As a mathematical physicist, I’ve always been in love with the Laplacian:

It shows up in many of the most fundamental equations of physics: the wave equation, the heat equation, Schrödinger’s equation… and Poisson’s equation:

which says how the density of matter, affects the gravitational potential

As I’ve grown interested in network theory, I’ve gotten more and more interested in ‘graph Laplacians’. These are discretized versions of the Laplacian, where we replace 3-dimensional space by a ‘graph’, meaning something like this:

You can get a lot of interesting information about a graph from its Laplacian. You can also set up discretized versions of all the famous equations I mentioned.

The new progress is a simple algorithm for very efficiently solving Poisson’s equation for graph Laplacians:

• Jonathan A. Kelner, Lorenzo Orecchia, Aaron Sidford, Zeyuan Allen Zhu, A simple, combinatorial algorithm for solving SDD systems in nearly-linear time.

Here’s a very clear explanation of the general idea, conveying some sense of why it’s so important, without any nasty equations:

• Larry Hardesty, Short algorithm, long-range consequences, MIT News, 1 March 2013.

It begins:

In the last decade, theoretical computer science has seen remarkable progress on the problem of solving graph Laplacians — the esoteric name for a calculation with hordes of familiar applications in scheduling, image processing, online product recommendation, network analysis, and scientific computing, to name just a few. Only in 2004 did researchers first propose an algorithm that solved graph Laplacians in “nearly linear time,” meaning that the algorithm’s running time didn’t increase exponentially with the size of the problem.

At this year’s ACM Symposium on the Theory of Computing, MIT researchers will present a new algorithm for solving graph Laplacians that is not only faster than its predecessors, but also drastically simpler.

This animation shows two different “spanning trees” for a simple graph, a grid like those used in much scientific computing. The speedups promised by a new MIT algorithm require “low-stretch” spanning trees (green), in which the paths between neighboring nodes don’t become excessively long (red).

I can’t beat this article at its own game… except to clarify that ‘solving graph Laplacians’ means solving Poisson’s equation with a graph Laplacian replacing the usual Laplacian.

So, let me just supplement this article with the nasty equations saying what a graph Laplacian actually is. Start with a graph. More precisely, start with a simple graph. Such a graph has a set of vertices and a set of edges such that

which says the edges are undirected, and

which says there are no loops.

The graph Laplacian is an operator that takes a function on the vertices of our graph,

and gives a new such function as follows:

The version of Poisson’s equation for this graph Laplacian is thus

But I should warn you: this operator has eigenvalues that are less than equal to zero, like the usual Laplacian People often insert a minus sign to make the eigenvalues ≥ 0.

There is a huge amount to say about graph Laplacians! If you want, you can learn more here:

• Michael William Newman, The Laplacian Spectrum of Graphs, Masters Thesis, Department of Mathematics, University of Manitoba, 2000.

I’ve been learning about some of their applications here:

• Ernesto Estrada, The Structure of Complex Networks: Theory and Applications, Oxford University Press, Oxford, 2011.

I hope sometime to summarize a bit of this material and push the math forward a bit. So, it was nice to see new progress on efficient algorithms for doing computations with graph Laplacians!

Lisa and I had dinner with Gregory Benford and his wife when I visited U.C. Irvine a couple of weekends ago, and he raised an interesting point. So far, radio searches for extraterrestrial life have only seen puzzling brief signals – not long transmissions. But what if this is precisely what we should expect?

A provocative example is Sullivan, et al. (1997). This survey lasted about 2.5 hours, with 190 1.2 minute integrations. With many repeat observations, they saw nothing that did not seem manmade. However, they “recorded intriguing, non-repeatable, narrowband signals, apparently not of manmade origin and with some degree of concentration toward the galactic plane…” Similar searches also saw one-time signals, not repeated (Shostak & Tarter, 1985; Gray & Marvel, 2001 Gray, 2001). These searches had slow times to revisit or reconfirm, often days (Tarter, 2001). Overall, few searches lasted more than hour, with lagging confirmation checks (Horowitz & Sagan, 1993). Another striking example is the “WOW” signal seen at the Ohio SETI site…

That’s a quote from a paper Benford wrote with his brother and nephew:

• Gregory Benford, James Benford, and Dominic Benford, Searching for cost optimized interstellar beacons.

They claim the cheapest way a civilization could communicate to lots of planets is a pulsed, broadband, narrowly focused microwave beam that scans the sky. So, for anyone receiving this signal, there would be a lot of time between pulses. That might explain some of the above mysteries, or this one:

As an example of using cost optimized beacon analysis for SETI purposes, consider in detail the puzzling transient bursting radio source, GCRT J17445-3009, which has extremely unusual properties. It was discovered in 2002 in the direction of the Galactic Center (1.25° south of GC) at 330 MHz in a VLA observation and subsequently re-observed in 2003 and 2004 in GMRT observations (Hyman, 2005, 2006, 2007). It is a pulsed coherent source, with the ‘burst’ lasting as much as 10 minutes, with 77-minute period. Averaged over all observations, Hyman et al. give a duty cycle of 7% (1/14), although since some observations may have missed part of bursts, the duty cycle might be as high as 13%.

Even if these are red herrings, it seems very smart to figure out the cheapest ways to transmit signals and use that to guess what signals we should look for. We can easily make the mistake of assuming all extraterrestrial civilizations who bother to send signals through space will be willing to beam signals of enormous power toward us all the time. That could be true of some, but not necessarily all.

The cost analysis is here:

• James Benford, Gregory Benford, Dominic Benford, Messaging with cost optimized interstellar beacons.

and you can see a summary in this talk by Gregory’s brother James, who works on high-power microwave technologies:

The summer before last, I invited Brendan Fong to Singapore to work with me on my new ‘network theory’ project. He quickly came up with a nice new proof of a result about mathematical chemistry. We blogged about it, and I added it to my book, but then he became a grad student at Oxford and got distracted by other kinds of networks—namely, Bayesian networks.

So, we’ve just now finally written up this result as a self-contained paper:

• John Baez and Brendan Fong, Quantum techniques for studying equilibria in chemical reaction networks.

Check it out and let us know if you spot mistakes or stuff that’s not clear!

The idea, in brief, is to use math from quantum field theory to give a somewhat new proof of the Anderson–Craciun–Kurtz theorem.

This remarkable result says that in many cases, we can start with an equilibrium solution of the ‘rate equation’ which describes the behavior of chemical reactions in a deterministic way in the limit of a large numbers of molecules, and get an equilibrium solution of the ‘master equation’ which describes chemical reactions probabilistically for any number of molecules.

The trick, in our approach, is to start with a chemical reaction network, which is something like this:

and use it to write down a Hamiltonian describing the time evolution of the probability that you have various numbers of each kind of molecule: A, B, C, D, E, … Using ideas from quantum mechanics, we can write this Hamiltonian in terms of annihilation and creation operators—even though our problem involves probability theory, not quantum mechanics! Then we can write down the equilibrium solution as a ‘coherent state’. In quantum mechanics, that’s a quantum state that approximates a classical one as well as possible.

All this is part of a larger plan to take tricks from quantum mechanics and apply them to ‘stochastic mechanics’, simply by working with real numbers representing probabilities instead of complex numbers representing amplitudes!

I should add that Brendan’s work on Bayesian networks is also very cool, and I plan to talk about it here and even work it into the grand network theory project I have in mind. But this may take quite a long time, so for now you should read his paper:

• Brendan Fong, Causal theories: a categorical perspective on Bayesian networks.

Natural philosophy (aka science) is distinguished from pure philosophy or mathematics by coupling theory to experiment. Engineering is distinguished from science in its focus on solving practical problems rather than merely coming up with more accurate models of the universe. Climate change will not be fixed by pure philosophy or argumentation. We need to use the methods of science and engineering to make progress towards a solution. The problem is complicated and involves not just climate dynamics and ecology, but psychology, economics and technology. Besides theory and experiment, we now have the tool of simulation. I propose a think-tank (or more properly, a think/do/simulate-tank) analogous to the Manhattan Project, which developed the first atomic bomb. However, this project would involve social and physical scientists, computer programmers, engineers, farmers and craftspeople who are trying to collaboratively solve the problem of how to provide food, shelter, water, clothes, medicine and recreation for a self contained village in a sustainable way. Sustainability has psychological dimensions, not just ecological. For example, it implies that people would want to keep living in this village, or similar villages. If we are interested in sustainability beyond the initial village, then sustainability implies replicability—that the village would inspire many other people to live similarly.

Initial outputs of this project would be well-founded suggestions regarding what kinds of production skills are needed and how to effectively network them, how many people, how much land, how much time spent on production in order to achieve village-scale independence and sustainability. An eventual outcome would be an actual demonstration of a functional village.

Why village? The word village is used here to mean a group of people who are economically networked in isolation from the rest of the global economy. It also implies choosing a particular geographic location, so not all outputs would be transferable to other locations, though with the initial simulation stage many locations could be tried.

Why economic isolation? Without putting a boundary on the experiment, the problem is too complex, even for simulation. Entropy reduction is the same reason cells have membranes and scientists have labs. The membrane could be permeable to sunlight, wind (and emissions) and water, but at first it might be simpler to keep it impermeable to economic exchange. In addition, it is easy to externalize all unsustainable practices without a membrane. But the size of the membrane is not predetermined. One possible conclusion might be that the village has to be the size of the whole earth. Another reason for starting with a village is that changes in biological (and probably other complex) systems always proceed from small populations that can spread out by replication. It is more practical to achieve a global change in lifestyle and technology starting with a small group of willing people who can then inspire others by example, rather than try to impose a change on a large population, the way fascist and communist experiments have proceeded. Another reason for keeping things smaller and more local is that a stronger feedback between production and consumption may arise, which would regulate unsustainable consumption, because the environmental, social and psychological costs of production are visible in the village, as opposed to hidden or abstracted from the consumers. There are other reasons for localization (e.g. resilience, freedom, more meaningful employment for more people, better relations among people or between people and nature), less directly related to climate change, and more speculative.

This is probably the place to admit my main bias. I am a Gandhist Luddite (who has a PhD in Physics, worked as a semiconductor engineer and a molecular biologist) , not the angry, machine-smashing kind, and I like not only to tinker with technology, but to think how it affects people and nature. I don’t think all technology can be equated with progress. I call this project the Luddite Manhattan Project (or or Localizing and Networking Basic Technology project) for that reason and because it parallels the project that produced the nuclear bomb. I think that the craftspeople and farmers would contribute more to this project than the scientists and engineers. I think that in the multidimensional optimization of technology, we have focused too much on efficiency (disregarding other human values) and that the industrial revolution was largely a mistake (though some good things came out of it, like global communication). If we focus on other human values, we can optimize technology better. I think that localism of basic-needs production (when coupled to non-technological things like democracy) is a constraint from which many other good things such as sustainability, full, meaningful employment, freedom, and good social relations would follow, though it too can be taken to extremes. Given my bias, I suspect that the kind of technology network that would be most sustainable would be pre-industrial, with a few modern innovations. If we really did the book-keeping accurately we would probably find that industrial production is unsustainable. Or rather we would find that pre-industrial production can be sustainable, while current industrial production is not (I leave open the possibility that industrial production might be sustainable in the future, with new innovations, but even then it tramples too many human values). But these conclusions would be outputs of the project, not pre-assumptions or inputs of the project. I welcome some discussion of these ideas, followed by computation, testing and implementation.

The technical part of the project is basically a networking problem. It would allow initial imports (in a way that would allow replicability—that is don’t hog a disproportionate fraction of resources into the village) into a specific location and then network existing technologies so that the system is self-sustaining. What one craftsperson produces, others in the village must use so that the village can continue in perpetuity. A blacksmith needs some fuel, but also customers who need his products and can exchange stuff that he needs. A cooper is mostly useless in the current industrial economy, but would probably find some use in a local village economy, where people need ways to store water and other liquids.

Here are some typical challenges and questions the project would face: How can antibiotics be made on a village scale with no external inputs? What can’t be made and can we find substitutes? Are there missing technology links and can we invent them, or do we need to start with another scenario? What food needs to be produced to provide basic caloric needs to all inhabitants of the village? How much area is required? How can water be captured and transported without plastic or rubber? How much carbon is emitted in production of everything? Where does garbage go? How can metals be recycled? Can plastic be produced? Can electronics be produced? Is there enough time for art, science, scholarship and other forms of edifying human activity? What kind of economic systems work? Is there an optimal one as far as sustainability, or is it a matter of personal preference? These are all questions that can be tackled, if we face them with curiosity and realism, instead of with fear and the kind of magical thinking that most people have towards technology and other things they don’t understand. I’ve heard that Leonardo Da Vinci was the last man to understand the technology of his age, but we have computers to help us.

It might be appropriate at this stage to mention that I do not advocate giving up entirely the industrial mode of production, or the global trade it requires. The Localizing and Networking Basic Technology project would address only food, shelter, water, medicine, all the subsidiary crafts necessary to sustain these, and a few edifying human activities like art, music and scholarship. Computers and internet hardware are almost certainly best left to industrial production, and so are cars, airplanes (but the need for these will drastically decrease if this project is successful), some of the parts for particle accelerators and fancy biotech equipment, etc.

The initial computational stage of the project could model itself on online multiplayer games like Warcraft and planning games like Sim City (I have tried to contact Will Wright, to no avail). I do not play these games (I prefer simple low tech games personally), but I see the usefulness of online collaboration and computation for this project, as a sort of in-silico evolution. Programmers and mathematicians could set up the software to allow both online collaboration and some central planning. I think the simplest solutions should be tried first, i.e. the most primitive technologies, like hunting and gathering. My educated guess is that they will be shown incapable of providing basic needs given the current world population. The same conclusion would probably follow for current industrial production, except the incapacity would be with regards to sustainability. I predict the sweet spot where both sustainability and capacity to “feed the world” (meaning provide a decent life) would be achieved by pre-industrial, agrarian and craft-based production.

I am totally willing to be proven wrong by this experiment about my anti-industrialization bias. With regards to scientific experimentation, there needs to be well posed hypotheses that can be proven wrong, and good controls. The engineering approach is an alternative. Who is willing to work on this project? Let’s make amends for unleashing the horror of the Bomb on the earth, tackle climate change realistically and have some technical fun. For further information please see:

• Iuval Clejan, Luddite Manhattan Project, first stage, 16 April 2012.

• Iuval Clejan, A proposal for funding a blueprint of a village-based technology ecosystem, 5 February 2012.

Suppose we take “applied mathematics” in an extremely broad sense that includes math developed for use in electrical engineering, population biology, epidemiology, chemistry, and many other fields. Suppose we look for mathematical structures that repeatedly appear in these diverse contexts — especially structures that aren’t familiar to pure mathematicians. What do we find? The answers may give us some clues about how to improve the foundations of mathematics!

This is what I’m talking about at the Category-Theoretic Foundations of Mathematics Workshop at U.C. Irvine this weekend.

You can see my talk slides here. You can click on any picture or anything written in blue in these slides to get more information — for example, references.

How can we bridge the gap between what we are doing about global warming and what we should be doing?

That’s what this paper is about:

• Kornelis Blok, Niklas Höhne, Kees van der Leun and Nicholas Harrison, Bridging the greenhouse-gas emissions gap, Nature Climate Change 2 (2012), 471-474.

According to the United Nations Environment Programme, we need to cut CO₂ emissions by about 12 gigatonnes/year by 2020 to hold global warming to 2 °C.

After the UN climate conference in Copenhagen, many countries made pledges to reduce CO₂ emissions. But by 2020 these pledges will cut emissions by at most 6 gigatonnes/year. Even worse, a lot of these pledges are contingent on other people meeting other pledges, and so on… so the confirmed value of all these pledges is only 3 gigatonnes/year.

The authors list 21 things that cities, large companies and individual citizens can do, which they claim will cut greenhouse gas emissions by the equivalent of 10 gigatonnes/year of CO₂ by 2020. For each initiative on their list, they claim:

(1) there is a concrete starting position from which a significant up-scaling until the year 2020 is possible;

(2) there are significant additional benefits besides a reduction of greenhouse-gas emissions, so people can be driven by self-interest or internal motivation, not external pressure;

(3) there is an organization or combination of organizations that can lead the initiative;

(4) the initiative has the potential to reach an emission reduction by about 0.5 Gt CO₂e by 2020.

21 Initiatives

Now I want to quote the paper and list the 21 initiatives. And here’s where I could use your help! For each of these, can you point me to one or more organizations that are in a good position to lead the initiative?

Some are already listed, but even for these I bet there are other good answers. I want to compile a list, and then start exploring what’s being done, and what needs to be done.

By the way, even if the UN estimate of the greenhouse-emissions gap is wrong, and even if all the numbers I’m about to quote are wrong, most of them are probably the right order of magnitude—and that’s all we need to get a sense of what needs to be done, and how we can do it.

Companies

1. Top 1,000 companies’ emission reductions. Many of the 1,000 largest greenhouse-gas-emitting companies already have greenhouse-gas emission-reduction goals to decrease their energy use and increase their long-term competitiveness, as well as to demonstrate their corporate social responsibility. An association such as the World Business Council for Sustainable Development could lead 30% of the top 1,000 companies to reduce energy-related emissions 10% below business as usual by 2020 and all companies to reduce their non-carbon dioxide greenhouse-gas emissions by 50%. Impact in 2020: up to 0.7 Gt CO₂e.

2. Supply-chain emission reductions. Several companies already have social and environmental requirements for their suppliers, which are driven by increased competitiveness, corporate social responsibility and the ability to be a front-runner. An organization such as the Consumer Goods Forum could stimulate 30% of companies to require their supply chains to reduce emissions 10% below business as usual by 2020. Impact in 2020: up to 0.2 Gt CO₂e.

3. Green financial institutions. More than 200 financial organizations are already members of the finance initiative of the United Nations Environment Programme (UNEP-FI). They are committed to environmental goals owing to corporate social responsibility, to gain investor certainty and to be placed well in emerging markets. UNEP-FI could lead the 20 largest banks to reduce the carbon footprint of 10% of their assets by 80%. Impact in 2020: up to 0.4 Gt of their assets by 80%. Impact in 2020: up to 0.4 Gt CO₂e.

4. Voluntary-offset companies. Many companies are already offsetting their greenhouse-gas emissions, mostly without explicit external pressure. A coalition between an organization with convening power, for example UNEP, and offset providers could motivate 20% of the companies in the light industry and commercial sector to calculate their greenhouse-gas emissions, apply emission-reduction measures and offset the remaining emissions (retiring the purchased credits). It is ensured that offset projects really reduce emissions by using the ‘gold standard’ for offset projects or another comparable mechanism. Governments could provide incentives by giving tax credits for offsetting, similar to those commonly given for charitable donations. Impact by 2020: up to 2.0 Gt CO₂e.

Other actors

5. Voluntary-offset consumers. A growing number of individuals (especially with high income) already offset their greenhouse-gas emissions, mostly for flights, but also through carbon-neutral products. Environmental NGOs could motivate 10% of the 20% of richest individuals to offset their personal emissions from electricity use, heating and transport at cost to them of around US$200 per year. Impact in 2020: up to 1.6 Gt CO₂e.

6. Major cities initiative. Major cities are large emitters of greenhouse gases and many have greenhouse-gas reduction targets. Cities are intrinsically highly motivated to act so as to improve local air quality, attractiveness and local job creation. Groups like the C40 Cities Climate Leadership Group and ICLEI — Local Governments for Sustainability could lead the 40 cities in C40 or an equivalent sample to reduce emissions 20% below business as usual by 2020, building on the thousands of emission-reduction activities already implemented by the C40 cities. Impact in 2020: up to 0.7 Gt CO₂e.

7. Subnational governments. Several states in the United States and provinces in Canada have already introduced support mechanisms for renewable energy, emission-trading schemes, carbon taxes and industry regulation. As a result, they expect an increase in local competitiveness, jobs and energy security. Following the example set by states such as California, these ambitious US states and Canadian provinces could accept an emission-reduction target of 15–20% below business as usual by 2020, as some states already have. Impact in 2020: up to 0.6 Gt CO₂e.

Energy efficiency

8. Building heating and cooling. New buildings, and increasingly existing buildings, are designed to be extremely energy efficient to realize net savings and increase comfort. The UN Secretary General’s Sustainable Energy for All Initiative could bring together the relevant players to realize 30% of the full reduction potential for 2020. Impact in 2020: up to 0.6 Gt CO₂e.

9. Ban of incandescent lamps. Many countries already have phase-out schedules for incandescent lamps as it provides net savings in the long term. The en.lighten initiative of UNEP and the Global Environment Facility already has a target to globally ban incandescent lamps by 2016. Impact in 2020: up to 0.2 Gt CO₂e.

10. Electric appliances. Many international labelling schemes and standards already exist for energy efficiency of appliances, as efficient appliances usually give net savings in the long term. The Collaborative Labeling and Appliance Standards Program or the Super-efficient Equipment and Appliance Deployment Initiative could drive use of the most energy-efficient appliances on the market. Impact in 2020: up to 0.6 Gt CO₂e.

11. Cars and trucks. All car and truck manufacturers put emphasis on developing vehicles that are more efficient. This fosters innovation and increases their long-term competitive position. The emissions of new cars in Europe fell by almost 20% in the past decade. A coalition of manufacturers and NGOs joined by the UNEP Partnership for Clean Fuels and Vehicles could agree to save one additional liter per 100 km globally by 2020 for cars, and equivalent reductions for trucks. Impact in 2020: up to 0.7 Gt CO₂e.

Energy supply

12. Boost solar photovoltaic energy. Prices of solar photovoltaic systems have come down rapidly in recent years, and installed capacity has increased much faster than expected. It created a new industry, an export market and local value added through, for example, roof installations. A coalition of progressive governments and producers could remove barriers by introducing good grid access and net metering rules, paving the way to add another 1,600 GW by 2020 (growth consistent with recent years). Impact in 2020: up to 1.4 Gt CO₂e.

13. Wind energy. Cost levels for wind energy have come down dramatically, making wind economically competitive with fossil-fuel-based power generation in many cases. The Global Wind Energy Council could foster the global introduction of arrangements that lead to risk reduction for investments in wind energy, with, for example, grid access and guarantees. This could lead to an installation of 1,070 GW by 2020, which is 650 GW over a reference scenario. Impact in 2020: up to 1.2 Gt CO₂e.

14. Access to energy through low-emission options. Strong calls and actions are already underway to provide electricity access to 1.4 billion people who are at present without and fulfill development goals. The UN Secretary General’s Sustainable Energy for All Initiative could ensure that all people without access to electricity get access through low-emission options. Impact in 2020: up to 0.4 Gt CO₂e.

15. Phasing out subsidies for fossil fuels. This highly recognized option to reduce emissions would improve investment in clean energy, provide other environmental, health and security benefits, and generate income. The International Energy Agency could work with countries to phase out half of all fossil-fuel subsidies. Impact in 2020: up to 0.9 Gt CO₂e.

Special sectors

16. International aviation and maritime transport. The aviation and shipping industries are seriously considering efficiency measures and biofuels to increase their competitive advantage. Leading aircraft and ship manufacturers could agree to design their vehicles to capture half of the technical mitigation potential. Impact in 2020: up to 0.2 Gt CO₂e.

17. Fluorinated gases (hydrofluorocarbons, perflourocarbons, SF6). Recent industry-led initiatives are already underway to reduce emissions of these gases originating from refrigeration, air-conditioning and industrial processes. Industry associations, such as Refrigerants, Naturally!, could work towards meeting half of the technical mitigation potential. Impact in 2020: up to 0.3 Gt CO2e.

18. Reduce deforestation. Some countries have already shown that it is strongly possible to reduce deforestation with an integrated approach that eliminates the drivers of deforestation. This has benefits for local air pollution and biodiversity, and can support the local population. Led by an individual with convening power, for example, the United Kingdom’s Prince of Wales or the UN Secretary General, such approaches could be rolled out to all the major countries with high deforestation emissions, halving global deforestation by 2020. Impact in 2020: up to 1.8 Gt CO₂e.

19. Agriculture. Options to reduce emissions from agriculture often increase efficiency. The International Federation of Agricultural Producers could help to realize 30% of the technical mitigation potential. (Well, at least it could before it collapsed, after this paper was written.) Impact in 2020: up to 0.8 Gt CO2e.

Air pollutants

20. Enhanced reduction of air pollutants. Reduction of classic air pollutants including black carbon has been pursued for years owing to positive impacts on health and local air quality. UNEP’s Climate and Clean Air Coalition To Reduce Short-Lived Climate Pollutants already has significant political momentum and could realize half of the technical mitigation potential. Impact in 2020: a reduction in radiative forcing impact equivalent to an emission reduction of greenhouse gases in the order of 1 Gt CO₂e, but outside of the definition of the gap.

21. Efficient cook-stoves. Cooking in rural areas is a source of carbon dioxide emissions. Furthermore, there are emissions of black carbon, which also leads to global warming. Replacing these cook-stoves would also significantly increase local air quality and reduce pressure on forests from fuel-wood demand. A global development organization such as the UN Development Programme could take the lead in scaling-up the many already existing programs to eventually replace half of the existing cook-stoves. Impact in 2020: a reduction in radiative forcing impact equivalent to an emission reduction of greenhouse gases of up to 0.6 Gt CO₂e, included in the effect of the above initiative and outside of the definition of the gap.

For more

For more, see the supplementary materials to this paper, and also:

• Niklas Höhne, Wedging the gap: 21 initiatives to bridge the greenhouse gas emissions gap.

The size of the emissions gap was calculated here:

• The Emissions Gap Report 2012, United Nations Environment Programme (UNEP).

If you’re in a rush, just read the executive summary.

The Perimeter Institute is a futuristic-looking place where over 250 physicists are thinking about quantum gravity, quantum information theory, cosmology and the like. Since I work on some of these things, I was recently invited to give the weekly colloquium there. But I took the opportunity to try to rally them into action:

• Energy and the Environment: What Physicists Can Do. Watch the video or read the slides.

Abstract. The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. While politics and economics pose the biggest challenges, physicists are in a good position to help make this transition a bit easier. After a quick review of the problems, we discuss a few ways physicists can help.

On the video you can hear me say a lot of stuff that’s not on the slides: it’s more of a coherent story. The advantage of the slides is that anything in blue, you can click on to get more information. So for example, when I say that solar power capacity has been growing annually by 75% in recent years, you can see where I got that number.

I was pleased by the response to this talk. Naturally, it was not a case of physicists saying “okay, tomorrow I’ll quit working on the foundations of quantum mechanics and start trying to improve quantum dot solar cells.” It’s more about getting them to see that huge problems are looming ahead of us… and to see the huge opportunities for physicists who are willing to face these problems head-on, starting now. Work on energy technologies, the smart grid, and ‘ecotechnology’ is going to keep growing. I think a bunch of the younger folks, at least, could see this.

However, perhaps the best immediate outcome of this talk was that Lee Smolin introduced me to Manjana Milkoreit. She’s at the school of international affairs at Waterloo University, practically next door to the Perimeter Institute. She works on “climate change governance, cognition and belief systems, international security, complex systems approaches, especially threshold behavior, and the science-policy interface.”

So, she knows a lot about the all-important human and political side of climate change. Right now she’s interviewing diplomats involved in climate treaty negotiations, trying to see what they believe about climate change. And it’s very interesting!

In my next post, I’ll talk about something she pointed me to. Namely: what we can do to hold the temperature increase to 2 °C or less, given that the pledges made by various nations aren’t enough.

Electronics	Mechanics
charge:	position:
current:	velocity:
flux linkage:	momentum:
voltage:	force:
inductance:	mass:
resistance:	damping coefficient:
inverse capacitance:	spring constant:

But this is just the first of a large set of analogies. Let me list some, so you can see how wide-ranging they are!

More analogies

People in system dynamics often use effort as a term to stand for anything analogous to force or voltage, and flow as a general term to stand for anything analogous to velocity or electric current. They call these variables and

To me it’s important that force is the time derivative of momentum, and velocity is the time derivative of position. Following physicists, I write momentum as and position as So, I’ll usually write effort as and flow as .

Of course, ‘position’ is a term special to mechanics; it’s nice to have a general term for the thing whose time derivative is flow, that applies to any context. People in systems dynamics seem to use displacement as that general term.

It would also be nice to have a general term for the thing whose time derivative is effort… but I don’t know one. So, I’ll use the word momentum.

Now let’s see the analogies! Let’s see how displacement , flow momentum and effort show up in several subjects:

	displacement:	flow:	momentum:	effort:
Mechanics: translation	position	velocity	momentum	force
Mechanics: rotation	angle	angular velocity	angular momentum	torque
Electronics	charge	current	flux linkage	voltage
Hydraulics	volume	flow	pressure momentum	pressure
Thermal Physics	entropy	entropy flow	temperature momentum	temperature
Chemistry	moles	molar flow	chemical momentum	chemical potential

We’d been considering mechanics of systems that move along a line, via translation, but we can also consider mechanics for systems that turn round and round, via rotation. So, there are two rows for mechanics here.

There’s a row for electronics, and then a row for hydraulics, which is closely analogous. In this analogy, a pipe is like a wire. The flow of water plays the role of current. Water pressure plays the role of electrostatic potential. The difference in water pressure between two ends of a pipe is like the voltage across a wire. When water flows through a pipe, the power equals the flow times this pressure difference—just as in an electrical circuit the power is the current times the voltage across the wire.

A resistor is like a narrowed pipe:

An inductor is like a heavy turbine placed inside a pipe: this makes the water tend to keep flowing at the same rate it’s already flowing! In other words, it provides a kind of ‘inertia’ analogous
to mass.

A capacitor is like a tank with pipes coming in from both ends, and a rubber sheet dividing it in two lengthwise:

When studying electrical circuits as a kid, I was shocked when I first learned that capacitors don’t let the electrons through: it didn’t seem likely you could do anything useful with something like that! But of course you can. Similarly, this gizmo doesn’t let the water through.

A voltage source is like a compressor set up to maintain a specified pressure difference between the input and output:

Similarly, a current source is like a pump set up to maintain a specified flow.

Finally, just as voltage is the time derivative of a fairly obscure quantity called ‘flux linkage’, pressure is the time derivative of an even more obscure quantity which has no standard name. I’m calling it ‘pressure momentum’, thanks to the analogy

Just as pressure has units of force per area, pressure momentum has units of momentum per area!

People invented this analogy back when they were first struggling to understand electricity, before electrons had been observed:

• Hydraulic analogy, Wikipedia.

The famous electrical engineer Oliver Heaviside pooh-poohed this analogy, calling it the “drain-pipe theory”. I think he was making fun of William Henry Preece. Preece was another electrical engineer, who liked the hydraulic analogy and disliked Heaviside’s fancy math. In his inaugural speech as president of the Institution of Electrical Engineers in 1893, Preece proclaimed:

True theory does not require the abstruse language of mathematics to make it clear and to render it acceptable. All that is solid and substantial in science and usefully applied in practice, have been made clear by relegating mathematic symbols to their proper store place—the study.

According to the judgement of history, Heaviside made more progress in understanding electromagnetism than Preece. But there’s still a nice analogy between electronics and hydraulics. And I’ll eventually use the abstruse language of mathematics to make it very precise!

But now let’s move on to the row called ‘thermal physics’. We could also call this ‘thermodynamics’. It works like this. Say you have a physical system in thermal equilibrium and all you can do is heat it up or cool it down ‘reversibly’—that is, while keeping it in thermal equilibrium all along. For example, imagine a box of gas that you can heat up or cool down. If you put a tiny amount of energy into the system in the form of heat, then its entropy increases by a tiny amount And they’re related by this equation:

where is the temperature.

Another way to say this is

where is time. On the left we have the power put into the system in the form of heat. But since power should be ‘effort’ times ‘flow’, on the right we should have ‘effort’ times ‘flow’. It makes some sense to call the ‘entropy flow’. So temperature, must play the role of ‘effort’.

This is a bit weird. I don’t usually think of temperature as a form of ‘effort’ analogous to force or torque. Stranger still, our analogy says that ‘effort’ should be the time derivative of some kind of ‘momentum’, So, we need to introduce temperature momentum: the integral of temperature over time. I’ve never seen people talk about this concept, so it makes me a bit nervous.

But when we have a more complicated physical system like a piston full of gas in thermal equilibrium, we can see the analogy working. Now we have

The change in energy of our gas now has two parts. There’s the change in heat energy , which we saw already. But now there’s also the change in energy due to compressing the piston! When we change the volume of the gas by a tiny amount we put in energy

Now look back at the first chart I drew! It says that pressure is a form of ‘effort’, while volume is a form of ‘displacement’. If you believe that, the equation above should help convince you that temperature is also a form of effort, while entropy is a form of displacement.

But what about the minus sign? That’s no big deal: it’s the result of some arbitrary conventions. is defined to be the outward pressure of the gas on our piston. If this is positive, reducing the volume of the gas takes a positive amount of energy, so we need to stick in a minus sign. I could eliminate this minus sign by changing some conventions—but if I did, the chemistry professors at UCR would haul me away and increase my heat energy by burning me at the stake.

Speaking of chemistry: here’s how the chemistry row in the analogy chart works. Suppose we have a piston full of gas made of different kinds of molecules, and there can be chemical reactions that change one kind into another. Now our equation gets fancier:

Here is the number of molecules of the ith kind, while is a quantity called a chemical potential. The chemical potential simply says how much energy it takes to increase the number of molecules of a given kind. So, we see that chemical potential is another form of effort, while number of molecules is another form of displacement.

But chemists are too busy to count molecules one at a time, so they count them in big bunches called ‘moles’. A mole is the number of atoms in 12 grams of carbon-12. That’s roughly

atoms. This is called Avogadro’s constant. If we used 1 gram of hydrogen, we’d get a very close number called ‘Avogadro’s number’, which leads to lots of jokes:

(He must be desperate because he looks so weird… sort of like a mole!)

So, instead of saying that the displacement in chemistry is called ‘number of molecules’, you’ll sound more like an expert if you say ‘moles’. And the corresponding flow is called molar flow.

The truly obscure quantity in this row of the chart is the one whose time derivative is chemical potential! I’m calling it chemical momentum simply because I don’t know another name.

Why are linear and angular momentum so famous compared to pressure momentum, temperature momentum and chemical momentum?

I suspect it’s because the laws of physics are symmetrical
under translations and rotations. When the assumptions of Noether’s theorem hold, this guarantees that the total momentum and angular momentum of a closed system are conserved. Apparently the laws of physics lack the symmetries that would make the other kinds of momentum be conserved.

This suggests that we should dig deeper and try to understand more deeply how this chart is connected to ideas in classical mechanics, like Noether’s theorem or symplectic geometry. I will try to do that sometime later in this series.

More generally, we should try to understand what gives rise to a row in this analogy chart. Are there are lots of rows I haven’t talked about yet, or just a few? There are probably lots. But are there lots of practically important rows that I haven’t talked about—ones that can serve as the basis for new kinds of engineering? Or does something about the structure of the physical world limit the number of such rows?

Mildly defective analogies

Engineers care a lot about dimensional analysis. So, they often make a big deal about the fact that while effort and flow have different dimensions in different rows of the analogy chart, the following four things are always true:

• has dimensions of action (= energy × time)
• has dimensions of energy
• has dimensions of energy
• has dimensions of power (= energy / time)

In fact any one of these things implies all the rest.

These facts are important when designing ‘mixed systems’, which combine different rows in the chart. For example, in mechatronics, we combine mechanical and electronic elements in a single circuit! And in a hydroelectric dam, power is converted from hydraulic to mechanical and then electric form:

One goal of network theory should be to develop a unified language for studying mixed systems! Engineers have already done most of the hard work. And they’ve realized that thanks to conservation of energy, working with pairs of flow and effort variables whose product has dimensions of power is very convenient. It makes it easy to track the flow of energy through these systems.

However, people have tried to extend the analogy chart to include ‘mildly defective’ examples where effort times flow doesn’t have dimensions of power. The two most popular are these:

	displacement:	flow:	momentum:	effort:
Heat flow	heat	heat flow	temperature momentum	temperature
Economics	inventory	product flow	economic momentum	product price

The heat flow analogy comes up because people like to think of heat flow as analogous to electrical current, and temperature as analogous to voltage. Why? Because an insulated wall acts a bit like a resistor! The current flowing through a resistor is a function the voltage across it. Similarly, the heat flowing through an insulated wall is about proportional to the difference in temperature between the inside and the outside.

However, there’s a difference. Current times voltage has dimensions of power. Heat flow times temperature does not have dimensions of power. In fact, heat flow by itself already has dimensions of power! So, engineers feel somewhat guilty about this analogy.

Being a mathematical physicist, a possible way out presents itself to me: use units where temperature is dimensionless! In fact such units are pretty popular in some circles. But I don’t know if this solution is a real one, or whether it causes some sort of trouble.

In the economic example, ‘energy’ has been replaced by ‘money’. So other words, ‘inventory’ times ‘product price’ has units of money. And so does ‘product flow’ times ‘economic momentum’! I’d never heard of economic momentum before I started studying these analogies, but I didn’t make up that term. It’s the thing whose time derivative is ‘product price’. Apparently economists have noticed a tendency for rising prices to keep rising, and falling prices to keep falling… a tendency toward ‘conservation of momentum’ that doesn’t fit into their models of rational behavior.

I’m suspicious of any attempt to make economics seem like physics. Unlike elementary particles or rocks, people don’t seem to be very well modelled by simple differential equations. However, some economists have used the above analogy to model economic systems. And I can’t help but find that interesting—even if intellectually dubious when taken too seriously.

An auto-analogy

Beside the analogy I’ve already described between electronics and mechanics, there’s another one, called ‘Firestone’s analogy’:

• F.A. Firestone, A new analogy between mechanical and electrical systems, Journal of the Acoustical Society of America 4 (1933), 249–267.

Alain Bossavit pointed this out in the comments to Part 27. The idea is to treat current as analogous to force instead of velocity… and treat voltage as analogous to velocity instead of force!

In other words, switch your ’s and ’s:

Electronics	Mechanics          (usual analogy)	Mechanics      (Firestone’s analogy)
charge	position:	momentum:
current	velocity:	force:
flux linkage	momentum:	position:
voltage	force:	velocity:

This new analogy is not ‘mildly defective’: the product of effort and flow variables still has dimensions of power. But why bother with another analogy?

It may be helpful to recall this circuit from last time:

It’s described by this differential equation:

We used the ‘usual analogy’ to translate it into classical mechanics problem, and we got a problem where an object of mass is hanging from a spring with spring constant and damping coefficient and feeling an additional external force

And that’s fine. But there’s an intuitive sense in which all three forces are acting ‘in parallel’ on the mass, rather than in series. In other words, all side by side, instead of one after the other.

Using Firestone’s analogy, we get a different classical mechanics problem, where the three forces are acting in series. The spring is connected to source of friction, which in turn is connected to an external force.

This may seem a bit mysterious. But instead of trying to explain it, I’ll urge you to read his paper, which is short and clearly written. I instead want to make a somewhat different point, which is that we can take a mechanical system, convert it to an electrical one following the usual analogy, and then convert back to a mechanical one using Firestone’s analogy. This gives us an ‘auto-analogy’ between mechanics and itself, which switches and

And although I haven’t been able to figure out why from Firestone’s paper, I have other reasons for feeling sure this auto-analogy should contain a minus sign. For example:

In other words, it should correspond to a 90° rotation in the plane. There’s nothing sacred about whether we rotate clockwise or counterclockwise; we can equally well do this:

But we need the minus sign to get a so-called symplectic transformation of the plane. And from my experience with classical mechanics, I’m pretty sure we want that. If I’m wrong, please let me know!

I have a feeling we should revisit this issue when we get more deeply into the symplectic aspects of circuit theory. So, I won’t go on now.

References

The analogies I’ve been talking about are studied in a branch of engineering called system dynamics. You can read more about it here:

• Dean C. Karnopp, Donald L. Margolis and Ronald C. Rosenberg, System Dynamics: a Unified Approach, Wiley, New York, 1990.

• Forbes T. Brown, Engineering System Dynamics: a Unified Graph-Centered Approach, CRC Press, Boca Raton, 2007.

• Francois E. Cellier, Continuous System Modelling, Springer, Berlin, 1991.

System dynamics already uses lots of diagrams of networks. One of my goals in weeks to come is to explain the category theory lurking behind these diagrams.

The role of differential equations

Last time we looked at stochastic Petri nets, which use a random event model for the reactions. Individual entities are represented by tokens that flow through the network. When the token counts get large, we observed that they can be approximated by continuous quantities, which opens the door to the application of continuous mathematics to the analysis of network dynamics.

A key result of this approach is the “rate equation,” which gives a law of motion for the expected sizes of the populations. Equilibrium can then be obtained by solving for zero motion. The rate equations are applied in chemistry, where they give the rates of change of the concentrations of the various species in a reaction.

But before discussing the rate equation, here I will talk about the mathematical form of this law of motion, which consists of differential equations. This form is naturally associated with deterministic systems involving continuous magnitudes. This includes the equations of motion for the sine wave:

and the graceful ellipses that are traced out by the orbits of the planets around the sun:

This post provides some mathematical context to programmers who have not worked on scientific applications. My goal is to get as many of you on board as possible, before setting sail with Petri net programming.

Three approaches to equations: theoretical, formula-based, and computational

Let’s first consider the major approaches to equations in general. We’ll illustrate with a Diophantine equation

where and are integer variables.

In the theoretical approach (aka “qualitative analysis”), we start with the meaning of the equation and then proceed to reason about its solutions. Here are some simple consequences of this equation. They can’t all be zero, can’t all be positive, can’t all be negative, can’t all be even, and can’t all be odd.

In the formula-based approach, we seek formulas to describe the solutions. Here is an example of a formula (which does not solve our equation):

Such formulas are nice to have, but the pursuit of them is diabolically difficult. In fact, for Diophantine equations, even the question of whether an arbitrarily chosen equation has any solutions whatsoever has been proven to be algorithmically undecidable.

Finally, in the computational approach, we seek algorithms to enumerate or numerically approximate the solutions to the equations.

The three approaches to differential equations

Let’s apply the preceding classification to differential equations.

Theoretical approach

A differential equation is one that constrains the rates at which the variables are changing. This can include constraints on the rates at which the rates are changing (second-order equations), etc. The equation is ordinary if there is a single independent variable, such as time, otherwise it is partial.

Consider the equation stating that a variable increases at a rate equal to its current value. The bigger it gets, the faster it increases. Given a starting value, this determines a process — the solution to the equation — which here is exponential growth.

Let be the value at time and let’s initialize it to 1 at time 0. So we have:

These are first-order equations, because the derivative is applied at most once to any variable. They are linear equations, because the terms on each side of the equations are linear combinations of either individual variables or derivatives (in this case all of the coefficients are 1). Note also that a system of differential equations may in general have zero, one, or multiple solutions. This example belongs to a class of equations which are proven to have a unique solution for each initial condition.

You could imagine more complex systems of equations, involving multiple dependent variables, all still depending on time. That includes the rate equations for a Petri net, which have one dependent variable for each of the population sizes. The ideas for such systems are an extension of the ideas for a single-variable system. Then, a state of the system is a vector of values, with one component for each of the dependent variables. For first-order systems, such as the rate equations, where the derivatives appear on the left-hand sides, the equations determine, for each possible state of the system, a “direction” and rate of change for the state of the system.

Now here is a simple illustration of what I called the theoretical approach. Can ever become negative? No, because it starts out positive at time 0, and in order to later become negative, it must be decreasing at a time when it is still positive. That is to say, , and . But that contradicts the assumption . The general lesson here is that we don’t need a solution formula in order to make such inferences.

For the rate equations, the theoretical approach leads to substantial theorems about the existence and structure of equilibrium solutions.

Formula-based approach

It is natural to look for concise formulas to solve our equations, but the results of this overall quest are largely negative. The exponential differential equation cannot be solved by any formula that involves a finite combination of simple operations. So the solution function must be treated as a new primitive, and given a name, say . But even when we extend our language to include this new symbol, there are many differential equations that remain beyond the reach of finite formulas. So an endless collection of primitive functions is called for. (As standard practice, we always include and its complex extensions to the trigonometric functions, as primitives in our toolbox.)

But the hard mathematical reality does not end here, because even when solution formulas do exist, finding them may call for an ace detective. Only for certain classes of differential equations, such as the linear ones, do we have systematic solution methods.

The picture changes, however, if we let the formulas contain an infinite number of operations. Then the arithmetic operators give a far-reaching base for defining new functions. In fact, as you can verify, the power series

which we view as an “infinite polynomial” over the time parameter t, exactly satisfies our equations for exponential motion, and This power series therefore defines By the way, applying it to the input 1 produces a definition for the transcendental number :

Computational approach

Let’s leave aside our troubles with formulas, and consider the computational approach. For broad classes of differential equations, there are approximation algorithms that be successfully applied.

For starters, any power series that satisfies a differential equation may work for a simple approximation method. If a series is known to converge over some range of inputs, then one can approximate the value at those points by stopping the computation after a finite number of terms.

But the standard methods work directly with the equations, provided that they can be put into the right form. The simplest one is called Euler’s method. It works over a sampling grid of points separated by some small number . Let’s take the case where we have a first-order equation in explicit form, which means that for some function

We begin with the initial value . Applying we get Then for the interval from 0 to ,we use a linear approximation for , by assuming that the derivative remains constant at That gives Next, and etc. Formally,

Applying this to our exponential equation, where we get:

Hence:

So the approximation method gives a discrete exponential growth, which converges to a continuous exponential in the limit as the mesh size goes to zero.

Note, the case we just considered has more generality than might appear at first, because (1) the ideas here are easily extended to systems of explicit first order equations, and (2) higher-order equations that are “explicit” in an extended sense—meaning that the highest-order derivative is expressed as a function of time, of the variable, and of the lower-order derivatives—can be converted into an equivalent system of explicit first-order equations.

The challenging world of differential equations

So, is our cup half-empty or half-full? We have no toolbox of primitive formulas for building the solutions to all differential equations by finite compositions. And even for those which can be solved by formulas, there is no general method for finding the solutions. That is how the cookie crumbles. But on the positive side, there is an array of theoretical tools for analyzing and solving important classes of differential equations, and numerical methods can be applied in many cases.

The study of differential equations leads to some challenging problems, such as the Navier-Stokes equations, which describe the flow of fluids.

These are partial differential equations involving flow velocity, pressure, density and external forces (such as gravity), all of which vary over space and time. There are non-linear (multiplicative) interactions between these variables and their spatial and temporal derivatives, which leads to complexity in the solutions.

At high flow rates, this complexity can produce chaotic solutions, which involve complex behavior at a wide range of resolution scales. This is turbulence. Here is an insightful portrait of turbulence, by Leonardo da Vinci, whose studies in turbulence date back to the 15th Century.

Turbulence, which has been described by Richard Feynman as the most important unsolved problem of classical physics, also presents a mathematical puzzle. The general existence of solutions to the Navier-Stokes equations remains unsettled. This is one of the “Millennium Prize Problems”, for which a one million dollar prize is offered: in three dimensions, given initial values for the velocity and scalar fields, does there exist a solution that is smooth and everywhere defined? There are also complications with grid-based numerical methods, which will fail to produce globally accurate results if the solutions contain details at a smaller scale than the grid mesh. So the ubiquitous phenomenon of turbulence, which is so basic to the movements of the atmosphere and the seas, remains an open case.

But fortunately we have enough traction with differential equations to proceed directly with the rate equations for Petri nets. There we will find illuminating equations, which are the subject of both major theorems and open problems. They are non-linear and intractable by formula-based methods, yet, as we will see, they are well handled by numerical methods.

• Milankovitch Cycles and the Earth’s Climate.

It’s a gentle introduction to these ideas, and it presents a lot of what Blake Pollard and I have said about Milankovitch cycles, in a condensed way. Of course when I give the talk, I’ll add more words, especially about the different famous ‘puzzles’.

If you have any corrections, please let me know!

I’m eager to visit Cal State Northridge and especially David Klein in their math department, since I’d like to incorporate some climate science in our math curriculum the way they’ve done there.