Bose Statistics and Classical Fields

Right now Kazimierz Rzążewski from the Center for Theoretical Physics at the Polish Academy of Sciences is giving a talk on “Bose statistics and classical fields”.

Abstract: Statistical properties of quantum systems are the heart of quantum statistical physics. Probability distributions of Bose-Einstein condensate are well understood for an ideal gas. In the presence of interactions only crude approximations are available. In this talk I will argue that now we have a powerful computational tool to study the statistics of weakly interacting Bose gas which is based on the so-called classical field approximation.

For a 3d ideal gas of bosonic atoms trapped in a 3d harmonic oscillator potential, the fraction of atoms in the ground state goes like

for T below a certain critical value, and 0 above that.

The grand canonical ensemble, where we assume the number of particles in our gas and its total energy are both variable, is a dubious method for Bose-Einstein condensates, because there’s no contact with a particle reservoir. The canonical ensemble is also fishy, where we assume the particle number is fixed both the total energy is variable, is also fishy. Why? Because there’s not contact with a heat reservoir, either. The microcanonical ensemble, where the energy and number of particles are both fixed, is closest to experimental reality.

We see this when we compute the fluctuations of the number of particles in the ground state. For the grand canonical ensemble, the standard deviation of the number of particles in the ground state becomes infinite at temperature below a certain value!

The fun starts when we move from the ideal gas to a weakly interacting gas. Most papers here consider particles trapped in a box, not in a harmonic oscillator — and they use the Bogoliubov approximation, which is exactly soluble for a box. This approximation involves a quadratic Hamiltonian that’s a sum of terms, one for each mode in the box. To set up this equation we need to use the Bogoliubov-deGennes equations.

As the temperature goes up, the Bogoliubov approximation breaks down… so we need a new approach.

Here is Rzążewski’s approach. A gas of bosons is described by a quantum field. But we can approximate the long-wavelength part of this quantum field by a classical field. Of course the basic idea here is not new. In our study of electromagnetism — this is what lets us approximate the quantum electromagnetic field by a classical field obeying the classical Maxwell equations. But the new part is setting up a theory that keeps some of the virtues of the quantum description, while approximating it with a classical one at low frequencies (i.e., large distance scales).

So: for modes below the cutoff we describe the system using annihilation and creation operators; for each mode above the cutoff we have 2d classical phase space. But: how to put in a nice ‘cutoff’ where we make the transition from the quantum field to the classical field?

Testing this problem on an exactly soluble model is a good idea: for example, the 1-dimensional ideal gas!

It turns out that by choosing the cutoff in an optimal way, the approximation is very good — not just for the 1d ideal gas, but also the 3d case, in both a harmonic potential and in a box. There is an analytic form for this optimal cutoff.

But more significant is the nonideal gas, where the particles repel each other. Here it’s easiest to start with the 1d case of a gas trapped in a harmonic oscillator potential. Now it’s more complicated. But we can simulate it numerically using the Metropolis algorithm!

We can also study ‘quasicondensates‘, where the coherence length is shorter than the size of the box, or the size of the cloud of atoms. (For example, in 2 dimensions, at temperatures above the Berezinskii-Kosterlitz-Thouless transition, there are lots of vortices in the gas, so the phase of the gas is nearly uniform only in small patches.)

Some papers:

• E. Witkowska, M. Gajda, and K. Rzążewski,
Bose statistics and classical fields, Phys. Rev. A 79 (2009), 033631.

• E. Witkowska, M. Gajda, and K. Rzążewski,
Monte Carlo method, classical fields and Bose statistics,
Opt. Comm. 283 (2010), 671-675.

• Z. Idziaszek, L. Zawitkowski, M. Gajda, and K. Rzążewski, Fluctuations of weakly interacting Bose-Einstein condensate, Europhysics Lett. 86 (2009), 10002.

As usual, I’d love it if an expert came along and explained anything more about these ideas. For example, I’m pretty vague about how exactly the Metropolis algorithm is used here.

Turning Renewable Energy into Fuels

Terry Bollinger wrote a comment that deserves to be a post of its own, because it could start an interesting discussion. Here it is:

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The theme of “what scientists can do to help save the planet” is a good one.

I’d like to introduce a topic that concerns me greatly these days: Well-intended efforts to produce cellulosic biofuels that, if applied in non-tropical climates (read “not in Brazil”), could end up taking over huge percentages of land and water resources without necessarily truly solving the problem of replacing fossil fuels.

At present, and quite ironically, it’s not even possible to produce biofuels without using a lot of petroleum in the process. Even more ominously, there have already been cases where people who didn’t have much to begin with are going without food because biofuels have run up the prices and taken over land that should have been used to feed them. That is a very scary trend, especially this early in the biofuels game. And if people who can at least express their need are going hungry, what does that leave as the most likely fate for forests and the animals that live in them? Again, a scary trend.

So here’s the scientific part of my query: Are there other options that would make more sense for fueling the mobile part of our global infrastructure? Good rechargeable batteries obviously are part of the answer, but surely there are more possibilities.

A wild example: Is there any way a fully chemical process (versus a land-using plant base one) could use concentrated, e.g. electrical, energy from renewable sources (or perhaps nuclear, sigh) to replicate what plants do using sunlight? That is, pull CO₂ from the air, add water, and generate high-energy-density hydrocarbon or carbohydrate fuels?

The point of such a wild idea would be to achieve the goal of biofuels — less use of fossil fuels — but in a way that avoids the awful environmental consequences of, in effect, using huge areas of land or water area as a very costly and inefficient way to collect renewable solar energy [1].

In short: Why not separate the energy collection component from the fuel-from-CO₂ part, then optimize both to achieve minimal global environmental impact?

I know the answer: Because it’s really hard to do. But that doesn’t necessarily mean it’s impossible.

Cheers,
Terry Bollinger

[1] Plants themselves are of course highly efficient at photosynthesis, especially some grasses. However, that’s the wrong metric; it’s like picking a gold nugget out of a ton of dross and then pretending the entire ton of dross has the value of gold. For biofuels, a realistic net cost metric would need to include how much land is used, the type of land used, what other uses of that land are being displaced (there is often a significant energy penalty when that is factored in correctly), and the average usable solar energy for the particular plants used. Under that kind of a full-cost metric, even the best cellulosic plant options amount to a very poor (and for some northern regions a potentially negative net benefit) way to collect solar energy.

Record High Temperatures

One swallow does not a summer make, nor does a hot day mean that global warming is underway… but since climate change deniers in the US made a big deal of the snowstorms this winter, despite the fact that global warming should increase the chance of such storms, I can’t resist pointing out this item from the blog of meteorologist Jeff Masters:

June 2010 features an unprecedented heat wave in Asia and North Africa

A withering heat wave of unprecedented intensity brought the hottest temperatures in recorded history to six nations in Asia and Africa, plus the Asian portion of Russia, in June 2010. At least two other Middle East nations came within a degree of their hottest temperatures ever in June.

The heat was the most intense in Kuwait, which recorded its hottest temperature in history on June 15 in Abdaly, according to the Kuwait Met office. The mercury hit 52.6°C (126.7°F). Kuwait’s previous all-time hottest temperature was 51.9°C (125.4°F), on July 27,2007, at Abdaly. Temperatures reached 51°C (123.8°F) in the capital of Kuwait City on June 15, 2010.

Iraq had its hottest day in history on June 14, 2010, when the mercury hit 52.0°C (125.6°F) in Basra. Iraq’s previous record was 51.7°C (125.1°F) set August 8, 1937, in Ash Shu’aybah.

Saudi Arabia had its hottest temperature ever on June 22, 2010, with a reading of 52.0°C (125.6°F) in Jeddah, the second largest city in Saudi Arabia. The previous record was 51.7°C (125.1°F), at Abqaiq, date unknown. The record heat was accompanied by a sandstorm, which caused eight power plants to go offline, resulting in blackouts to several Saudi cities.

In Africa, Chad had its hottest day in history on June 22, 2010, when the temperature reached 47.6°C (117.7°F) at Faya. The previous record was 47.4°C (117.3°F) at Faya on June 3 and June 9, 1961.

Niger tied its record for hottest day in history on June 22, 2010, when the temperature reached 47.1°C (116.8°F) at Bilma. That record stood for just one day, as Bilma broke the record again on June 23, when the mercury topped out at 48.2°C (118.8°F). The previous record was 47.1°C on May 24, 1998, also at Bilma.

Sudan recorded its hottest temperature in its history on June 25 when the mercury rose to 49.6°C (121.3°F) at Dongola. The previous record was 49.5°C (121.1°F) set in July 1987 in Aba Hamed.

The Asian portion of Russia recorded its highest temperature in history on June 25, when the mercury hit 42.3°C (108.1°F) at Belogorsk, near the Amur River border with China. The previous record was 41.7°C (107.1°F) at nearby Aksha on July 21, 2004. (The record for European Russia is 43.8°C–110.8°F–set on August 6, 1940, at Alexandrov Gaj near the border with Kazakhstan.

Two other countries came within a degree of their all time hottest temperature on record during the heat wave. Bahrain had its hottest June temperature ever, 46.9°C, on June 20, missing the all-time record of 47.5°C (117.5°F), set July 14, 2000. Temperatures in Quatar reached 48.8°C (119.8°F) on June 20. Quatar’s all-time record hottest temperature was 49.6°C (121.3°F) set on July 9, 2000. All of these records are unofficial, and will need to be verified by the World Meteorological Organization (WMO.) The source for the previous all-time records listed here is the book Extreme Weather by Chris Burt. According to Mr. Burt, the only other time as many as six nations set their all-time highest temperature marks in a single month was during the European heat wave of August 2003.

Perhaps more important than these scattered jaw-dropping hot spots are the following facts from the US National Climatic Data Center:

The world land surface temperature June 2010 anomaly of 1.07°C (1.93°F) was the warmest on record, surpassing the previous June record set in 2005 by 0.12°C (0.22°F). The anomalous warm conditions that affected large portions of each inhabited continent also contributed to the warmest June worldwide land and ocean surface temperature since records began in 1880. The previous June record was set in 2005. Separately, the worldwide ocean surface temperatures during June 2010 were 0.54°C (0.97°F) above the 20th century average—the fourth warmest June on record.

In fact, the whole year has been hot…

But even more important are the long-term trends…

Of course, you need to read the paper to understand how this graph was made.

Quantum Steganography

Besides talking about environmental issues, I’d also like to use this blog to talk about my day job at the Centre for Quantum Technologies. I hope this isn’t too distracting…

I’d like to try live-blogging a talk here. Today there’s a talk by Bilal Shaw of the University of Southern California about a paper he wrote with Todd Brun on Quantum Steganography.

“Steganography” is the art of hiding information by embedding it in a seemingly innocent message. In case you’re wondering – and I’ve got the kind of mind that can’t help wondering – the word “steganography” actually is etymologically related to the word “stegosaurus”. They both go back to words meaning “cover” or “roof”. Some other words with the same root are “thatch”, “deck” -and even “detect”, which is like “de-deck”: to take the lid off something!

Steganography is an ancient art, still thriving today. For example, that Russian spy ring they just caught were embedding secret data in publicly visible websites. The advantage of steganography over ordinary cryptography is that if you do it right, it doesn’t draw attention to itself. See this picture?

Remove all but the two least significant bits of each color component and you’ll get a picture that’s almost black. But then make that picture 85 times brighter and here’s what you’ll see:

All this is purely classical, of course. But what fiendish tricks can we play using quantum mechanics? Can we hide Schrödinger’s cat in a seemingly innocent tree?

Bilal’s paper describes a few recipes for quantum steganography. Alas, I’m not good enough at cryptography and live-blogging to beautifully deliver an instant summary of how they work. But roughly, the idea is to fake the effects of mildly “depolarizing” channel, one that introduces some errors into the qubits you’re transmitting, pushing pure states closer to the center of the Bloch sphere, where pure noise lives. You can’t introduce too many errors, since this would make the error rate suspiciously high to someone spying on our transmissions. So, there’s a kind of tradeoff here…

I’d be happy for an expert to give a better description!

News About the Younger Dryas

I don’t want to write anything really interesting here until the technology gets upgraded…

… but I figure I might as well start puttle little comments about ecological issues here, instead of on my diary.

So:

• Chris Colose, Revisiting the Younger Dryas, RealClimate, July 17, 2010.

The Younger Dryas was, among other things, a sudden cooling event in Europe shortly after the end of the last ice age. In only 20 years, the temperature in Europe dropped about 7 Celsius! It stayed cold for about 1,300 years. In Greenland, the temperature went down 15 Celsius. And then, at the end of the Younger Dryas, temperatures in Europe bounced back just as fast.

Sudden climate changes of this magnitude could have a huge impact on human civilization – just imagine glaciers in the Lake District in England. So, it’s worth learning all we can about this episode. Indeed, some people have suggested that freshwater from melting ice was what brought on the Younger Dryas, by disrupting ocean circulation in the northern Atlantic… which raises the specter of a repeat of this incident sometime soon! Luckily, the chances of that now seem very low. But it’s still good to understand this stuff.

If you haven’t learned a bit about Heinrich events (when icebergs drop lots of rocks on to the floor of the northern Altantic), the Bølling-Allerød warm period that came right before the Younger Dryas, the Last Glacial Maximum or LGM around 20,000 year ago, and the Atlantic meridional overturning circulation or AMOC, Chris Colose’s comments may seem a bit dry and jargonesque. But I find them fascinating!

For one thing, I hadn’t known that people were finding evidence of Younger-Dryas-like episodes at the end of earlier glacial periods, suggesting that these events are in some sense routine, rather than something that requires a freak event like a comet impact to explain. (Yes, some people have argued that a comet was to blame.)

Welcome to Azimuth

Hello!  This is John Baez’s new blog.

I’m a mathematical physicist.  I teach at U.C. Riverside, but tonight my wife and I are flying to Singapore.  For two years she’ll be teaching at the philosophy department of the National University of Singapore, and I’ll be working at the Centre for Quantum Technologies.   This will be a good time to change gears and try something new.  I’ve been working on n-categories and fundamental physics, but now I want to work on more practical things, too.

Why?  I keep realizing more and more that the Earth is in serious trouble! The deep secrets of math and physics are endlessly engrossing — but they can wait, and other things can’t.

I hope we talk about many things here: from math to physics to earth science, biology, computer science, economics, and the technologies of today and tomorrow – but in general, centered around the theme of what scientists can do to help save the planet.

That sounds pretty ambitious, verging on pompous. Right now, though, it’s midnight and I’m sitting in the airport lounge in Los Angeles waiting for a 1:40 am flight to Hong Kong, and then a connection to Singapore. So, that’s all for now. I just felt like kicking off this new blog…