Azimuth News (Part 5)

11 June, 2016

I’ve been rather quiet about Azimuth projects lately, because I’ve been too busy actually working on them. Here’s some of what’s happening:

Jason Erbele is finishing his thesis, entitled Categories in Control: Applied PROPs. He successfully gave his thesis defense on Wednesday June 8th, but he needs to polish it up some more. Building on the material in our paper “Categories in control”, he’s defined a category where the morphisms are signal flow diagrams. But interestingly, not all the diagrams you can draw are actually considered useful in control theory! So he’s also found a subcategory where the morphisms are the ‘good’ signal flow diagrams, the ones control theorists like. For these he studies familiar concepts like controllability and observability. When his thesis is done I’ll announce it here.

Brendan Fong is also finishing his thesis, called The Algebra of Open and Interconnected Systems. Brendan has already created a powerful formalism for studying open systems: the decorated cospan formalism. We’ve applied it to two examples: electrical circuits and Markov processes. Lately he’s been developing the formalism further, and this will appear in his thesis. Again, I’ll talk about it when he’s done!

Blake Pollard and I are writing a paper called “A compositional framework for open chemical reaction networks”. Here we take our work on Markov processes and throw in two new ingredients: dynamics and nonlinearity. Of course Markov processes have a dynamics, but in our previous paper when we ‘black-boxed’ them to study their external behaviour, we got a relation between flows and populations in equilibrium. Now we explain how to handle nonequilibrium situations as well.

Brandon Coya, Franciscus Rebro and I are writing a paper that might be called “The algebra of networks”. I’m not completely sure of the title, nor who the authors will be: Brendan Fong may also be a coauthor. But the paper explores the technology of PROPs as a tool for describing networks. As an application, we’ll give a new shorter proof of the functoriality of black-boxing for electrical circuits. This new proof also applies to nonlinear circuits. I’m really excited about how the theory of PROPs, first introduced in algebraic topology, is catching fire with all the new applications to network theory.

I expect all these projects to be done by the end of the summer. Near the end of June I’ll go to the Centre for Quantum Technologies, in Singapore. This will be my last summer there. My main job will be to finish up the two papers that I’m supposed to be writing.

There’s another paper that’s already done:

Kenny Courser has written a paper “A bicategory of decorated cospans“, pushing Brendan’s framework from categories to bicategories. I’ll explain this very soon here on this blog! One goal is to understand things like the coarse-graining of open systems: that is, the process of replacing a detailed description by a less detailed description. Since we treat open systems as morphisms, coarse-graining is something that goes from one morphism to another, so it’s naturally treated as a 2-morphism in a bicategory.

So, I’ve got a lot of new ideas to explain here, and I’ll start soon! I also want to get deeper into systems biology.



In the fall I’ve got a couple of short trips lined up:

• Monday November 14 – Friday November 18, 2016 – I’ve been invited by Yoav Kallus to visit the Santa Fe Institute. From the 16th to 18th I’ll attend a workshop on Statistical Physics, Information Processing and Biology.

• Monday December 5 – Friday December 9 – I’ve been invited to Berkeley for a workshop on Compositionality at the Simons Institute for the Theory of Computing, organized by Samson Abramsky, Lucien Hardy, and Michael Mislove. ‘Compositionality’ is a name for how you describe the behavior of a big complicated system in terms of the behaviors of its parts, so this is closely connected to my dream of studying open systems by treating them as morphisms that can be composed to form bigger open systems.

Here’s the announcement:

The compositional description of complex objects is a fundamental feature of the logical structure of computation. The use of logical languages in database theory and in algorithmic and finite model theory provides a basic level of compositionality, but establishing systematic relationships between compositional descriptions and complexity remains elusive. Compositional models of probabilistic systems and languages have been developed, but inferring probabilistic properties of systems in a compositional fashion is an important challenge. In quantum computation, the phenomenon of entanglement poses a challenge at a fundamental level to the scope of compositional descriptions. At the same time, compositionally has been proposed as a fundamental principle for the development of physical theories. This workshop will focus on the common structures and methods centered on compositionality that run through all these areas.

I’ll say more about both these workshops when they take place.


Interview (Part 2)

21 March, 2016

Greg Bernhardt runs an excellent website for discussing physics, math and other topics, called Physics Forums. He recently interviewed me there. Since I used this opportunity to explain a bit about the Azimuth Project and network theory, I thought I’d reprint the interview here. Here is Part 2.

 

Tell us about your experience with past projects like “This Week’s Finds in Mathematical Physics”.

I was hired by U.C. Riverside back in 1989. I was lonely and bored, since Lisa was back on the other coast. So, I spent a lot of evenings on the computer.

We had the internet back then—this was shortly after stone tools were invented—but the world-wide web hadn’t caught on yet. So, I would read and write posts on “newsgroups” using a program called a “news server”. You have to imagine me sitting in front of an old green­-on­-black cathode ray tube monitor with a large floppy disk drive, firing up the old modem to hook up to the internet.

In 1993, I started writing a series of posts on the papers I’d read. I called it “This Week’s Finds in Mathematical Physics”, which was a big mistake, because I couldn’t really write one every week. After a while I started using it to explain lots of topics in math and physics. I wrote 300 issues. Then I quit in 2010, when I started taking climate change seriously.

Share with us a bit about your current projects like Azimuth and the n­-Café.

The n­-Category Café is a blog I started with Urs Schreiber and the philosopher David Corfield back in 2006, when all three of us realized that n­-categories are the big wave that math is riding right now. We have a bunch more bloggers on the team now. But the n­-Café lost some steam when I quit work in n­-categories and Urs started putting most of his energy into two related projects: a wiki called the nLab and a discussion group called the nForum.

In 2010, when I noticed that global warming was like a huge wave crashing down on our civilization, I started the Azimuth Project. The goal was to create a focal point for scientists and engineers interested in saving the planet. It consists of a team of people, a blog, a wiki and a discussion group. It was very productive for a while: we wrote a lot of educational articles on climate science and energy issues. But lately I’ve realized I’m better at abstract math. So, I’ve been putting more time into working with my grad students.

What about climate change has captured your interest?

That’s like asking: “What about that huge tsunami rushing toward us has captured your interest?”

Around 2004 I started hearing news that sent chills up my spine ­ and what really worried me is how few people were talking about this news, at least in the US.

I’m talking about how we’re pushing the Earth’s climate out of the glacial cycle we’ve been in for over a million years, into brand new territory. I’m talking about things like how it takes hundreds or thousands of years for CO2 to exit the atmosphere after it’s been put in. And I’m talking about how global warming is just part of a bigger phenomenon: the Anthropocene. That’s a new geological epoch, in which the biosphere is rapidly changing due to human influences. It’s not just the temperature:

• About 1/4 of all chemical energy produced by plants is now used by humans.

• The rate of species going extinct is 100­–1000 times the usual background rate.

• Populations of large ocean fish have declined 90% since 1950.

• Humans now take more nitrogen from the atmosphere and convert it into nitrates than all other processes combined.

8­-9 times as much phosphorus is flowing into oceans than the natural background rate.

This doesn’t necessarily spell the end of our civilization, but it is something that we’ll all have to deal with.

So, I felt the need to alert people and try to dream up strategies to do something. That’s why in 2010 I quit work on n­-categories and started the Azimuth Project.

Carbon Dioxide Variations

You have life experience on both US coasts. Which do you prefer and why?

There are some differences between the coasts, but they’re fairly minor. The West Coast is part of the Pacific Rim, so there’s more Asian influence here. The seasons are less pronounced here, because winds in the northern hemisphere blow from west to east, and the oceans serve as a temperature control system. Down south in Riverside it’s a semi­-desert, so we can eat breakfast in our back yard in January! But I live here not because I like the West Coast more. This just happens to be where my wife Lisa and I managed to get a job.

What I really like is getting out of the US and seeing the rest of the world. When you’re at cremation ritual in Bali, or a Hmong festival in Laos, the difference between regions of the US starts seeming pretty small.

But I wasn’t a born traveler. When I spent my first summer in England, I was very apprehensive about making a fool of myself. The British have different manners, and their old universities are full of arcane customs and subtle social distinctions that even the British find terrifying. But after a few summers there I got over it. First, all around the world, being American gives you a license to be clueless. If you behave any better than the worst stereotypes, people are impressed. Second, I spend most of my time with mathematicians, who are incredibly forgiving of bad social behavior as long as you know interesting theorems.

By now I’ve gotten to feel very comfortable in England. The last couple of years I’ve spent time at the quantum computation group at Oxford–the group run by Bob Coecke and Samson Abramsky. I like talking to Jamie Vicary about n­categories and physics, and also my old friend Minhyong Kim, who is a number theorist there.

I was also very apprehensive when I first visited Paris. Everyone talks about how the waiters are rude, and so on. But I think that’s an exaggeration. Yes, if you go to cafés packed with boorish tourists, the waiters will treat you like a boorish tourist—so don’t do that. If you go to quieter places and behave politely, most people are friendly. Luckily Lisa speaks French and has some friends in Paris; that opens up a lot of opportunities. I don’t speak French, so I always feel like a bit of an idiot, but I’ve learned to cope. I’ve spent a few summers there working with Paul­-André Melliès on category theory and logic.

Yau Ma Tei Market - Hong Kong

Yau Ma Tei Market – Hong Kong

I was also intimidated when I first spent a summer in Hong Kong—and even more so when I spent a summer in Shanghai. Lisa speaks Chinese too: she’s more cultured than me, and she drags me to interesting places. My first day walking around Shanghai left me completely exhausted: everything was new! Walking down the street you see people selling frogs in a bucket, strange fungi and herbs, then a little phone shop where telephone numbers with lots of 8’s cost more, and so on: it’s a kind of cognitive assault.

But again, I came to enjoy it. And coming back to California, everything seemed a bit boring. Why is there so much land that’s not being used? Where are all the people? Why is the food so bland?

I’ve spent the most time outside the US in Singapore. Again, that’s because my wife and I both got job offers there, not because it’s the best place in the world. Compared to China it’s rather sterile and manicured. But it’s still a fascinating place. They’ve pulled themselves up from a British colonial port town to a multi­cultural country that’s in some ways more technologically advanced than the US. The food is great: it’s a mix of Chinese, Indian, Malay and pretty much everything else. There’s essentially no crime: you can walk around in the darkest alley in the worst part of town at 3 am and still feel safe. It’s interesting to live in a country where people from very different cultures are learning to live together and prosper. The US considers itself a melting-pot, but in Singapore they have four national languages: English, Mandarin, Malay and Tamil.

Most of all, it’s great to live in places where the culture and politics is different than where I grew up. But I’m trying to travel less, because it’s bad for the planet.

You’ve gained some fame for your “crackpot index”. What were your motivations for developing it? Any new criteria you’d add?

After the internet first caught on, a bunch of us started using it to talk about physics on the usenet newsgroup sci.physics.

And then, all of a sudden, crackpots around the world started joining in!

Before this, I don’t think anybody realized how many people had their own personal theories of physics. You might have a crazy uncle who spent his time trying to refute special relativity, but you didn’t realize there were actually thousands of these crazy uncles.

As I’m sure you know here at Physics Forums, crackpots naturally tend to drive out more serious conversations. If you have some people talking about the laws of black hole thermodynamics, and some guy jumps in and says that the universe is a black hole, everyone will drop what they’re doing and argue with that guy. It’s irresistible. It reminds me of how when someone brings a baby to a party, everyone will start cooing to the baby. But it’s worse.

When physics crackpots started taking over the usenet newsgroup sci.physics, I discovered that they had a lot of features in common. The Crackpot Index summarizes these common features. Whenever I notice a new pattern, I add it.

For example: if someone starts comparing themselves to Galileo and says the physics establishment is going after them like the Inquisition, I guarantee you that they’re a crackpot. Their theories could be right—but unfortunately, they’ve got delusions of grandeur and a persecution complex.

It’s not being wrong that makes someone a crackpot. Being a full­-fledged crackpot is the endpoint of a tragic syndrome. Someone starts out being a bit too confident that they can revolutionize physics without learning it first. In fact, many young physicists go through this stage! But the good ones react to criticism by upping their game. The ones who become crackpots just brush it off. They come up with an idea that they think is great, and when nobody likes it, they don’t say “okay, I need to learn more.” Instead, they make up excuses: nobody understands me, maybe there’s a conspiracy at work, etc. The excuses get more complicated with each rebuff, and it gets harder and harder for them to back down and say “whoops, I was wrong”.

When I wrote the Crackpot Index, I thought crackpots were funny. Alexander Abian claimed all the world’s ills would be cured if we blew up the Moon. Archimedes Plutonium thinks the Universe is a giant plutonium atom. These ideas are funny. But now I realize how sad it is that someone can start with an passion for physics and end up in this kind of trap. They almost never escape.

Who are some of your math and physics heroes of the past and of today?

Wow, that’s a big question! I think every scientist needs to have heroes. I’ve had a lot.

Marie Curie

Marie Curie

When I was a kid, I was in love with Marie Curie. I wanted to marry a woman like her: someone who really cared about science. She overcame huge obstacles to get a degree in physics, discovered not one but two new elements, often doing experiments in her own kitchen—and won not one but two Nobel prizes. She was a tragic figure in many ways. Her beloved husband Pierre, a great physicist in his own right, slipped and was run over by a horse­-drawn cart, dying instantly when the wheels ran over his skull. She herself probably died from her experiments with radiation. But this made me love her all the more.

Later my big hero was Einstein. How could any physicist not have Einstein as a hero? First he came up with the idea that light comes in discrete quanta: photons. Then, two months later, he used Brownian motion to figure out the size of atoms. One month after that: special relativity, unifying space and time! Three months later, the equivalence between mass and energy. And all this was just a warmup for his truly magnificent theory of general relativity, explaining gravity as the curvature of space and time. He truly transformed our vision of the Universe. And then, in his later years, the noble and unsuccessful search for a unified field theory. As a friend of mine put it, what matters here is not that he failed: what matters is that he set physics a new goal, more ambitious than any goal it had before.

Later it was Feynman. As I mentioned, my uncle gave me Feynman’s Lectures on Physics. This is how I first learned Maxwell’s equations, special relativity, quantum mechanics. His way of explaining things with a minimum of jargon, getting straight to the heart of every issue, is something I really admire. Later I enjoyed his books like Surely You Must Be Joking. Still later I learned enough to be impressed by his work on QED.

But when you read his autobiographical books, you can see that he was a bit too obsessed with pretending to be a fun­-loving ordinary guy. A fun­-loving ordinary guy who just happens to be smarter than everyone else. In short, a self­-absorbed showoff. He could also be pretty mean to women—and in that respect, Einstein was even worse. So our heroes should not be admired uncritically.

Alexander Grothendieck

Alexander Grothendieck

A good example is Alexander Grothendieck. I guess he’s my main math hero these days. To solve concrete problems like the Weil conjectures, he avoided brute force techniques and instead developed revolutionary new concepts that gently dissolved those problems. And these new concepts turned out to be much more important than the problems that motivated him. I’m talking about abelian categories, schemes, topoi, stacks, things like that. Everyone who really wants to understand math at a deep level has got to learn these concepts. They’re beautiful and wonderfully simple—but not easy to master. You have to really change your world view to understand them, just like general relativity or quantum mechanics. You have to rewire your neurons.

At his peak, Grothendieck seemed almost superhuman. It seems he worked almost all day and all night, bouncing his ideas off the other amazing French algebraic geometers. Apparently 20,000 pages of his writings remain unpublished! But he became increasingly alienated from the mathematical establishment and eventually disappeared completely, hiding in a village near the Pyrenees.

Which groundbreaking advances in science and math are you most looking forward to?

I’d really like to see progress in figuring out the fundamental laws of physics. Ideally, I’d like to know the Theory of Everything. Of course, we don’t even know that there is one! There could be an endless succession of deeper and deeper realizations to be had about the laws of physics, with no final answer.

If we ever do discover the Theory of Everything, that won’t be the end of the story. It could be just the beginning. For example, next we could ask why this particular theory governs our Universe. Is it necessary, or contingent? People like to chat about this puzzle already, but I think it’s premature. I think we should find the Theory of Everything first.

Unfortunately, right now fundamental physics is in a phase of being “stuck”. I don’t expect to see the Theory of Everything in my lifetime. I’d be happy to see any progress at all! There are dozens of very basic things we don’t understand.

When it comes to math, I expect that people will have their hands full this century redoing the foundations using ∞-categories, and answering some of the questions that come up when you do this. The crowd working on “homotopy type theory” is making good progress–but so far they’re mainly thinking about ∞-groupoids, which are a very special sort of ∞-category. When we do all of math using ∞-categories, it will be a whole new ballgame.

And then there’s the question of whether humanity will figure out a way to keep from ruining the planet we live on. And the question of whether we’ll succeed in replacing ourselves with something more intelligent—or even wiser.

The Milky Way and Andromeda Nebula after their first collision, 4 billion years from now

The Milky Way and Andromeda Nebula after their first collision, 4 billion years from now

Here’s something cool: red dwarf stars will keep burning for 10 trillion years. If we, or any civilization, can settle down next to one of those, there will be plenty of time to figure things out. That’s what I hope for.

But some of my friends think that life always uses up resources as fast as possible. So one of my big questions is whether intelligent life will develop the patience to sit around and think interesting thoughts, or whether it will burn up red dwarf stars and every other source of energy as fast as it can, as we’re doing now with fossil fuels.

What does the future hold for John Baez? What are your goals?

What the future holds for me, primarily, is death.

That’s true of all of us—or at least most of us. While some hope that technology will bring immortality, or at least a much longer life, I bet most of us are headed for death fairly soon. So I try to make the most of the time I have.

I’m always re­-evaluating what I should do. I used to spend time thinking about quantum gravity and n­-categories. But quantum gravity feels stuck, and n­-category theory is shooting forward so fast that my help is no longer needed.

Climate change is hugely important, and nobody really knows what to do about it. Lots of people are trying lots of different things. Unfortunately I’m no better than the rest when it comes to the most obvious strategies—like politics, or climate science, or safer nuclear reactors, or better batteries and photocells.

The trick is finding things you can do better than other people. Right now for me that means thinking about networks and biology in a very abstract way. I’m inspired by this remark by Patten and Witkamp:

To understand ecosystems, ultimately will be to understand networks.

So that’s my goal for the next five years or so. It’s probably not be the best thing anyone can do to prepare for the Middle Anthropocene. But it may be the best thing I can do: use the math I know to help people understand the biosphere.

It may seem like I keep jumping around: from quantum gravity to n-categories to biology. But I keep wanting to think about networks, and how they change in time.

At some point I hope to retire and become a bit more of a self­-indulgent wastrel. I could write a fun book about group theory in geometry and physics, and a fun book about the octonions. I might even get around to spending more time on music!

John Baez in Namo Gorge, Gansu

John Baez


Azimuth News (Part 4)

1 January, 2016

 

Happy New Year!

I’ve been rethinking my approach to life. Nothing major, just some small course corrections.

Blogging

First, I’ve decided to stop posting so much on Google+, and post more here on Azimuth. I explain why here:

New Year’s Resolution.

I want a vibrant, lively online environment where people talk about things in serious, almost obsessive ways. At times this blog has been like that. So was the n-Category Café, back when a core group of people were all focused on roughly the same thing.

Lately I’ve been posting very little here on Azimuth. Instead, I’ve been working hard with a team of grad students to figure out how network theory can organize our understanding of circuits, control theory and nonequilibrium thermodynamics. My more general posts on ecological issues and science in general have been going to Google+. I now suspect it’s bad to split up my blogging that way. If I write more stuff in the same place, I hope more people will come here to talk.

Research

Second, I have slowly come to realize that my talents lie in highly theoretical work—for example, network theory—rather than data-driven work like climate science, or the practical task of figuring out what to do about global warming. I still think that adapting to the Anthropocene and developing an ‘ecotechnic civilization’ is the challenge of the century for
civilization as a whole and scientists in particular. But I think the more urgent aspects of this—the ones that need to be done really soon—are not the ones I’m good at. The stuff I’m good at will help later, if at all. And yet, it doesn’t pay for me to do things that other people are already doing better.

There’s been a big change since I started this blog in 2010! Back then, it seemed only a few people knew quite how serious global warming would be, so I felt the need to shout an alarm. This December, thousands of politicians from around the world met in Paris to do something about it—and while they haven’t done enough yet, they all know the basic facts that had me so worried: for example, that we need to leave most of the world’s fossil fuels unburnt in the ground, in part because if we don’t, the effects will last for thousands of years:


More generally, there’s been a huge shift towards recognizing that we have to change our habits, quickly. In 2013, Copenhagen announced it will try to go carbon-neutral by 2025. In October of this year, California passed a law requiring the state to obtain 50% of its electricity from renewables by 2035. In December, San Diego became the largest city in the USA to require that all of the city’s power to come from renewables—again, by 2035.

And so on. The battle has not been won, but it has been joined. Countries, localities, cities and individuals around the world are volunteering to tackle this problem.

I can serve as a cheerleader for this trend, but I’m not really a politician, an engineer or even an experimentalist. Mathematicians, too, have their part to play. So, this year I’ll keep talking about network theory, and push it toward the ‘green mathematics’ I’ve been dreaming about, by getting serious about its applications to biology and ecology.

More on that soon!


Azimuth News (Part 3)

6 February, 2015

post by David Tanzer

Here are some notes from the back offices of the Azimuth project. After a long and productive stay as the Azimuth tech guy, Andrew Stacey is moving along and passing the baton to me. As part of this change, we’ve relocated the servers to a new Azimuth hosted account, and updated the forum software.

The forum is now at a new location:

https://forum.azimuthproject.org

This is where we collaborate on writing wiki and blog articles, on research and education projects, and on software development and systems issues. It’s also a fun place to chat with other professionals in a wide range of science-related fields.

So come on down to the forum! If you want to post, just apply for an account there. Acceptance criteria are minimal. A sincere desire to help goes a long way.

Important:  please use your full name, using “camel case” capitalization e.g. DavidTanzer, as your userid.  I will then put the spaces into your user ID.  (We want the spaces, but the registration form blocks them.)  The point is that we want to present ourselves as we really are.


The Stochastic Resonance Program (Part 1)

10 May, 2014

guest post by David Tanzer

At the Azimuth Code Project, we are aiming to produce educational software that is relevant to the Earth sciences and the study of climate. Our present software takes the form of interactive web pages, which allow you to experiment with the parameters of models and view their outputs. But to fully understand the meaning of a program, we need to know about the concepts and theories that inform it. So we will be writing articles to explain both the programs themselves and the math and science behind them.

In this two-part series, I’ll explain this program:

Stochastic resonance.

Check it out—it runs on your browser! It was created by Allan Erskine and Glyn Adgie. In the Azimuth blog article Increasing the Signal-to-Noise Ratio with More Noise, Glyn Adgie and Tim van Beek give a nice explanation of the idea of stochastic resonance, which includes some clear and exciting graphs.

My goal today is give a compact, developer-oriented introduction to stochastic resonance, which will set the context for the next blog article, where I’ll dissect the program itself. By way of introduction, I am a software developer with research training in computer science. It’s a new area for me, and any clarifications will be welcome!

The concept of stochastic resonance

Stochastic resonance is a phenomenon, occurring under certain circumstances, in which a noise source may amplify the effect of a weak signal. This concept was used in an early hypothesis about the timing of ice-age cycles, and has since been applied to a wide range of phenomena, including neuronal detection mechanisms and patterns of traffic congestion.

Suppose we have a signal detector whose internal, analog state is driven by an input signal, and suppose the analog states are partitioned into two regions, called “on” and “off” — this is a digital state, abstracted from the analog state. With a light switch, we could take the force as the input signal, the angle as the analog state, and the up/down classification of the angle as the digital state.

Consider the effect of a periodic input signal on the digital state. Suppose the wave amplitude is not large enough to change the digital state, yet large enough to drive the analog state close to the digital state boundary. Then, a bit of random noise, occurring near the peak of an input cycle, may “tap” the system over to the other digital state. So we will see a probability of state-transitions that is synchronized with the input signal. In a complex way, the noise has amplified the input signal.

But it’s a pretty funky amplifier! Here is a picture from the Azimuth library article on stochastic resonance:

Stochastic resonance has been found in the signal detection mechanisms of neurons. There are, for example, cells in the tails of crayfish that are tuned to low-frequency signals in the water caused by predator motions. These signals are too weak to cross the firing threshold for the neurons, but with the right amount of noise, they do trigger the neurons.

See:

Stochastic resonance, Azimuth Library.

Stochastic resonance in neurobiology, David Lyttle.

Bistable stochastic resonance and Milankovitch theories of ice-age cycles

Stochastic resonance was originally formulated in terms of systems that are bistable — where each digital state is the basin of attraction of a stable equilibrium.

An early application of stochastic resonance was to a hypothesis, within the framework of bistable climate dynamics, about the timing of the ice-age cycles. Although it has not been confirmed, it remains of interest (1) historically, (2) because the timing of ice-age cycles remains an open problem, and (3) because the Milankovitch hypothesis upon which it rests is an active part of the current research.

In the bistable model, the climate states are a cold, “snowball” Earth and a hot, iceless Earth. The snowball Earth is stable because it is white, and hence reflects solar energy, which keeps it frozen. The iceless Earth is stable because it is dark, and hence absorbs solar energy, which keeps it melted.

The Milankovitch hypothesis states that the drivers of climate state change are long-duration cycles in the insolation — the solar energy received in the northern latitudes — caused by periodic changes in the Earth’s orbital parameters. The north is significant because that is where the glaciers are concentrated, and so a sufficient “pulse” in northern temperatures could initiate a state change.

Three relevant astronomical cycles have been identified:

• Changing of the eccentricity of the Earth’s elliptical orbit, with a period of 100 kiloyears

• Changing of the obliquity (tilt) of the Earth’s axis, with a period of 41 kiloyears

• Precession (swiveling) of the Earth’s axis, with a period of 23 kiloyears

In the stochastic resonance hypothesis, the Milankovitch signal is amplified by random events to produce climate state changes. In more recent Milankovitch theories, a deterministic forcing mechanism is used. In a theory by Didier Paillard, the climate is modeled with three states, called interglacial, mild glacial and full glacial, and the state changes depend on the volume of ice as well as the insolation.

See:

Milankovitch cycle, Azimuth Library.

Mathematics of the environment (part 10), John Baez. This gives an exposition of Paillard’s theory.

Bistable systems defined by a potential function

Any smooth function with two local minima can be used to define a bistable system. For instance, consider the function V(x) = x^4/4 - x^2/2:

To define the bistable system, construct a differential equation where the time derivative of x is set to the negative of the derivative of the potential at x:

dx/dt = -V'(x) = -x^3 + x = x(1 - x^2)

So, for instance, where the potential graph is sloping upward as x increases, -V'(x) is negative, and this sends X(t) ‘downhill’ towards the minimum.

The roots of V'(x) yield stable equilibria at 1 and -1, and an unstable equilibrium at 0. The latter separates the basins of attraction for the stable equilibria.

Discrete stochastic resonance

Now let’s look at a discrete-time model which exhibits stochastic resonance. This is the model used in the Azimuth demo program.

We construct the discrete-time derivative, using the potential function, a sampled sine wave, and a normally distributed random number:

\Delta X_t = -V'(X_t) * \Delta t + \mathrm{Wave}(t) + \mathrm{Noise}(t) =
X_t (1 - X_t^2) \Delta t + \alpha * \sin(\omega t) + \beta * \mathrm{GaussianSample}(t)

where \Delta t is a constant and t is restricted to multiples of \Delta t.

This equation is the discrete-time counterpart to a continuous-time stochastic differential equation.

Next time, we will look into the Azimuth demo program itself.


2014 on Azimuth

31 December, 2013

 

Happy New Year! We’ve got some fun guest posts lined up for next year, including:

Marc Harper, Relative entropy in evolutionary dynamics.

Marc Harper uses ideas from information theory in his work on bioinformatics and evolutionary game theory. This article explains some of his new work. And as a warmup, it explains how relative entropy can serve as a Lyapunov function in evolution!

This includes answering the question:

“What is a Lyapunov function, and why should I care?”

The brief answer, in case you’re eager to know, is this. A Lyapunov function is something that always increases—or always decreases—as time goes on. Examples include entropy and free energy. So, a Lyapunov function can be a way of making the 2nd law of thermodynamics mathematically precise! It’s also a way to show things are approaching equilibrium.

The overall goal here is applying entropy and information theory to better understand the behavior of biological and ecological systems. And in April 2015, Marc Harper and I are helping run a workshop on this topic! We’re doing this with John Harte, an ecologist who uses maximum entropy methods to predict the distribution, abundance and energy usage of species. It should be really interesting!

But back to blog articles:

Manoj Gopalkrishnan, Lyapunov functions for complex-balanced systems.

Manoj Gopalkrishnan is a mathematician at the Tata Institute of Fundamental Research in Mumbai who works on problems coming from chemistry and biology. This post will explain his recent paper on a Lyapunov function for chemical reactions. This function is closely related to free energy, a concept from thermodynamics. So again, one of the overall goals is to apply entropy to better understand living systems.

Since some evolutionary games are isomorphic to chemical reaction networks, this post should be connected to Marc’s. But there’s some mental work left to make the connection—for me, at least. It should be really cool when it all fits together!

Alastair Jamieson-Lane, Stochastic cross impact balance analysis.

Alastair Jamieson-Lane is a mathematician in the master’s program at the University of British Columbia. Very roughly, this post is about a method for determining which economic scenarios are more likely. The likely scenarios get fed into things like the IPCC climate models, so this is important.

This blog article has an interesting origin. Vanessa Schweizer has a bachelor’s degree in physics, a masters in environmental studies, and a PhD in engineering and public policy. She now works at the University of Waterloo on long-term decision-making problems.

A while back, I met Vanessa at a workshop called What Is Climate Change and What To Do About It?, at the Balsillie School of International Affairs, which is in Waterloo. She described her work with Alastair Jamieson-Lane and the physicist Matteo Smerlak on stochastic cross impact balance analysis. It sounded really interesting, something I’d like to work on. So I solicited some blog articles from them. I hope this is just the first!

So: Happy New Year, and good reading!

Also: we’re always looking for good guest posts here on Azimuth, and we have a system for helping you write them. So, if you know something interesting about environmental or energy issues, ecology, biology or chemistry, consider giving it a try!

If you read some posts here, especially guest posts, you’ll get an idea of what we’re looking for. David Tanzer, a software developer in New York who is very active in the Azimuth Project these days, made an organized list of Azimuth blog posts here:

Azimuth Blog Overview.

You can see the guest posts listed by author. This overview is also great for catching up on old posts!


Azimuth Blog Overview

6 September, 2013

We’ve got lots of series of articles on this blog. Some people say it’s a bit overwhelming. So David Tanzer of the Azimuth Project had a good idea: create an organized list of the articles on this blog, to make them easier to find. Here it is:

Azimuth Blog overview.

You can also find a link to this on top of the “ALSO READ THESE” list at the right-hand side of this blog!

Needless to say, this could be improved in many ways. Don’t say how: just do it!


Follow

Get every new post delivered to your Inbox.

Join 3,307 other followers