OK, so after three decades of surpluses, we can certainly afford it, but it’s nevertheless an alarming sign of the times: Canada’s first trade deficit since, what was it, 1974 I believe. It is not a pretty thought.
Feb 112009
Feb 092009
So you smash up a perfectly good airplane, dunk a bunch of passengers in frigid water, and lose their luggage, and what do they give you? The keys to New York City, that’s what.
Keys to the City
Then again, perhaps the fact that it wasn’t your fault and that everyone actually came out alive and mostly unharmed had something to do with it.
Feb 072009
I remain troubled by this business with black holes.
In particular, the zeroth law. Many authors, such as Wald, say that the zeroth law states that a body’s temperature is constant at equilibrium. I find this formulation less than satisfactory. Thermodynamics is about equilibrium systems to begin with, so it’s not like you have a choice to measure temperatures in a non-equilibrium system; temperature is not even defined there! A proper formulation for the zeroth law is between systems: the idea that an equilibrium exists between systems 1 and 2 expressed in the form of a function f(p1, V1, p2, V2) being zero. Between systems 2 and 3, we have g(p2, V2, p3, V3) = 0, and between systems 3 and 1, we have h(p3, V3, p1, V1) = 0. The zeroth law says that if f(p1, V1, p2, V2) = 0 and g(p2, V2, p3, V3) = 0, then h(p3, V3, p1, V1) = 0. From this, the concept of empirical temperature can be obtained. I don’t see the analog of this for black holes… can we compare two black holes on the basis of J and Ω (which take the place of V and p) and say that they are in “equilibrium”? That makes no sense to me.
On the other hand, if you have a Pfaffian in the form of dA + B dC, there always exists an integrating denominator X (in this simple case, one doesn’t even need Carathéodory’s principle and assume the existence of irreversible processes) such that X dY = dA + B dC. So simply writing down dM – Ω dJ already gives rise to an equation in the form X dY = dM – Ω dJ. That κ and A serve nicely as X and Y may be no more than an interesting coincidence.
But then there is the area theorem such that dA > 0 (just like dS > 0). Is that another coincidence?
And then there is Hawking radiation. The temperature of which is proportional to the surface gravity, T = κ/2π, which is what leads to the identification S = A/4. Too many coincidences?
I don’t know. I can see why this black hole thermodynamics business is not outright stupid, but I remain troubled.
Feb 072009
I just saw a bus from my window. It stopped at a bus stop. A person got off it, and the bus then continued.
A perfectly ordinary sight in a first world city (a G8 capital no less!), unless you consider that Ottawa was without public transportation for the past two months because of a stupid and senseless strike that accomplished nothing.
Feb 062009
I’m thinking about quantum computers today.
Quantum computers are supposed to be “better” than ordinary digital computers in that they’re able to solve, in polynomial time, many problems that an ordinary digital computer can only solve in exponential time. This has enormous practical implications: notably, many cryptographic methods are based on the fact that there are mathematical problems that can only be solved in exponential time, rendering it impractical to break an encryption key by computer using any “brute force” method. However, if a quantum computer could solve the same problem in polynomial time, a “brute force” method may be practical.
But the thing is, quantum computers are not exactly unique in this respect. Any good old analog computer from the 1950s can also solve the same problems in polynomial time. At least, in principle.
And that’s the operative phrase here: in principle. An analog computer, which represents data in the form of continuous quantities such as lengths, currents, voltages, angles, etc., is limited by its accuracy: even the best analog computer rarely has an accuracy better than one part in a thousand. Not exactly helpful when you’re trying to factorize 1000-digit numbers, for instance.
A quantum computer also represents data in the form of a continuous quantity: the (phase of the) wave function. Like an analog computer, a quantum computer is also limited in accuracy: this limitation is known as decoherence, when the wave function collapses into one of its eigenstates, as if a measurement had been performed.
So why bother with quantum computers, then? Simple: it is widely believed that it is possible to restore coherence in a quantum computer. If this is indeed possible, then a quantum computer is like an analog computer on steroids: any intermediate calculations could be carried out to arbitrary precision, only the final measurement (i.e., reading out the result) would be subject to a classical measurement error, which is not really a big issue when the final result, for instance, is a yes/no type result.
So that’s what quantum computing boils down to: “redundant qubits” that can ensure that coherence is maintained throughout a calculation. Many think that this can be done… I remain somewhat skeptical.
Feb 032009
According the CNN, it is confirmed by the Pentagon: Iran successfully launched an orbital satellite.
This is a tremendous accomplishment for a nation that exists in economic isolation.
On the other hand, it is a cause for tremendous concern: the missile belongs to a nation that has been openly advocating the destruction of Israel, and is likely in the advanced stages of a nuclear weapons program.
I guess what it boils down to is two questions: 1) Are the ayatollahs crazy enough to try to nuke Israel or lob an ICBM over the Atlantic? 2) Are other parties worried enough to start a major war by launching a preemptive strike against Iran?
If the answer is a yes to either of these questions, lots of people will die and lots of unpleasant things will happen to lots of other people.
Feb 032009
I’m reading Robert Wald’s book, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, and I am puzzled. According to Wald, the black hole equivalent of the First Law reads (for a Kerr black hole):
(1/8π)κdA = dM – ΩdJ,
where κ is the surface gravity, A is the area of the event horizon, M is the mass, Ω is the angular velocity of the event horizon, and J is the black hole’s angular momentum.
The analogy with thermodynamics is obvious if one write the First Law as
TdS = dU + pdV,
where T is the temperature, S is the entropy, U is the internal energy, p is the pressure, and V is the volume. Further, as per the black hole area theorem, which Wald proves, A always increases, in analogy with the thermodynamical entropy.
But… if I am to take this analogy seriously, then I am reminded of the fact that in a thermodynamical system the temperature is determined as a function of pressure and volume, i.e., there is a function f such that T = f(p, V). Is there an analogue of this in black hole physics? Is the surface gravity κ fully determined as a function of Ω and J? It is not obvious to me that this is the case, and Wald doesn’t say. Yet without it, there is no zeroth law and no thermodynamics. He does mention the zeroth law in the context of a single black hole having uniform surface gravity, but that’s not good enough. It doesn’t tell me how the surface gravity can be calculated from Ω and J alone, nor does it tell me anything about more than one black hole being involved, whereas in thermodynamics, the zeroth law is about multiple thermodynamical systems being in thermal equilibrium.
Another puzzling aspect is that the area theorem has often been quoted as “proof” that a black hole cannot evaporate. Yet again, if I take the analogy with thermodynamics seriously, the Second Law applies only to closed systems that exchange neither matter nor energy with their environment; it is, in fact, quite possible to reduce S in an open system, otherwise your fridge would not work. So if a black hole can exchange energy and matter with its environment, perhaps it can evaporate after all.
Moreover, for the analogy to be complete, we’d also be required to have
8π∂M/dA = κ,
∂M/∂J = Ω,
just as in ordinary thermodynamics, we have T = ∂U/∂S and p = –∂U/∂V. So, do these relationships hold for black holes?
I guess I’ll go to ArXiv and read some recent papers on black hole thermodynamics.
Feb 022009
These are not unusual pictures for us up here in the Great White North:
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Trouble is, these pictures are not from Ottawa, Toronto, or London, Ontario. They are from London, England, where it hasn’t stopped snowing yet.
Jan 302009
I’m reading a 40-year old book, Methods of Thermodynamics by Howard Reiss. I think I bought it after reading a recommendation on Amazon.com, describing this book as one of the few that takes the idea of axiomatic thermodynamics seriously, and treats it without mixing in concepts from statistical physics or quantum mechanics.
It is a very good book. Not only does it deliver on its promise, it also raises some issues that would not have occurred to me otherwise. For instance, the idea that a so-called equation of state does not fully describe the state of a material, even an ideal gas. You cannot derive U = CvT from the equation of state. You cannot that the internal energy U is a linear function of the temperature T, it has to be postulated.
One thing you can derive from the ideal gas equation of state alone is that an adiabatic expansion must be isothermal. As an ideal gas expands and its volume increases while its pressure decreases, its temperature remains constant. It also made me think again about the cosmological equation of state… cosmologists often play with idealized cases (e.g., dust-filled universe, radiation-filled universe) but until now, I never considered the possibility that even in these idealized cases, the equations of state do not full describe the stuff that they supposedly represent.
Jan 302009
Our paper about the thermal analysis of Pioneer 10 and 11 was accepted for publication by Physical Review and it is now on ArXiv.
I think it is an interesting paper. First, it derives from basic principles equations of the thermal recoil force. This is not usually in heat transfer textbooks, as those are more concerned about energy exchange than about momentum. We also derive the infamous factor of 2/3 for a Lambertian (diffuse) surface.
More notably, we make a direct connection between the thermal power of heat sources and the recoil force. The thermal power of heat sources within a spacecraft is usually known very well, and may also be telemetered. So, if a simple formalism exists that gives the recoil force as a function of thermal power, we have a very meaningful way to connect telemetry and trajectory analysis. This is indeed what my “homebrew” orbit determination code does, using Pioneer telemetry and Doppler data together.
No results yet… the paper uses simulated Pioneer 10 data, precisely to avoid jumping to a premature conclusion. We can jump to conclusions once we’re done analyzing all the data using methods that include what’s in this paper… until then, we have to keep an open mind.
Jan 292009
In two days, I got two notices of papers being accepted, among them our paper about the possible relationship between modified gravity and the origin of inertia. I am most pleased, because the journal accepting it (MNRAS Letters) is quite prestigious and the paper was a potentially controversial one. The other paper is about Pioneer, and was accepted by Physical Review D. Needless to say, I am pleased.
Jan 272009
Long before blogs, long before the Web even, there was an Internet and people communicated via public forums (fora?), Usenet foremost among them.
Yet I stopped using Usenet about a decade ago. Here is a good example as to why. Excerpts from an exchange:
“You will have more success on Usenet if you learn and follow the normal Usenet posting conventions.“
“About posting conventions: where did I stray from them? I do indeed want to respect the list rules.“
“Have a look at <http://cfaj.freeshell.org/google/>“
“Got it: thanks.“
“You failed to appropriately quote the message that you are responding to. See the FAQ and the more detailed explanation of posting style that it links to. Then, if the explanation provided is not sufficiently clear, ask for clarification.“
“I am afraid that you have not yet ‘got it’. You have gone from not quoting the message you are responding to, to top-posting and failing to appropriately trim the material that you are quoting.“
“If you had been told what you did wrong, that would, hopefully, eliminate one class of error from your future posts. You were told where to read about conventions, which *should* eliminate *all* of the well-known errors.“
You are forgiven if you thought that the thread from which I excerpted these snotty remarks was about Usenet’s “netiquette”. But it wasn’t. It was all in response to a very polite and sensible question about ways to implement a destructor in JavaScript.
I guess my views are rather clear on the question as to which people harm Usenet more: those who stray from flawless “netiquette”, or those who feel obliged to lecture them. I have yet to understand why it is proper “netiquette” to flood a topic with such lectures instead of limiting responses to the topic at hand, and responding only when one actually knows the answer. I guess that would be too helpful, and helping other people without scolding them is not proper “netiquette”?
Jan 272009
I’ve read a lot about the coming “digital dark age”, when much of the written record produced by our digital society will no longer be readable due to changing data formats, obsolete hardware, or deteriorating media.
But perhaps, just perhaps, the opposite is happening. Material that is worth preserving may in fact be more likely to survive, simply because it’ll exist in so many copies.
For instance, I was recently citing two books in a paper: one by d’Alembert, written in 1743, and another by Mach, from 1883. Is it pretentious to cite books that you cannot find at any library within a 500-mile radius?
Not anymore, thanks, in this case, to Google Books:
Jean Le Rond d’ Alembert: Traité de dynamique
Ernst Mach: Die Mechanik in ihrer Entwickelung
And now, extra copies of these books exist on my server, as I downloaded and I am preserving the PDFs. Others may do the same, and the books may survive so long as computers exist, as copies are being made and reproduced all the time.
Sometimes, it’s really nice to live in the digital world.
Jan 262009
The other day, I put my latest (well, I actually did it last summer, but it’s the latest that has seen the light of day) Pioneer paper on ArXiv.org; it is not about new results (yet), just a confirmation of the Pioneer anomaly using independently developed code, and a demonstration that a jerk term may be present in the data.
Jan 252009
Often, I wondered: who designed the graphical elements, like the fonts and icons that appear on my computer screen?
Finally, I know the name of one of these people. She is Susan Kare, and her work appeared in the original Macintosh, Windows 3.0, OS/2, even Facebook. I came across her name as I was reading about the 25th anniversary of the Macintosh and clicked a link that took me to a 12-year old article from The New York Times that Ms. Kare has on her Web site.
Jan 242009
Once again, I am studying classical thermodynamics. Axiomatic thermodynamics to be precise, none of this statistical physics business (which is interesting on its own right, but it is quite a different topic.)
The more I learn about it, the more I find thermodynamics incredibly fascinating. Why is it so different from other areas of physics? Perhaps I now have an answer that may be trivial to some, but eluded me until now.
Most of physics is described by functions of coordinates and time. This is true even in the case of general relativity, even as the coordinate system itself may be curved, the curvature (the metric) is described as a function of space-time coordinates.
In contrast, there are no coordinates in axiomatic thermodynamics, only states. States are decribed by state variables, and usually you have these in excess. For instance, the state of one mole of an ideal gas is described by any two of the three variables p (pressure), V (volume) and T (temperature); once two of these are known, the third is given by the ideal gas equation of state, pV = KT, where K is a constant.
Notice that there is no independent variable. The variables p, V, and T are not written as functions of time. Nor should they be, since axiomatic thermodynamics is really equilibrium thermodynamics, and when a system is in equilibrium, it is not changing, its state is constant.
So why is it not called thermostatics? What does dynamics have to do with stationary states? As it turns out, thermodynamics is the science of fitting a square peg in a round hole, as having just established that it’s a science of static states, it nevertheless goes on to explain how states can change… so long as all the intermediate states can exist as static states on their own right, such as when you’re heating a gas slowly enough so that its temperature is more or less uniform at all times, and its state is well approximated by thermodynamic variables.
The zeroeth law states that an empirical temperature exists that is associative: systems that have the same temperature form equivalence classes.
The first law defines the (infinitesimal) quantity of heat dQ as the sum of changes in internal energy (dU) and mechanical work (p dV). An important thing about dQ is that there may not be a Q; in the jargon of differential forms, dQ is a Pfaffian that may not be exact.
The second law uses the assumption of irreversibility and Carathéodory’s theorem to show that there is an integrating denominator T and a function S such that dQ = T dS. (Presto, we have entropy.) Further, T is uniquely determined up to a multiplicative constant.
Combined, the two laws can be written in the form dU = T dS − p dV. After that, much of what is in the textbooks about classical thermodynamics can be written compactly in the form of the Jacobian determinant ∂(T, S)/∂(p, V) = 1.
Given that I know all this, why do I still find myself occasionally baffled by the simplest thermodynamic problems, such as convincing myself that when an isolated system of ideal gas expands, its temperature remains constant? (It does, the math says so, textbooks say so, but still…) There is something uniquely non-trivial about axiomatic thermodynamics.
Jan 222009
The other day, arXiv.org split a popular category, astro-ph, into six subcategories. This is convenient… astro-ph, the astrophysics archive, was getting rather large, and the split into sub-categories makes it easier to find papers that are relevant to one’s specialization.
On the other hand… it also means that one is less likely to read papers that are not directly relevant to one’s specialization, but may be interesting, eye-opening, and may help to broaden one’s horizons. Is this a good thing?
There are no easy answers of course… the number of papers just on arXiv.org is mind-boggling (they proudly announced that they’ve passed the half million paper milestone on October, with thousands of new papers added every month) and no one has the time to read them all. Hmmm, perhaps I should have spent more time applauding a recent initiative by Physical Review, their This Week in Physics newsletter and associated Web site.
Jan 202009
While we were celebrating President Obama, the Bank of Canada made its move: the Canadian prime rate is now lower than ever, at 1%. The expectation is that the economy will not fare well in coming months.
Being the holder of a variable rate mortgage, I have no reason to complain. Still, it’s an unsettling development.
Jan 192009
There is ceasefire in Gaza. Perhaps it will hold for a while.
It may have been precipitated by Israel’s desire to wrap up its military operations before Obama is inaugurated, anticipating that the new administration in Washington will be a lot less sympathetic toward, well, if not necessarily Israel’s cause then certainly the methods that Israel chooses to advance its cause.
It may have been the result of rising doubt and anger among Israelis themselves, those who realize that the history of the Gaza strip did not begin with the Qassam rockets.
Or perhaps they began asking questions like that asked by Time Magazine: Can Israel survive its assault on Gaza?
Or maybe it was compassion. The other day, the assault on Gaza turned from an abstract litany of numbers (last I heard, over 1,100 killed, more than 4,000 wounded, many of them civilians) into something personal, as an Israeli-trained Palestinian doctor, peace activist to boot, who regularly reported on Israeli television via cell phone, was reporting the shelling of his own house and the loss of three of his daughters on live TV.
The biggest irony of any “war on terror” (not just that of Bush, not just that of Olmert) is how, contrary to the stated intent of the war’s leaders, such a war flawlessly serves the terrorists’ interests. Such wars are based on the blatant ignorance of their leaders, leaders who believe that the terrorist is motivated by hatred and a desire to kill. While they are not free of hatred and bloodlust, they are motivated by something else altogether: by a desire to change the political course through their acts of terror. When our leaders declare a “war on terror” in response, they accomplish precisely what the terrorist wants… which is why Bush and Olmert ended harming the interests of their own countries to an extent far greater than anything the terrorists could have accomplished by themselves.
Gavrilo Princip knew this. When he assassinated the arch-duke of Austria-Hungary in 1914, his hatred of a despised leader was outweighed by his the hope that the assassination will change the existing world order and free Serbia from Austrian rule. Princip became one of the most influential (and most successful!) persons of history, not so much as a result of his own actions, but as a result of the predictably stupid reaction of Vienna’s fossilized political leadership. Why are we—the U.S., the Western world, Israel—so bent on repeating that mistake?
Jan 182009
“John Moffat is not crazy.” These are the opening words of Dan Falk’s new review of John’s book, Reinventing Gravity, which (the review, that is) appeared in the Globe and Mail today. It is an excellent review, and it was a pleasure to see that the sales rank of John’s book immediately went up on amazon.ca. As to the opening sentence… does that mean that I am not crazy either, having worked with John on his gravity theory?