Posted on September 17th, 2012 No comments
According to Wikipedia, Physical Review Letters’ “focus is rapid dissemination of significant, or notable, results of fundamental research on all topics related to all fields of physics. This is accomplished by rapid publication of short reports, called ‘Letters’”. That’s what I always thought, so I was somewhat taken aback to find an article in last week’s issue entitled, “Encouraging Moderation: Clues from a Simple Model of Ideological Conflict.” Unfortunately, you can’t see the whole thing without a subscription, but here’s the abstract:
Some of the most pivotal moments in intellectual history occur when a new ideology sweeps through a society, supplanting an established system of beliefs in a rapid revolution of thought. Yet in many cases the new ideology is as extreme as the old. Why is it then that moderate positions so rarely prevail? Here, in the context of a simple model of opinion spreading, we test seven plausible strategies for deradicalizing a society and find that only one of them significantly expands the moderate subpopulation without risking its extinction in the process.
That’s physics?! Not according to any of the definitions in my ancient copy of Webster’s Dictionary. Evidently some new ones have cropped up since it was published, and nobody bothered to inform me. In any case, tossing in this kind of stuff doesn’t exactly enhance the integrity of the field. If you don’t have access to the paper, I would not encourage you to visit your local university campus to have a look. I doubt the effort would be worth it.
Where should I start? In the first place, the authors simply assume that “moderate” is to be conflated with “good”, without bothering to offer a coherent definition of “moderate.” In the context of U.S. politics, for example, the term is practically useless. People with an ideological ax to grind tend to consider themselves “moderate,” and their opponents “extreme.” Conservatives refer to the mildest of their opponents as “extreme left wing,” and liberals refer to the most milque-toast of their opponents as “ultra right wing.” Consider, for example, a post about the Muhammad film flap that just appeared on a website with the moniker, “The Moderate Voice.” I don’t doubt that it might be termed “moderate” in the academic milieu from which papers such as the one we are discussing usually emanate, but it wouldn’t pass the smell test as such among mainstream conservatives, and has already been dismissed in those quarters as the fumings of the raving extremist hacks of the left. Back in the 30′s, it was a commonplace and decidedly ”moderate” opinion among the authors who contributed articles to The New Republic, the American Mercury, the Atlantic, and the other prestigious intellectual journals of the day was that capitalism was breathing its last, and should be replaced with a socialist system of one stripe or another as soon as possible. Obviously, what passes as ”moderate” isn’t constant, even over relatively short times. Is the Tea Party Movement moderate? Certainly not as far as most university professors are concerned, but decidedly so among mainstream conservatives.
According to the authors, the types of ideological swings they refer to occur in science as well as politics. One wonders what “moderation” would look like in such cases. Perhaps the textbooks would inform us that only half the species on earth evolved, and God created the rest, or that, while oxygen is necessary to keep a fire burning, phlogiston is necessary to start one, or that only the most visible stars are imbedded in a crystal ball surrounding the earth known as the “firmament,” while the other half are actually many light years away.
Undeterred by such considerations, the authors created a simple mathematical model that is supposed to reflect the dynamics of ideological change. Just as the economic models are all infallible for predicting the behavior of Homo economicus, it is similarly effective at predicting the behavior of what one might call Homo ideologicus. As for Homo sapiens, not so much. There is no attempt whatsoever to incorporate even the most elementary aspects of human nature in the model. It is inhabited by “speakers” and “listeners,” who are identified as either AB, the inhabitants of the moderate middle ground, or A and B, the extemists on either side of it. For good measure, there is also an Ac, inhabited by “committed” and intransigent followers of A. The subpopulations in these groups are, in turn, labeled nA, nB, nAB, and p. Only moderate listeners can be converted to one of the extremes, and vice versa, although we are reliably informed that, for example, the Nazis found some of their most fertile recruiting grounds among the Communists at the opposite extreme, and certainly not just among German moderates. With the assumptions noted above, and setting aside trivialities such as units of measure, the authors come up with “dynamic equations” such as,
nA = (p + nA)nAB – nAnB
nB = nBnAB – (p + nA)n
There are variations, complete with parameters to account for “stubbornness” and “evangelism.” There are any number of counterintuitive assumptions implicit in the models, such as that all speakers are equally effective at convincing others to change sides, opinions about given issues are held independently of opinions about other issues, although this is almost never the case among people who care about the issues one way or the other, that a metric for deciding what is the moderate “good” and what the extreme “evil” will always be available to the philosopher kings who apply the models, etc. The models were tested on “real social networks,” and (surprise, surprise) the curves derived from a judicious choice of nA, nB, etc., were in nice agreement with predictions.
According to the authors,
Since we present no formal evidence that the dynamics of (the equations noted above) do actually occur in practice, our work could alternatively be viewed as posing this model and its subsequent generalizations as interesting in their own right.
While I heartily concur with the first part of the sentence, I suggest that the model and its subsequent generalizations might be of more enduring interest to sociologists than physicists. Perhaps the editors of Phys Rev Letters and their reviewers will consider that possibility the next time a similar paper is submitted, and kindly direct the authors to a more appropriate journal.
Posted on July 7th, 2012 No comments
It’s been over a century since Max Planck came up with the idea that electromagnetic energy could only be emitted in fixed units called quanta as a means of explaining the observed spectrum of light from incandescent light bulbs. Starting from this point, great physicists such as Bohr, de Broglie, Schrödinger, and Dirac developed the field of quantum mechanics, revolutionizing our understanding of the physical universe. By the 1930′s it was known that matter, as well as electromagnetic energy, could be described by wave equations. In other words, at the level of the atom, particles do not behave at all as if they were billiard balls on a table, or, in general, in the way that our senses portray physical objects to us at a much larger scale. For example, electrons don’t act like hard little balls flying around outside the nuclei of atoms. Rather, it is necessary to describe where they are in terms of probability distributions, and how they act in terms of wave functions. It is impossible to tell at any moment exactly where they are, a fact formalized mathematically in Heisenberg’s famous Uncertainty Principle. All this has profound implications for the very nature of reality, most of which, even after the passage of many decades, are still unknown to the average lay person. Among other things, it follows from all this that there are two basic types of elementary particles; fermions and bosons. It turns out that they behave in profoundly different ways, and that the idiosyncrasies of neither of them can be understood in terms of classical physics.
Sometimes the correspondence between mathematics and physical reality seems almost magical. So it is with the math that predicts the existence of fermions and bosons. When it was discovered that particles at the atomic level actually behave as waves, a brilliant Austrian scientist named Erwin Schrödinger came up with a now-famous wave equation to describe the phenomenon. Derived from a few elementary assumptions based on some postulates derived by Einstein and others relating the wavelength and frequency of matter waves to physical quantities such as momentum and energy, and the behavior of waves in general, the Schrödinger equation could be solved to find wave functions. It was found that these wave functions were complex numbers, that is, they had a real component, and an “imaginary” component that was a multiple of i, the square root of minus one. For example, such a number might be written down mathematically as x + iy. Each such number has a complex conjugate, found by changing the sign of the complex term. The complex conjugate of the above number is, therefore, x – iy. Max born found that the probability of finding a physical particle at any given point in space and time could be derived from the product of a solution to Schrödinger’s equation and its complex conjugate.
So far, so good, but eventually it was realized that there was a problem with describing particles in this way that didn’t arise in classical physics; you couldn’t tell them apart! Elementary particles are, after all, indistinguishable. One electron, for example, resembles every other electron like so many peas in a pod. Suppose you could put two electrons in a glass box, and set them in motion bouncing off the walls. Assuming you had very good eyes, you wouldn’t have any trouble telling the two of them apart if they behaved like classical billiard balls. You would simply have to watch their trajectories as they bounced around in the box. However, they don’t behave like billiard balls. Their motion must be described by wave functions, and wave functions can overlap, making it impossible to tell which wave function belongs to which electron! Trying to measure where they are won’t help, because the wave functions are changed by the very act of measurement.
All this was problematic, because if elementary particles really were indistinguishable in that way, they also had to be indistinguishable in the mathematical equations that described their behavior. As noted above, it had been discovered that the physical attributes of a particle could be determined in terms of the product of a solution to Schrödinger’s equation and its complex conjugate. Assuming for the moment that the two electrons in the box didn’t collide or otherwise interact with each other, that implies that the solution for the two particle system would depend on the product of the solution for both particles and their complex conjugates. Unfortunately, the simple product didn’t work. If the particles were labeled and the labels switched around in the solution, the answer came out different. The particles were distinguishable! What to do?
Well, Schrödinger’s equation has a very useful mathematical property. It is linear. What that means in practical terms is that if the products of the wave functions for the two particle system is a solution, then any combination of the products will also be a solution. It was found that if the overall solution was expressed as the product of the two wave functions plus their product with the labels of the two particles interchanged, or of the product of the two wave functions minus their product with the labels interchanged, the resulting probability density function was not changed by changing around the labels. The particles remained indistinguishable!
The solution to the Schrödinger equation, referred to mathematically as an eigenfunction, is called symmetric in the plus case, and antisymmetric in the minus case. It turns out, however, that if you do the math, particles act in very different ways depending on whether the plus sign or the minus sign is used. And here’s where the magic comes in. So far with just been doing math, right? We’ve just been manipulating symbols to get the math to come out right. Well, as the great physicist, Richard Feynman, once put it, “To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature.” So it is in this case. The real particles act just as the math predicts, and in ways that are completely unexplainable in terms of classical physics! Particles that can be described by an antisymmetric eigenfunction are called fermions, and particles that can be described by an symmetric eigenfunction are called bosons.
How do they actually differ? Well, for reasons I won’t go into here, the so-called exclusion principle applies to fermions. There can never be more than one of them in exactly the same quantum state. Electrons are fermions, and that’s why they are arranged in different levels as they orbit the nucleus of an atom. Bosons behave differently, and in ways that can be quite spectacular. Assuming a collection of bosons can be cooled to a low enough temperature they will tend to all condense into the same low energy quantum state. As it happens, the helium atom is a boson. When it is cooled below a temperature of 2.18 degrees above absolute zero, it shows some very remarkable large scale quantum effects. Perhaps the weirdest of these is superfluidity. In this state, it behaves as if it had no viscosity at all, and can climb up the sides of a container and siphon itself out over the top!
No one really knows what matter is at a fundamental level, or why it exists at all. However, we do know enough about it to realize that our senses only tell us how it acts at the large scales that matter to most living creatures. They don’t tell us anything about its essence. It’s unfortunate that now, nearly a century after some of these wonderful discoveries about the quantum world were made, so few people know anything about them. It seems to me that knowing about them and the great scientist who made them adds a certain interest and richness to life. If nothing else, when physicists talk about the Higgs boson, it’s nice to have some clue what they’re talking about.