And what,

you might be asking yourselves,

has he been doing all these recent months instead of writing high-octane science friction and science fact here on the intarwubs?

Frozen carbon dioxide turns directly into a gas. How sublime! The dry ice is so cold that it causes water vapor in the air to condense, forming a fog.

Answer: All sorts of zany things! During a recent Really Free Market hosted by Occupy Durham, I had the opportunity to do another chemistry show.  Like the demonstration in my CO2 Problems video, I used soapy water and phenol red pH indicator to help illustrate the properties of frozen carbon dioxide. The color change is particularly dramatic, and is a good tie-in to the environmental effects of CO2. The greenhouse effect seems harder to demonstrate effectively – if anyone has a good way of demonstrating the idea, let me know!

“]

dry ice and phenol red, bubblin' away... { pix courtesy of Specious }

One thing I showed in this demo which wasn’t in CO2 Problems is the strange noises that dry ice makes in response to metal. If you try to cut a piece of dry ice with a knife, or press a paperclip into it, the ice will make a horrible screeching shriek. It’s most dramatic if you put a larger chunk of dry ice into a metal pot – it will scream and skitter around! My explanation? The warm, thermally conductive metal speeds up the sublimation of CO2 near its edge; the expanding gas pushes the metal away briefly and then the pressure buildup dissipates, bringing the metal back in contact with the ice. This oscillation makes the screeching noise. Try it out yourself and see if you think I’m right!

If you are new to climate science, you might be wondering what, exactly, this ‘temperature anomaly’ thing is that you keep hearing about. I know I was a bit confused at first! This post explains the concept, using a real-world example.

Absolute temperatures (yearly averaged) from two sites in the UK: one urban (St. James Park, green) and one rural (Rothamsted, red). Although the urban site is consistently warmer, the two sites show the same warming trend. But is there a way to compare them directly? Data from Jones et al. 2008, kindly provided by Dr. Jones.

Cities tend to be warmer than their surrounding countrysides, a fact known as the urban heat island effect (UHI). This occasionally is offered as an alternative explanation for greenhouse warming, but it fails on closer inspection. We can use data from Jones et al. (2008) [PDF] to see one reason UHI can’t explain observed warming. One time series is from St. James Park, in the city of London; the other is from nearby Rothamsted, a rural site some tens of miles away. As you can see, the urban location is consistently about 2 C warmer; however, the warming is nearly identical at both sites (a strongly significant 0.03 deg C/year). Jones et al. note:

“… the evolution of the time series is almost identical. As for trends since 1961 all sites give similar values …  in terms of anomalies from a common base period, all sites would give similar values.”

This gives us a hint about what a temperature anomaly is: it’s a measure of how much warmer or colder a temperature is compared to some reference point. The anomaly usually gets calculated by selecting a convenient period of time, calculating its average, and subtracting the average from each point in the time series:

Anomaly = Data – Mean (over a common base period)

I’ve taken the absolute temperature data graphed above and calculated temperature anomalies relative to the 20 year period between 1970 and 1990, marked out by the horizontal bars. I calculate the average value of each time series during this period and then subtract that value from each point in the time series. Graphically, this takes the two zigzag lines in the above image and slides them and the horizontal bars vertically, so that both bars are aligned with the horizontal axis. This sets the average of both base periods to zero. When we do this, we do indeed see that the temperature anomalies not only tell the same 0.03C/year story as before, they’re virtually identical:

The same data in the first figure, except that each time series has been re-aligned so that their average value over 1970-1990 is zero. This realignment shows that, beside the systematic warming caused by UHI, the average temperatures in the two locations unfold almost identically.

In this example, the difference was only a few degrees between the two sites. But to talk about the change in global temperatures, you need to deal with even bigger temperature differences – think the Arctic vs the Sahara. Temperature anomalies allow us to measure the trend in global temperature, in the face of these large regional temperature differences.

nomnomnomnomnomnomnomnom

Jones, P., Lister, D., & Li, Q. (2008). Urbanization effects in large-scale temperature records, with an emphasis on China Journal of Geophysical Research, 113 (D16) DOI: 10.1029/2008JD009916

True Stories. Click for a non blackedout site.

If you are unfamiliar with these lovely bits of legislation, they would effectively mean the end of the internet as we know it. If someone posts a link on my site which supposedly violates copyright law, TopOc can disappear – for good. Given the notoriously itchy trigger finger on certain copyright holders, that should scare the pants off you.

Tor, an piece of anonymity software developed by the US Navy for use in repressive countries, would effectively be outlawed. Indeed, the proponents of SOPA and PIPA believe that it would be effective because it is based on censorship techniques which have been used effectively in Syria, China, and Uzbekistan.

Additionally, this legislation would seriously compromise internet security.

Perhaps most disturbingly, members of the US Congress have rejected expert testimony critical of SOPA and PIPA, deriding the critics as ‘nerds’. Considering the poor quality of testimony that they are willing to entertain, this is a real slap in the face.

What can you do? Call your Senators and Representatives! Tell them to keep the IntarTubes free!

Regularly scheduled programming will resume shortly, I swear. The next post will be about the concept of a temperature anomaly – stay tuned! Additionally, I apologize for the relative lack of citations; it is not in my nature to make unsupported assertions. But I got a late start and, well, most of my sources are also participating in the blackout so it would sort of be a moot point. 

UPDATE: …aaaaand we’re back. Thanks to everyone who participated; we’re making a difference!

Some people think that the existence of workarounds for the blackout is somehow a problem for it. On the contrary, that people are finding and using them is a further success of the action. When people use these hacks, it puts them in direct contact with the inner workings of the technology they depend on, and this understanding is as critical for maintaining internet freedom (and freedom in general) as our legal system. Every n00b who is introduced to caches or proxies by the blackout is a success for the world’s first cyber-strike, a success in addition to its influence on policymakers.

Back to writing…

So what is on the TopOc horizon for 2012?

• More hard-hitting commentary!
• More sassing of people who don’t understand graphs!
• Updates on previous projects!
• Audiovisual delights!
• More sweet hax!
• Fractals and fungaloids!
• Pentagons and pentagrams!
• More dry ice! (The shark puppet will also return.)

Here is a mushroom to tide you over while you wait…

It's like a fungal satellite dish!

In the meantime, enjoy this pleasing image

Praying mantis.... or preying mantis??? Clearly the tyrannosaur of the insect world. Photo via Ildar Sagdejev; clix four phool.

Yay!

“How can we expect to predict future climate when we can’t predict the weather?”

“How can we claim to know the half-life of a radioactive element when we can’t predict when a given atom will decay?”

To those familiar with chaos, this shouldn’t come as a surprise. Lorenz didn’t just discover apparent disorder in his model, but a deeper, eerie structure lurking in the noise.

The Lorenz Attractor: wibbly-wobbly mess of the millenium. Three simulation runs (red, green, blue) are shown; they start close together but quickly spin off on different trajectories, demonstrating sensitivity to initial conditions. Nonetheless, the trajectories quickly converge on an intricate structure in the phase space, called an 'attractor'. The attractor doesn't vary with initial conditions, but is instead a feature of the Lorenz equations themselves. Image generated with code from TitanLab - click to check them out

You may remember that the Lorenz equations relate three variables (X, Y, Z), which vary over time. In the above image, I’ve plotted the evolution of three runs of the Lorenz model by putting a dot at each (X(t), Y(t), Z(t)) coordinate, at every time t in the given interval. The three runs start very close together in this three-dimensional ‘phase space’, but quickly diverge.

However, despite their different individual behaviors, these runs are confined to a structure in phase space, known as the Lorenz attractor – an attractor, because all trajectories converge on it, regardless of their initial conditions. If you perturb the system by bouncing it off the attractor, it quickly settles back into the same loops through phase space. Lorenz (1963) described it:

“Additional numerical solutions indicate that other trajectories, originating at points well removed from these surfaces, soon meet these surfaces. The surfaces therefor appear to be composed of all points lying on limiting trajectories.”

Like the statistical measures we saw last time, this attractor reflects the system parameters, and is independent of initial conditions.

Even if the models of fluid turbulence exhibit the Butterfly Effect, it doesn’t follow that climate can’t be modeled. But do the weather models even exhibit that effect? Not necessarily. Dr. Jagadish Shukla ran weather simulations with different initial conditions and compared the results. He found that

“even for very large differences in the initial conditions, the model solutions for the seasonal mean tropical circulation and rainfall do not diverge as would be expected in a chaotic system but instead converge to nearly identical values.”

The initial values chosen were from historical weather scenarios; simulation outputs agreed well not only with each other but also with the observed weather in these scenarios*. More powerful a determinant than initial conditions was sea surface temperature; predicting atmospheric conditions becomes a problem of predicting SSTs. However, Shukla argues that this problem is fairly tractable:

“at least for certain regions of the tropics, the potential exists for making dynamic forecasts of climate anomalies several seasons in advance. [...] It is now clear that certain aspects of the climate system have far more predictability than was previously recognized.”

From Shukla (1998). The top two figures are weather simulations with different starting points: one starts with atmospheric conditions from December 1988 (top) and one from December 1982 (middle). In spite of these different starting points, the models converged on very similar solutions. Moreover, both models agree fairly well with the observed weather in spring 1983 - probably because it includes sea surface temperatures from the end of 1982.

The discovery of extreme sensitivity to initial conditions launched a revolution in math and science, but it doesn’t preclude climate modelling. We may never be able to predict the weather very well far into the future, but perhaps that sort of precisely detailed forecast isn’t what we really want from climatology. In his history of chaos theory, James Gleick mused:

“Only the most naive scientist believes that the perfect model is the one that perfectly represents reality. Such a model would have the same drawbacks as a map as large and detailed as the city it represents, a map depicting every park, every street, every building, every tree, every pothole, every inhabitant, and every map. Were such a map possible, its specificity would defeat its purpose: to generalize and abstract. Mapmakers highlight such features as their clients choose. Whatever their purpose, maps and models must simplify as much as they mimic the world.” (Gleick p.278-279)

oOo

Shukla, J. (1998). Predictability in the Midst of Chaos: A Scientific Basis for Climate Forecasting Science, 282 (5389), 728-731 DOI: 10.1126/science.282.5389.728

Lorenz, Edward N. (1963). Deterministic Nonperiodic Flow Journal of the Atmospheric Sciences, 20 (2)

James Gleick. Chaos: Making a New Science. 1987.

Lorenz equations modeled with software adapted from that available from TitanLab.

A companion article at ArkFab shares my thoughts on peer review in regards to this project and DIY/community/citizen science in general. 

At long last, the much-anticipated booklet, “CO2 Trouble: Ocean Acidification, Dr. Everett, and Congressional Science Standards” is available and approved for human consumption! Download and share HERE (or at Scribd HERE).

In this document, I have bundled, updated, and expanded my series of essays debunking the congressional testimony of Dr. John Everett regarding the environmental chemistry of carbon dioxide.

It has been designed to be a fairly short (less than 30 pages, including images, appendicies, etc.) and accessible read. It has been challenging but fun to write; I have had to learn a lot about GIMP, Python, Scribus, social networking, and of course ocean acidification to get to this point.

It was also very useful for me as an opportunity to go back through my earlier remarks and double-check my work. For example, I later realized that the documentation which Dr. Everett provides for his CO2 data in part two is ambiguous: Although the citation for the rate data is referred to as “Recent Global CO2”, the URL provided links to the longer record as measured at Mauna Loa Observatory. This confusion had led me in the past to make incorrect claims about some of the figures he presents. Ultimately it was inconsequential to my argument, but it was frustrating to have to deal with such ambiguities. On the other hand, this led me into comparing the Mauna Loa record with the global record (Appendix B) which was an interesting exercise.

In researching this project, I also came across new phenomena I wasn’t previously aware of. For example, while I was calculating historical rates of CO2 change, I ran though the 1000-year Law Dome record and saw this:

Pre-1750 CO2 rates, based upon the Law Dome ice cores. There appears to be both a clear periodicity and a major perturbation in this record. I wonder why? Clyx for teh Datumz!

Not only does there appear to be a centennial-scale oscillation in the preindustrial CO2 accumulation rate, there is a clear perturbation around 1500-1700. Intriguingly, these dates roughly correspond to the start of the Little Ice Age. In fact, some have pointed to the decline in CO2 as measured by the Law Dome data as evidence for a speculative but intriguing explanation for the LIA: depopulation caused by the Black Death reversed land use trends, causing reforestation of agricultural land and removing carbon from the atmosphere. I am unfamiliar with the oscillation, however – if you know more about it, or would like to help me  crunch these numbers more thoroughly, let me know

Stay tuned for more updates – coming up is an austere, printer-friendly version, a zine version, press information, and more. And be sure to check out the report, which contains unreleased material. Velociraptors figure prominently (really!)

A part of my John Everett series – read more: 0/I – II.0 – II.5 – II.75 –  III.0 – III.3 - IV.0 – IV.4 – IV.8 – V – VII – VIII – Full Report 

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Last time, we saw that some mathematical systems are so sensitive to initial conditions that even very small uncertainties in their initial state can snowball, causing even very similar states to evolve very differently. The equations describing fluid turbulence are examples of such a system; Lorenz’s discovery of extreme sensitivity to initial conditions ended hopes for long term weather forecasting. Because the state of the weather can only be known so well, the small errors and uncertainties will quickly build up, rendering weather simulations useless for looking more than a few days ahead of time.

But Lorenz’s discovery doesn’t have much impact on climate modelling, contrary to the claims of some climate skuptix. Climate is not weather, and modelling is not forecasting.

Weather refers to the state of the atmosphere at a particular time and place: What temperature is it? Is it raining? How hard is the wind blowing, and in which direction? Climate, on the other hand, is defined in terms of the statistical behavior of these quantities:

“Climate in a narrow sense is usually defined as the average weather, or more rigorously, as the statistical description in terms of the mean and variability of relevant quantities over a period of time ranging from months to thousands or millions of years. [...] Climate change refers to a change in the state of the climate that can be identified (e.g., by using statistical tests) by changes in the mean and/or the variability of its properties, and that persists for an extended period, typically decades or longer. ” IPCC

Many climate skuptik talking points derive from confusing these two quantities, in much the same way that a gambler might win a few hands of poker and decide that they are on a roll.

Although it is generally not possible to predict a specific future state of the weather (there is no telling what temperature it will be in Oregon on December 21 2012), it is still possible to make statistical claims about the climate (it is very likely that Oregon’s December 2012 temperatures will be colder than its July 2012 temperatures). It is very likely that the reverse will be true in New Zealand. It is safe to conclude that precipitation will be more frequent in the Amazon than in the Sahara, even if you can’t tell exactly when and where that rain will fall.

In fact, Lorenz’s groundbreaking paper, ‘Deterministic Nonperiodic Flow’, would seem to endorse this sort of statistical approach to understanding fluid dynamics:

“Because instantaneous turbulent flow patterns are so irregular, attention is often confined to the statistics of turbulence, which, in contrast to the details of turbulence, often behave in a regular well-organized manner.” (Lorenz 1963)

Let’s take a closer look.

Fig. 1. Three solutions of the Lorenz equations, starting at virtually identical points. Although the solutions are similar at first, they rapidly decouple around T=12.

The Lorenz equations consist of three variables describing turbulent fluid flow (X,Y, and Z), and three controlling parameters (r, b, and s). The equations are differential equations, meaning that a variable is described in terms of how it changes over time- saying ‘Johnny is driving west at 60 miles per hour’ is a simple differential equation. In order to solve a DiffEq, you need an initial condition – “Johnny started in Chicago” is an initial condition; without knowing that, you can’t say where she will be after driving for three hours.

Here’s the equations:

 X’ = s * (Y – X)

 Y’ = X * (r  – Z) – Y

 Z’ = X * Y – b * Z

… where V’ is the time derivative of variable V.

Its ironic that the same people who criticize climate models for their supposed lack of realism are relying on results derived from a such an austere model, described by James Gleick:

‘The sun beat down through a sky that had never seen clouds. The winds swept across an earth as smooth as glass. Night never came, and autumn never gave way to winter. It never rained.’ – (Gleick 1987 p. 11)

Fig. 1, above, demonstrates this model’s sensitivity to initial conditions. Even though the three simulation runs start off almost exactly the same, they slowly drift, and then spontaneously fly apart, quickly losing any similarity to each other. This quick catastrophic divergence might seem like it would doom a climate model, but remember that climate is defined statistically, and even very dissimilar runs can have consistent statistical behavior.

Fig. 2. The mean value of the variable Z in the Lorenz equations. At each value of the Rayleigh number, the Lorenz system is solved with 10 different starting conditions, with s = 10 and b = 8/3.

Here’s an example. In Fig 2, I’ve varied the parameter r (more on that in a second – for now, concentrate on the canonical Lorenz value of r=28) and then solved the Lorenz equations with a number of different initial conditions. Then, I found the average value of the output of variable Z (interpreted as a measure of the heat profile in the turbulent fluid). The values of mean(Z) are very tightly clustered, meaning that the initial conditions didn’t influence the average very much.

On the other hand, the mean is strongly determined by the parameter r. r is called the Rayleigh number (Strogatz 1994), and it describes how easily a fluid will convect. Similarly, climatic quantities such as globally averaged temperature are determined by system parameters such as greenhouse gas concentrations much more than they are by initial conditions

Fig. 3. The standard deviations of a Lorenz variable from an ensemble of simulations, as a function of the Rayleigh number. They're not as tightly clustered as the means were, but they still appear well-behaved - the 'climate' of the Lorenz model can be described in spite of chaos!

Here’s another example. I’ve run the simulations again, and taken the variable X (interpreted as the intensity of convection). Then I calculate the standard deviation of the time series. At a given Rayleigh number (eg, 28) the standDev is not as well clustered as the average was in Fig 2, so initial conditions play a larger role here. Still, the statistics of X seem to be clustered pretty tightly, and the parameter r seems to have a strong(er?) control on the statistics. There is a clear increase in standDev(X) over 20 < r < 45, and a tight linear increase over 35 < r < 45. There also appears to be a lessened dependence on initial conditions in 35 < r < 45 and there seems to be a significant dip in 30 < r < 35. We can also see a sharp transition around r = 25; this corresponds to a critical value of the Rayleigh number ~24.74, the point at which the stability of steady convection changes. (Lorenz 1963)

I took a chaotic model, ran several simulations with different starting conditions, and considered the statistical properties of the ‘ensemble’ of all the simulations. This procedure is standard in climate modelling. My favorite example is in “Is the climate warming or cooling?” (Easterling & Wehner 2009). This paper explores the question: does a decade-long downturn in global temperatures indicate that global warming has stopped? To answer this question, the authors looked at a 20th century temperature record, observing that there are several ten-year intervals containing no trend, or even a slight cooling trend, despite a long-term warming. (Fig 4; this might sound familiar )

Fig. 4. The probability that a given decade will have a particular trend. These probabilities are calculated from ensembles of model runs and 20th century temperature reconstructions. As you'd expect, the preindustrial simulations, lacking today's greenhouse forcing, show equal probabilities for warming and cooling trends on the decade scale. Observed and predicted climate change shifts the probabilities towards warming, but the odds of a decade showing no trend, or even a cooling trend, remains significant. Notice also the good agreement between the statistics derived from the model runs for the 20th century and those derived from the instrumental record. Sauce is (Easterling & Wehner 2009)

They also took an ensemble of climate simulations – some hindcasts which replayed variations of the preindustrial era or the 20th century, some forecasts of the 21st century – and they looked at the probability of a decade having a particular trend (a statistical property of the collection). They found (Fig. 4):

“The observed record shows a very similar distribution to the 20th century simulations, especially considering that only one version of the observed record was used in this analysis adding credence to the conclusions in the IPCC AR4 that the observed warming since 1950 is very likely due to increasing greenhouse gases. Finally for the simulations of the entire 21st century there is still about a 5% chance of a negative decadal trend, even in the absence of any simulated volcanic eruptions. If we restrict the period to the first half of the 21st century the probability increases to about 10% revealing that the trend in surface air temperature has its own positive trend in the A2 emissions scenario.”

The Lorenz equations are a canonical example of extreme sensitivity to initial conditions. However, their chaotic nature is not an obstacle to making statistical claims about their behavior. Climate is defined statistically, so the fact the weather is chaotic is not a barrier to climate modelling.

Lorenz, Edward N. (1963). Deterministic Nonperiodic Flow Journal of the Atmospheric Sciences, 20 (2)

Regarding climate models, physician and science fiction writer Michael Crichton had this to say:

“Since climate may be a chaotic system—no one is sure—these predictions are inherently doubtful, to be polite.” (Aliens Cause Global Warming)

What does he mean when he says that climate may be chaotic, and what impact does this have on climate modelling?

Flash back to the early 1960s. Meteorologist Edward Lorenz was studying a bare-bones weather model, consisting of three differential equations. Give the model an initial state and the differential equations would describe how the state changes over time, in much the same way that you can predict where Johnny will be in three hours’ time, given that he starts in Chicago and is driving west at 60 miles per hour. The hope was that with a big enough computer, a powerful enough model, and an accurately measured state of the atmosphere, the weather could one day be predicted far in advance.

Lorenz, the story goes, found a run of the model which interested him, and sat down to replay the simulation. He entered the initial conditions and set the model in motion, only to watch in bewilderment as the replay rapidly diverged from the original simulation.

"From nearly the same starting point, Edward Lorenz saw his computer weather produce patterns that grew farther and farther apart until all resemblance disappeared" (Image and caption from Chaos: Making a New Science, by James Gleick, 1987, p.17)

Lorenz tore his code apart looking for the error, only to realise that the error had been in his assumptions. In a distinctly Crichtonesque twist, the computer worked with numbers to six decimal places (0.123456) but only printed out values to three decimal places (0.123) in order to save space. It was these shortened number which Lorenz entered as the initial conditions for his model. Surely those last digits were inconsequential; after all, they were but a few hundred parts per million, comparable to the atmospheric concentration of the trace gas carbon dioxide.

Oh, but the consequences! Its roots stretched back to earlier anomalies and the term ‘chaos’ would not be introduced for another decade, but it was Lorenz’s observation which heralded the beginnings of chaos theory.

Lorenz had discovered that even very small changes in the state of a chaotic system can quickly and radically change the way that the system develops over time. This property is known as extreme sensitivity to initial conditions, also called the ‘Butterfly Effect’ because it suggested neglecting an event as small as the flapping of a butterfly’s wings could be enough to derail a weather forecast. There is more to chaotic systems than the Butterfly Effect, but this characteristic is one of their best know properties. Lorenz’s work put and end to hopes of long-term weather forecasting. The state of the atmosphere could only be known so well, and even the smallest of imprecisions would lead the simulations to catastrophic failure.

‘Nobody believes a weather prediction twelve hours ahead. Now we’re asked to believe a prediction that goes out 100 years into the future? And make financial investments based on that prediction? Has everybody lost their minds?’ – Crichton

But does chaos theory signal doom for climate modelling? Stay tuned for part II….

You can find the current paper here.

A while back, I got the idea to investigate how the entropy of a poker tournament evolves with time. In thermodynamics, entropy is a measure of how ‘spread out’ energy is amongst the states available to it. When the energy in a system is concentrated in one place (like a hot cup of coffee in a cold room), the entropy of the system is low. When the energy is spread out (a few hours later, both the room and the coffee are the same temperature) the entropy of the system is high.  Although originally defined for distributions of physical energy, entropy can be defined more generally to study arbitrary distributions – for example the distribution of capital, in the form of chips, between players in a poker tournament.

Just by looking at the formal structure of the game, you can tell some things about how entropy behaves. For example, it is formally required that entropy falls to zero with time. On the one hand, this is a fancy way of saying, ‘one person will eventually win the tournament’; on the other hand, it is interesting to consider that this is the exact opposite of what happens in the physical, thermodynamic world. The entropy of a closed thermodynamic system necessarily increases with time: hot coffee in a cold room will cool down, but warm coffee in a warm room will never heat up. However, the entropy of a closed poker table necessarily decreases. It has a second law of thermodynamics that runs in the opposite direction from ours.

Beyond a bottom-up analytical approach, I wanted to see how real-life tournaments behave. Although online gaming has generated a wealth of data, accessing the data is difficult, and I could find only one other paper which investigated the phenomenon. This was ‘Universal statistical properties of poker tournaments’ by Clement Sire (Sire 2007). The author notes that most of the game-theoretic work on poker has been on largely restricted to optimal betting strategies in head-to-head tournaments. Sire builds a relatively simple model of player behavior: a player bets according to a simple evaluation of their hand and the table, and goes all-in if their hand is evaluated to be above a certain quality. This model predicts that tournaments will have certain statistical properties; this prediction is born out in real-life tournaments.

I was only able to get two suitable datasets, so it’s hard to draw solid conclusions about what is going on. However,  there are interesting observations to be made. Here’s a visualisation of one tournament:

Tournament fortunes and entropy for one set of data. The top graph shows the holdings of each player; the bottom graph shows the entropy of the tournament as a function of time (green). The red lines are the upper and lower bounds for the tournament; this is a function of the number of players, which is in turn a function of time.

For one thing, the entropy remains close to its theoretical maximum value, generally ~90% of the absolute maximum. In the tournament pictured, entropy appears to increase to a maximum, and then slowly decline, before the loss of a player abruptly changes the distribution of chips (the sudden changes in the stair-step of the max/min entropy.)  Furthermore, when the tournament entropy is normalized by its maximum entropy, there is a significant upwards trend (p = 0.012). Over the course of the tournament, the entropy increases towards its theoretical maximum. Additionally, it is interesting to me that, in between the losses of players, entropy appears to increase, reach a maximum, and then decrease again before collapsing. (It’s more clear in this image) I interpret this as the redistribution of the winnings of the leaving player (eg, of cyan to black and then to the rest of the table in hands 1-25) followed by a concentration of chips which eventually pushes a player out (yellow vs. the rest, hands 25-40).

However, none of these observations held in the second tournament. One possibility is that, because the second tournament was faster paced, players were eliminated much faster, and these frequent perturbations are obscuring the pattern. On the other hand, it’s entirely possible that the first tournament was a fluke. The only way to resolve this question is with more data!

One reason I am interested in this question has to do with a series of papers written by Arto Annilla from the University of Helsinki. He’s shown that protein folding, genomics, abiogenesis, ecological succession – pretty much every aspect of nature – is not merely constrained by the second law of thermodynamics, but a direct consequence of it. Most relevant to this project, I think, is his analysis of economies and ecosystems. The ultimate goal of each, he argues, is not only to increase entropy to a maximum, but to do so as fast as possible. To the extent that poker tournaments can be thought of as a toy model of an economy, they may provide empirical insights into thermoeconomics. Of course, we’ve already seen that tournament entropy is formally constrained to decrease with time, though it would be interesting to see the behavior of a tournament which is not driven by a rising minimum bet, as these are. The first tournament may show an upward trend after perturbation from quasiequilibrium conditions, and the relative entropy may show a tournament-scale increase. (Or, it may not. Argh! Why oh why must n=2?!)

Dr. Annilla’s discussion of economics also applies to ecosystems, and I often think about tournaments in ecological terms, with the chips representing environmental resources which various species (players) compete for. Indeed, the top graph in the above image is modelled after a common visual representation of relative abundance of bacterial species in microbiomes:

The relative abundances of (and hence, the distribution of resources across) different bacterial taxa on a leaf. The horizontal axis is time; each vertical bar is a snapshot of the bacterial community on the leaves of a cottonwood tree, at points in time over the course of a year. clix for suace11! '

In this context it is interesting to me how often, at least in the slow tournament, there is a continuous decline in a dying population until the player is forced to go all in and loses (E.g. yellow and green in the first image.) This strongly reminds me of species whose population is slowly reduced until it’s pushed over a critical threshold, and goes extinct.

I have always been interested in the tendency of multiple interacting systems, sometimes trivially simple ones, can create a supersystem with complex behavior. Cellular automata, the nervous system, and ant colonies are all examples. Might a poker tournament be viewed as a collection of automata, interacting along thermodynamic lines? (Sire 2007) notes,

Although a priori governed by human laws (bluff, prudence, aggressiveness…), we shall find that some of their interesting properties can be quantitatively described.

What sort of self-organizing properties might a collection of interacting automata have when equipped with poker strategies, even very simple ones such as those used by Sire?

There’s a lot of interesting questions left to explore (for example, is it more meaningful to normalize entropy by division, or to add the change in entropy caused by player dropout? The latter could be physically interpretted as the entropy of a closed system including both players and explayers who take away a certain amount of entropy when they leave the table.) I’ve had some questions raised about the validity and meaning of the distribution I used to calculate the entropy (though I think that I did it right). I’ve also gotten to have fun conversations with people about interesting problems. And I’ve had sweet theme songs to back me up.

If you are interested and want to hear more or have ideas, please comment! And definitely, if you or someone you know have tournament histories and would like to contribute them, please contact me at ThermoPoker(at)gmail.com.

~~~—~~~

Clément Sire (2007). Universal statistical properties of poker tournaments J. Stat. Mech. (2007) P08013 arXiv: physics/0703122v3

Annila, Arto (2009). Economies Evolve by Energy Dispersal Entropy, 11 (4), 606-633 DOI: 10.3390/e11040606