the daily or monthly fluctuations of stock prices. In other words, reacting to fluctuations and trying to understand the market with a reductionist approach will eventually fail to account for every factor that affects prices, and as a rule of complex systems, even minor factors have big consequences. Additionally, you often pay more in fees for the large amount of trading that is performed. My intuition is that this analogy applies, at least loosely, to federal spending. It may simply be impossible to effect a detailed budget plan that helps the economy in the long run. Any attempt to do so might work, but it's cost in man-hours and other resources may lead to other problems. Across the board cuts imposed by sequestration might be the best approach we have to reducing spending within the government.

Finally, let's keep in mind that the purpose of a government is to secure the welfare of its people. The simplest question we can ask is this: which of the two options—sequestration or a detailed budget reduction plan—will ultimately make people happier? I haven't been convinced yet which one is best for U.S. citizens.

I'm going to end this post for now in favor of other work I need to do, but I think that there are many other subtleties here that are worth considering. I welcome any and all comments, thoughts, and questions as we try to sort this out before January.