Entropic Forces

 

In 2009, Erik Verlinde argued that gravity is an entropic force. This created a big stir—and it helped him win about $6,500,000 in prize money and grants! But what the heck is an ‘entropic force’, anyway?

Entropic forces are nothing unusual: you’ve felt one if you’ve ever stretched a rubber band. Why does a rubber band pull back when you stretch it? You might think it’s because a stretched rubber band has more energy than an unstretched one. That would indeed be a fine explanation for a metal spring. But rubber doesn’t work that way. Instead, a stretched rubber band mainly has less entropy than an unstretched one—and this too can cause a force.

You see, molecules of rubber are like long chains. When unstretched, these chains can curl up in lots of random wiggly ways. ‘Lots of random ways’ means lots of entropy. But when you stretch one of these chains, the number of ways it can be shaped decreases, until it’s pulled taut and there’s just one way! Only past that point does stretching the molecule take a lot of energy; before that, you’re mainly decreasing its entropy.

So, the force of a stretched rubber band is an entropic force.

But how can changes in either energy or entropy give rise to forces? That’s what I want to explain. But instead of talking about force, I’ll start out talking about pressure. This too arises both from changes in energy and changes in entropy.

Entropic pressure — a sloppy derivation

If you’ve ever studied thermodynamics you’ve probably heard about an ideal gas. You can think of this as a gas consisting of point particles that almost never collide with each other—because they’re just points—and bounce elastically off the walls of the container they’re in. If you have a box of gas like this, it’ll push on the walls with some pressure. But the cause of this pressure is not that slowly making the box smaller increases the energy of the gas inside: in fact, it doesn’t! The cause is that making the box smaller decreases the entropy of the gas.

To understand how pressure has an ‘energetic’ part and an ‘entropic’ part, let’s start with the basic equation of thermodynamics:

What does this mean? It means the internal energy of a box of stuff changes when you heat or cool it, meaning that you change its entropy but also when you shrink or expand it, meaning that you change its volume Increasing its entropy raises its internal energy at a rate proportional to its temperature Increasing its volume lowers its internal energy at a rate proportional to its pressure

We can already see that both changes in energy, and entropy, can affect Pressure is like force—indeed it’s just force per area—so we should try to solve for

First let’s do it in a sloppy way. One reason people don’t like thermodynamics is that they don’t understand partial derivatives when there are lots different coordinate systems floating around—which is what thermodynamics is all about! So, they manipulate these partial derivatives sloppily, feeling a sense of guilt and unease, and sometimes it works, but other times it fails disastrously. The cure is not to learn more thermodynamics; the cure is to learn about differential forms. All the expressions in the basic equation are differential forms. If you learn what they are and how to work with them, you’ll never get in trouble with partial derivatives in thermodynamics—as long as you proceed slowly and carefully.

But let’s act like we don’t know this! Let’s start with the basic equation

and solve for First we get

This is fine. Then we divide by and get

This is not so fine: here the guilt starts to set in. After all, we’ve been told that we need to use ‘partial derivatives’ when we have functions of several variables—and the main fact about partial derivatives, the one that everybody remembers, is that these are written with with curly d’s, not ordinary letter d’s. So we must have done something wrong. So, we make the d’s curly:

But we still feel guilty. First of all, who gave us the right to make those d’s curly? Second of all, a partial derivative like makes no sense unless is one of a set of coordinate functions: only then we can talk about how much some function changes as we change while keeping the other coordinates fixed. The value of actually depends on what other coordinates we’re keeping fixed! So what coordinates are we using?

Well, it seems like one of them is and the other is… we don’t know! It could be or or or perhaps even This is where real unease sets in. If we’re taking a test, we might in desperation think something like this: “Since the easiest things to control about our box of stuff are its volume and its temperature, let’s take these as our coordinates!” And then we might write

And then we might do okay on this problem, because this formula is in fact correct! But I hope you agree that this is an unsatisfactory way to manipulate partial derivatives: we’re shooting in the dark and hoping for luck.

Entropic pressure and entropic force

So, I want to show you a better way to get this result. But first let’s take a break and think about what it means. It means there are two possible reasons a box of gas may push back with pressure as we try to squeeze it smaller while keeping its temperature constant. One is that the energy may go up:

will be positive if the internal energy goes up as we squeeze the box smaller. But the other reason is that entropy may go down:

will be positive if the entropy goes down as we squeeze the box smaller, assuming

Let’s turn this fact into a result about force. Remember that pressure is just force per area. Say we have some stuff in a cylinder with a piston on top. Say the the position of the piston is given by some coordinate and its area is Then the stuff will push on the piston with a force

and the change in the cylinder’s volume as the piston moves is

Then

gives us

So, the force consists of two parts: the energetic force

and the entropic force:

Energetic forces are familiar from classical statics: for example, a rock pushes down on the table because its energy would decrease if it could go down. Entropic forces enter the game when we generalize to thermal statics, as we’re doing now. But when we set these entropic forces go away and we’re back to classical statics!

Entropic pressure—a better derivation

Okay, enough philosophizing. To conclude, let’s derive

in a less sloppy way. We start with

which is true no matter what coordinates we use. We can choose 2 of the 5 variables here as local coordinates, generically at least, so let’s choose and Then

and similarly

Using these, our equation

becomes

If you know about differential forms, you know that the differentials of the coordinate functions, namely and form a basis of 1-forms. Thus we can equate the coefficients of in the equation above and get:

and thus:

which is what we wanted! There should be no bitter aftertaste of guilt this time.

The big picture

That’s almost all I want to say: a simple exposition of well-known stuff that’s not quite as well-known as it should be. If you know some thermodynamics and are feeling mildly ambitious, you can now work out the pressure of an ideal gas and show that it’s completely entropic in origin: only the first term in the right-hand side above is nonzero. If you’re feeling a lot more ambitious, you can try to read Verlinde’s papers and explain them to me. But my own goal was not to think about gravity. Instead, it was to ponder a question raised by Allen Knutson: how does the ‘entropic force’ idea fit into my ruminations on classical mechanics versus thermodynamics?

It seems to fit in this way: as we go from classical statics (governed by the principle of least energy) to thermal statics at fixed temperature (governed by the principle of least free energy), the definition of force familiar in classical statics must be adjusted. In classical statics we have

where

is the energy as a function of some coordinates on the configuration space of our system, some manifold But in thermal statics at temperature our system will try to minimize, not the energy but the Helmholtz free energy

where

is the entropy. So now we should define force by

and we see that force has an entropic part and an energetic part:

When the entropic part goes away and we’re back to classical statics!

---

I’m subject to the natural forces. – Lyle Lovett

The Faculty of 1000

As of this minute, 1890 scholars have signed a pledge not to cooperate with the publisher Elsevier. People are starting to notice. According to this Wired article, the open-access movement is “catching fire”:

• David Dobbs, Testify: the open-science movement catches fire, Wired, 30 January 2012.

Now is a good time to take more substantial actions. But what?

Many things are being discussed, but it’s good to spend a bit of time thinking about the root problems and the ultimate solutions.

The world-wide web has made journals obsolete: it would be better to put papers on freely available archives and then let boards of top scholars referee them. But how do we get to this system?

In math and physics we have the arXiv, but nobody referees those papers. In biology and medicine, a board called the Faculty of 1000 chooses and evaluates the best papers, but there’s no archive: they get those papers from traditional journals.

Whoops—never mind! That was yesterday. Now the Faculty of 1000 has started an archive!

• Rebecca Lawrence, F1000 Research – join us and shape the future of scholarly communication, F1000, 30 January 2012.

• Ivan Oransky, An arXiv for all of science? F1000 launches new immediate publication journal, Retraction Watch, 30 January 2012.

This blog article says “an arXiv for all science”, but it seems the new F1000 Research archive is just for biology and medicine. So now it’s time for the mathematicians and physicists to start catching up.

A Quantum Hammersley–Clifford Theorem

I’m at this workshop:

• Sydney Quantum Information Theory Workshop: Coogee 2012, 30 January – 2 February 2012, Coogee Bay Hotel, Coogee, Sydney, organized by Stephen Bartlett, Gavin Brennen, Andrew Doherty and Tom Stace.

Right now David Poulin is speaking about a quantum version of the Hammersley–Clifford theorem, which is a theorem about Markov networks. Let me quickly say a bit about what he proved! This will be a bit rough, since I’m doing it live…

The mutual information between two random variables is

The conditional mutual information between three random variables is

It’s the average amount of information about learned by measuring when you already knew

All this works for both classical (Shannon) and quantum (von Neumann) entropy. So, when we say ‘random variable’ above, we
could mean it in the traditional classical sense or in the quantum sense.

If then has the following Markov property: if you know learning tells you nothing new about In condensed matter physics, say a spin system, we get (quantum) random variables from measuring what’s going on in regions, and we have short range entanglement if when corresponds to some sufficiently thick region separating the regions and We’ll get this in any Gibbs state of a spin chain with a local Hamiltonian.

A Markov network is a graph with random variables at vertices (and thus subsets of vertices) such that whenever is a subset of vertices that completely ‘shields’ the subset from the subset : any path from to goes through a vertex in a

The Hammersley–Clifford theorem says that in the classical case we can get any Markov network from the Gibbs state

of a local Hamiltonian and vice versa. Here a Hamiltonian is local if it is a sum of terms, one depending on the degrees of freedom in each clique in the graph:

Hayden, Jozsa, Petz and Winter gave a quantum generalization of one direction of this result to graphs that are just ‘chains’, like this:

o—o—o—o—o—o—o—o—o—o—o—o

Namely: for such graphs, any quantum Markov network is the Gibbs state of some local Hamiltonian. Now Poulin has shown the same for all graphs. But the converse is, in general, false. If the different terms in a local Hamiltonian all commute, its Gibbs state will have the Markov property. But otherwise, it may not.

For some related material, see:

• David Poulin, Quantum graphical models and belief propagation.

How to Cut Carbon Emissions and Save Money

McKinsey & Company is a management consulting firm. In 2010 they released this ‘carbon abatement cost curve’ for the whole world:

Click it to see a nice big version. So, they’re claiming:

• By 2030 we can cut CO₂ emissions about 15 gigatonnes per year while saving lots of money.

• By 2030 can cut CO₂ emissions by up to 37 gigatonnes per year before the total cost—that is, cost minus savings—becomes positive.

The graph is cute. The vertical axis of the graph says how many euros per tonne it would cost to cut CO₂ emissions by 2030 using various measures. The horizontal axis says how many gigatonnes per year we could reduce CO₂ emissions using these measures.

So, we get lots of blue rectangles. If a rectangle is below the horizontal axis, its area says how many euros per year we’d save by implementing that measure. If it’s above the axis, its area says how much that measure would cost.

I believe the total blue area below the axis equals the total blue area above the axis. So if we do all these things, the total cost is zero.

37 gigatonnes of CO₂ is roughly 10 gigatonnes of carbon: remember, there’s a crucial factor of here. In 2004, Pacala and Socolow argued that the world needs to find ways to cut carbon emissions by about 7 gigatonnes/year by 2054 to keep emissions flat until this time. By now we’d need 9 gigatonnes/year.

If so, it seems the measures shown here could keep carbon emissions flat worldwide at no net cost!

But as usual, there are at least a few problems.

Problem 1

Is McKinsey’s analysis correct? I don’t know. Here’s their report, along with some others:

• McKinsey & Company, Impact of the financial crisis on carbon economics: Version 2.1 of the global greenhouse gas abatement cost curve, 2010.

For more details it’s good to read version 2.0:

• McKinsey & Company, Pathways to a low carbon economy: Version 2 of the global greenhouse gas abatement cost curve, 2009.

They’re free if you fill out some forms. But it’s not easy to check these things. Does anyone know papers that try to check McKinsey’s work? I find it’s more fun to study a problem like this after you see two sides of the same story.

Problem 2

I said ‘no net cost’. But if you need to spend a lot of money, the fact that I’m saving a lot doesn’t compensate you. So there’s the nontrivial problem of taking money that’s saved on some measures and making sure it gets spent on others. Here’s where ‘big government’ might be required—which makes some people decide global warming is just a political conspiracy, nyeh-heh-heh.

Is there another way to make the money transfer happen, without top-down authority?

We could still get the job about half-done at a huge savings, of course. McKinsey says we could cut CO₂ emissions by 15 gigatonnes per year doing things that only save money. That’s about 4 gigatonnes of carbon per year! We could at least do that.

Problem 3

Keeping carbon emissions flat is not enough. Carbon dioxide, once put in the atmosphere, stays there a long time—though individual molecules come and go. As the saying goes, carbon is forever. (Click that link for more precise information.)

So, even Pacala and Socolow say keeping carbon emissions flat is a mere stopgap before we actually reduce carbon emissions, starting in 2054. But some more recent papers seem to suggest Pacala and Socolow were being overly optimistic.

Of course it depends on how much global warming you’re willing to tolerate! It also depends on lots of other things.

Anyway, this paper claims that if we cut global greenhouse gas emissions in half by 2050 (as compared to what they were in 1990), there’s a 12–45% probability that the world will get at least 2 °C warmer than its temperature before the industrial revolution:

• Malte Meinshausen et al, Greenhouse-gas emission targets for limiting global warming to 2 °C, Nature 458 (2009), 1158–1163.

Abstract: More than 100 countries have adopted a global warming limit of 2 °C or below (relative to pre-industrial levels) as a guiding principle for mitigation efforts to reduce climate change risks, impacts and damages. However, the greenhouse gas (GHG) emissions corresponding to a specified maximum warming are poorly known owing to uncertainties in the carbon cycle and the climate response. Here we provide a comprehensive probabilistic analysis aimed at quantifying GHG emission budgets for the 2000–50 period that would limit warming throughout the twenty-first century to below 2 °C, based on a combination of published distributions of climate system properties and observational constraints. We show that, for the chosen class of emission scenarios, both cumulative emissions up to 2050 and emission levels in 2050 are robust indicators of the probability that twenty-first century warming will not exceed 2 °C relative to pre-industrial temperatures.

Limiting cumulative CO₂ emissions over 2000–50 to 1,000 Gt CO₂ yields a 25% probability of warming exceeding 2 °C—and a limit of 1,440 Gt CO2 yields a 50% probability—given a representative estimate of the distribution of climate system properties. As known 2000–06 CO₂ emissions were 234 Gt CO₂, less than half the proven economically recoverable oil, gas and coal reserves can still be emitted up to 2050 to achieve such a goal. Recent G8 Communiques envisage halved global GHG emissions by 2050, for which we estimate a 12–45% probability of exceeding 2 °C—assuming 1990 as emission base year and a range of published climate sensitivity distributions. Emissions levels in 2020 are a less robust indicator, but for the scenarios considered, the probability of exceeding 2 °C rises to 53–87% if global GHG emissions are still more than 25% above 2000 levels in 2020.

This paper says we’re basically doomed to suffer unless we revamp society:

• Ted Trainer, Can renewables etc. solve the greenhouse problem? The negative case, Energy Policy 38 (2010), 4107–4114.

Abstract: Virtually all current discussion of climate change and energy problems proceeds on the assumption that technical solutions are possible within basically affluent-consumer societies. There is however a substantial case that this assumption is mistaken. This case derives from a consideration of the scale of the tasks and of the limits of non-carbon energy sources, focusing especially on the need for redundant capacity in winter. The first line of argument is to do with the extremely high capital cost of the supply system that would be required, and the second is to do with the problems set by the intermittency of renewable sources. It is concluded that the general climate change and energy problem cannot be solved without large scale reductions in rates of economic production and consumption, and therefore without transition to fundamentally different social structures and systems.

It’s worth reading because it uses actual numbers, not just hand-waving. But it seeks much more than keeping carbon emissions flat until 2050; that’s one reason for the dire conclusions.

It’s worth noting this rebuttal, which says that everything about Trainer’s paper is fine except a premature dismissal of nuclear power:

• Barry Brook, Could nuclear fission energy, etc., solve the greenhouse problem? The affirmative case, Energy Policy, available online 16 December 2011.

To get your hands on Brook’s paper you either need a subscription or you need to email him. You can do that starting from his blog article about the paper… which is definitely worth reading:

• Barry Brook, Could nuclear fission energy, etc., solve the greenhouse problem? The affirmative case, BraveNewClimate, 14 January 2012.

According to Brook, we can keep global warming from getting too bad if we get really serious about nuclear power.

Of course, these three papers are just a few of many. I’m still trying to sift through the information and figure out what’s really going on. It’s hard. It may be impossible. But McKinsey’s list of ways to cut carbon emissions and save money points to some things we start doing right now.

Ban Elsevier

Please take the pledge not to do business with Elsevier. 404 scientists have done it so far:

• The cost of knowledge.

You can separately say you

1) won’t publish with them,
2) won’t referee for them, and/or
3) won’t do editorial work for them.

At least do number 2): how often can you do something good by doing less work? When a huge corporation relies so heavily on nasty monopolistic practices and unpaid volunteer labor, they leave themselves open to this.

This pledge website is the brainchild of Tim Gowers, a Fields medalist and prominent math blogger:

• Tim Gowers, Elsevier: my part in its downfall and http://thecostofknowledge.com.

In case you’re not familiar with the Elsevier problem, here’s something excerpted from my website. This does not yet mention Elsevier’s recent support of the Research Works Act, which would try to roll back the US government’s requirement that taxpayer-funded medical research be made freely available online. Nor does it mention the fake medical journals created by Elsevier, where what looked like peer-reviewed papers were secretly advertisements paid for by drug companies! Nor does it mention the Chaos, Solitons and Fractals fiasco. Indeed, it’s hard keeping up with Elsevier’s dirty deeds!

The problem and the solutions

The problem of highly priced science journals is well-known. A wave of mergers in the publishing business has created giant firms with the power to extract ever higher journal prices from university libraries. As a result, libraries are continually being forced to cough up more money or cut their journal subscriptions. It’s really become a crisis.

Luckily, there are also two counter-trends at work. In mathematics and physics, more and more papers are available from a free electronic database called the arXiv, and journals are beginning to let papers stay on this database even after they are published. In the life sciences, PubMed Central plays a similar role.

There are also a growing number of free journals. Many of these are peer-reviewed, and most are run by academics instead of large corporations.

The situation is worst in biology and medicine: the extremely profitable spinoffs of research in these subjects has made it easy for journals to charge outrageous prices and limit the free nature of discourse. A non-profit organization called the Public Library of Science was formed to fight this, and circulated an open letter calling on publishers to adopt reasonable policies. 30,000 scientists signed this and pledged to:

publish in, edit or review for, and personally subscribe to only those scholarly and scientific journals that have agreed to grant unrestricted free distribution rights to any and all original research reports that they have published, through PubMed Central and similar online public resources, within 6 months of their initial publication date.

Unsurprisingly, the response from publishers was chilly. As a result, the Public Library of Science started its own free journals in biology and medicine, with the help of a 9 million dollar grant from the Gordon and Betty Moore Foundation.

A number of other organizations are also pushing for free access to scholarly journals, such as Create Change, the Scholarly Publishing and Academic Resources Coalition, and the Budapest Open Access Initiative, funded by George Soros.

Editorial boards are beginning to wise up, too. On August 10, 2006, all the editors of the math journal Topology resigned to protest the outrageous prices of the publisher, Reed Elsevier. In August of this year, the editorial board of the Springer journal K-Theory followed suit. The Ecole Normale Superieure has also stopped having Elsevier publish the journal Annales Scientifiques de l’École Normale Supérieure.

So, we may just win this war! But only if we all do our part.

What we can do

What can we do to keep academic discourse freely available to all? Here are some things:

1. Don’t publish in overpriced journals.

2. Don’t do free work for overpriced journals (like refereeing and editing).

3. Put your articles on the arXiv or a similar site before publishing them.

4. Only publish in journals that let you keep your articles on the arXiv or a similar site.

5. Support free journals by publishing in them, refereeing for them, editing them… even starting your own!

6. Help make sure free journals and the arXiv stay free.

7. Help start a system of independent ‘referee boards‘ for arXiv papers. These can referee papers and help hiring, tenure and promotion committees to assess the worth of papers, eliminating the last remaining reasons for the existence of traditional for-profit journals.

The nice thing is that most of these are easy to do! Only items 5 through 7 require serious work. As for item 4, a lot of math and physics journals not only let you keep your article on the arXiv, but let you submit it by telling them its arXiv number! In math it’s easy to find these journals, because there’s a public list of them.

Of course, you should read the copyright agreement that you’ll be forced to sign before submitting to a journal or publishing a book. Check to see if you can keep your work on the arXiv, on your own website, etcetera. You can pretty much assume that any rights you don’t explicitly keep, your publisher will get. Eric Weisstein didn’t do this, and look what happened to him: he got sued and spent over a year in legal hell!

Luckily it’s not hard to read these copyright agreements: you can get them off the web. An extensive list is available from Sherpa, an organization devoted to free electronic archives.

If you think maybe you want to start your own journal, or move an existing journal to a cheaper publisher, read Joan Birman’s article about this. Go to the Create Change website and learn what other people are doing. Also check out SPARC—the Scholarly Publishing and Academic Resources Coalition. They can help. And try the Budapest Open Access Initiative—they give out grants.

You can also support the Public Library of Science or join the Open Archives Initiative.

Also: if you like mathematics, tell your librarian about Mathematical Sciences Publishers, a nonprofit organization run by mathematicians for the purpose of publishing low-cost, high-quality math journals.

Which journals are overpriced?

In 1997 Robion Kirby urged mathematicians not to submit papers to, nor edit for, nor referee for overpriced journals. I think this suggestion is great, and it applies not just to mathematics but all disciplines. There is really no good reason for us to donate our work to profit-making corporations who sell it back to us at exorbitant prices! Indeed in climate science this has a terrible effect: crackpot bloggers distribute their misinformation free of charge, while lots of important serious climate science papers are hidden, available only to people who work at institutions with expensive subscriptions.

But how can you tell if a journal is overpriced? In mathematics, Up-to-date information on the rise of journal prices is available from the American Mathematical Society. They even include an Excel spreadsheet that lets you do your own calculations with this data! Some of this information is nicely summarized on a webpage by Ulf Rehmann. Using these tools you can make up your own mind which journals are too expensive to be worth supporting with your free volunteer labor.

What about other subjects? I don’t know. Maybe you do?

When I first learned how bad the situation was, I started by boycotting all journals published by Reed Elsevier. This juggernaut was formed by merger of Reed Publishing and Elsevier Scientific Press in 1993. In August 2001 it bought Harcourt Press—which in turn owned Academic Press, which ran a journal I helped edit, Advances in Mathematics. I don’t work for that journal anymore! The reason is that Reed Elsevier is a particularly bad culprit when it comes to charging high prices. You can see this from the above lists of journal prices, and you can also see it in the business news. In 2002, Forbes magazine wrote:

If you are not a scientist or a lawyer, you might never guess which company is one of the world’s biggest in online revenue. Ebay will haul in only $1 billion this year. Amazon has $3.5 billion in revenue but is still, famously, losing money. Outperforming them both is Reed Elsevier, the London-based publishing company. Of its $8 billion in likely sales this year, $1.5 billion will come from online delivery of data, and its operating margin on the internet is a fabulous 22%.

Credit this accomplishment to two things. One is that Reed primarily sells not advertising or entertainment but the dry data used by lawyers, doctors, nurses, scientists and teachers. The other is its newfound marketing hustle: Its CEO since 1999 has been Crispin Davis, formerly a soap salesman.

But Davis will have to keep hustling to stay out of trouble. Reed Elsevier has fat margins and high prices in a business based on information—a commodity, and one that is cheaper than ever in the internet era. New technologies and increasingly universal access to free information make it vulnerable to attack from below. Today pirated music downloaded from the web ravages corporate profits in the music industry. Tomorrow could be the publishing industry’s turn.

Some customers accuse Reed Elsevier of price gouging. Daniel DeVito, a patent lawyer with Skadden, Arps, Slate, Meagher & Flom, is a fan of Reed’s legal-search service, but he himself does free science searches on the Google site before paying for something like Reed’s ScienceDirect—and often finds what he’s looking for at no cost. Reed can ill afford to rest.

Why should we slave away unpaid to keep Crispin Davis and his ilk rolling in dough? There’s really no good reason.

Sneaky tricks

To fight against the free journals and the arXiv, publishing companies are playing sneaky tricks like these:

• Proprietary Preprint Archives. Examples included ChemWeb and something they called "The Mathematics Preprint Server". The latter was especially devious, because mathematicians used to call the arXiv "the mathematics preprint server".

However, the Mathematics Preprint Server didn’t fool many smart people, so lots of the papers they got were crap, like a supposed proof of Goldbach’s conjecture, and a claim that the rotation of a galactic supercluster is due to a "topological defect" in spacetime. Eventually Elsevier gave up and stopped accepting new papers on their preprint server. Now it’s a laughable shadow of its former self. Similarly, ChemWeb was sold off.

• Web Spamming. More recently, publishers have tried a new trick: “web spamming”, also known as “search engine spamming” or “cloaking”. The company gives search engine crawlers access to full-text articles — but when you try to read these articles, you get a "doorway page" demanding a subscription or payment. Sometimes you’ll even be taken to a page that has nothing to do with the paper you thought you were about to see!

Culprits include Springer, Reed Elsevier, and the Institute of Electrical and Electronic Engineers. The last one seems to have quit — but check out their powerpoint presentation on this subject, courtesy of Carl Willis.

If you see pages like this, report them to Google or your favorite search engine.

• Journal Bundling. Worse still is the strategy of "bundling" subscriptions into huge all-or-nothing packages, so libraries can’t save money by ceasing to subscribe to a single journal. It’s a clever trap, especially because these bundled subscriptions look like a good deal at first. The cost becomes apparent only later. Now universities libraries are being bankrupted as the prices of these bundles keep soaring. The library of my own university, U.C. Riverside, barely has money for any books anymore!

Luckily, people are catching on. In 2003, Cornell University bravely dropped their subscription to 930 Elsevier journals. Four North Carolina universities have joined the revolt, and the University of California has also been battling Elsevier. For other actions universities have taken, read Peter Suber’s list.

• Legal bullying. Large corporations like to scare people by means of threats of legal action backed up by deep pockets. A classic example is the lawsuit launched by Gordon and Breach against the American Physical Society for publishing lists of journal prices. Luckily they lost this suit.

• Hiring a Dr. Evil lookalike as their PR consultant.

Click either of the pictures for an explanation.

I, Robot

On 13 February 2012, I will give a talk at Google in the form of a robot. I will look like this:

My talk will be about “Energy, the Environment and What We Can Do.” Since I think we should cut unnecessary travel, I decided to stay here in Singapore and use a telepresence robot instead of flying to California.

I thank Mike Stay for arranging this at Google, and I especially thank Trevor Blackwell and everyone else at Anybots for letting me use one of their robots!

I believe Google will film this event and make a video available. But I hope reporters attend, because it should be fun, and I plan to describe some ways we can slash carbon emissions.

More detail: I will give this talk at 4 pm Monday, February 13, 2012 in the Paramaribo Room on the Google campus (Building 42, Floor 2). Visitors and reporters are invited, but they need to check in at the main visitor’s lounge in Building 43, and they’ll need to be escorted to and from the talk, so someone will pick them up 10 or 15 minutes before the talk starts.

Energy, the Environment and What We Can Do

Abstract: Our heavy reliance on fossil fuels is causing two serious problems: global warming, and the decline of cheaply available oil reserves. Unfortunately the second problem will not cancel out the first. Each one individually seems extremely hard to solve, and taken
together they demand a major worldwide effort starting now. After an overview of these problems, we turn to the question: what can we do about them?

I also need help from all of you reading this! I want to talk about solutions, not just problems—and given my audience, and the political deadlock in the US, I especially want to talk about innovative solutions that come from individuals and companies, not governments.

Can changing whole systems produce massive cuts in carbon emissions, in a way that spreads virally rather than being imposed through top-down directives? It’s possible. Curtis Faith has some inspiring thoughts on this:

I’ve been looking on various transportation and energy and environment issues for more than 5 years, and almost no one gets the idea that we can radically reduce consumption if we look at the complete systems. In economic terms, we currently have a suboptimal Nash Equilibrium with a diminishing pie when an optimal expanding pie equilibrium is possible. Just tossing around ideas a a very high level with back of the envelope estimates we can get orders of magnitude improvements with systemic changes that will make people’s lives better if we can loosen up the grip of the big corporations and government.

To borrow a physics analogy, the Nash Equilibrium is a bit like a multi-dimensional metastable state where the system is locked into a high energy configuration and any local attempts to make the change revert to the higher energy configuration locally, so it would require sufficient energy or energy in exactly the right form to move all the different metastable states off their equilibrium either simultaneously or in a cascade.

Ideally, we find the right set of systemic economic changes that can have a cascade effect, so that they are locally systemically optimal and can compete more effectively within the larger system where the Nash Equilibrium dominates. I hope I haven’t mixed up too many terms from too many fields and confused things. These terms all have overlapping and sometimes very different meaning in the different contexts as I’m sure is true even within math and science.

One great example is transportation. We assume we need electric cars or biofuel or some such thing. But the very assumption that a car is necessary is flawed. Why do people want cars? Give them a better alternative and they’ll stop wanting cars. Now, what that might be? Public transportation? No. All the money spent building a 2,000 kg vehicle to accelerate and decelerate a few hundred kg and then to replace that vehicle on a regular basis can be saved if we eliminate the need for cars.

The best alternative to cars is walking, or walking on inclined pathways up and down so we get exercise. Why don’t people walk? Not because they don’t want to but because our cities and towns have optimized for cars. Create walkable neighborhoods and give people jobs near their home and you eliminate the need for cars. I live in Savannah, GA in a very tiny place. I never use the car. Perhaps 5 miles a week. And even that wouldn’t be necessary with the right supplemental business structures to provide services more efficiently.

Or electricity for A/C. Everyone lives isolated in structures that are very inefficient to heat. Large community structures could be air conditioned naturally using various techniques and that could cut electricity demand by 50% for neighborhoods. Shade trees are better than insulation.

Or how about moving virtually entire cities to cooler climates during the hot months? That is what people used to do. Take a train North for the summer. If the destinations are low-resource destinations, this can be a huge reduction for the city. Again, getting to this state is hard without changing a lot of parts together.

These problems are not technical, or political, they are economic. We need the economic systems that support these alternatives. People want them. We’ll all be happier and use far less resources (and money). The economic system needs to be changed, and that isn’t going to happen with politics, it will happen with economic innovation. We tend to think of our current models as the way things are, but they aren’t. Most of the status quo is comprised of human inventions, money, fractional reserve banking, corporations, etc. They all brought specific improvements that made them more effective at the time they were introduce because of the conditions during those times. Our times too are different. Some new models will work much better for solving our current problems.

Your idea really starts to address the reason why people fly unnecessarily. This change in perspective is important. What if we went back to sailing ships? And instead of flying we took long leisurely educational seminar cruises on modern versions of sail yachts? What if we improved our trains? But we need to start from scratch and design new systems so they work together effectively. Why are we stuck with models of cities based on the 19th-century norms?

We aren’t, but too many people think we are because the scope of their job or academic career is just the piece of a system, not the system itself.

System level design thinking is the key to making the difference we need. Changes to the complete systems can have order of magnitude improvements. Changes to the parts will have us fighting for tens of percentages.

Do you know good references on ideas like this—preferably with actual numbers? I’ve done some research, but I feel I must be missing a lot of things.

This book, for example, is interesting:

• Michael Peters, Shane Fudge and Tim Jackson, editors, Low Carbon Communities: Imaginative Approaches to Combating Climate Change Locally, Edward Elgar Publishing Group, Cheltenham, UK, 2010.

but I wish it had more numbers on how much carbon emissions were cut by some of the projects they describe: Energy Conscious Households in Action, the HadLOW CARBON Community, the Transition Network, and so on.

Classical Mechanics versus Thermodynamics (Part 2)

I showed you last time that in many branches of physics—including classical mechanics and thermodynamics—we can see our task as minimizing or maximizing some function. Today I want to show how we get from that task to symplectic geometry.

So, suppose we have a smooth function

where is some manifold. A minimum or maximum of can only occur at a point where

Here the differential which is a 1-form on If we pick local coordinates in some open set of then we have

and these derivatives are very interesting. Let’s see why:

Example 1. In classical mechanics, consider a particle on a manifold Suppose the particle starts at some fixed position at some fixed time. Suppose that it ends up at the position at time Then the particle will seek to follow a path that minimizes the action given these conditions. Assume this path exists and is unique. The action of this path is then called Hamilton’s principal function, Let

and assume Hamilton’s principal function is a smooth function

We then have

where are local coordinates on

is called the momentum in the ith direction, and

is called the energy. The minus signs here are basically just a mild nuisance. Time is different from space, and in special relativity the difference comes from a minus sign, but I don’t think that’s the explanation here. We could get rid of the minus signs by working with negative energy, but it’s not such a big deal.

Example 2. In thermodynamics, consider a system with the internal energy and volume Then the system will choose a state that maximizes the entropy given these constraints. Assume this state exists and is unique. Call the entropy of this state Let

and assume the entropy is a smooth function

We then have

where is the temperature of the system, and is the pressure. The slight awkwardness of this formula makes people favor other setups.

Example 3. In thermodynamics there are many setups for studying the same system using different minimum or maximum principles. One of the most popular is called the energy scheme. If internal energy increases with increasing entropy, as usually the case, this scheme is equivalent to the one we just saw.

In the energy scheme we fix the entropy and volume Then the system will choose a state that minimizes the internal energy given these constraints. Assume this state exists and is unique. Call the internal energy of this state Let

and assume the entropy is a smooth function

We then have

where

is the temperature, and

is the pressure. You’ll note the formulas here closely resemble those in Example 1!

Example 4. Here are the four most popular schemes for thermodynamics:

• If we fix the entropy and volume the system will choose a state that minimizes the internal energy

• If we fix the entropy and pressure the system will choose a state that minimizes the enthalpy

• If we fix the temperature and volume the system will choose a state that minimizes the Helmholtz free energy

• If we fix the temperature and pressure the system will choose a state that minimizes the Gibbs free energy

These quantities are related by a pack of similar-looking formulas, from which we may derive a mind-numbing little labyrinth of Maxwell relations. But for now, all we need to know is that all these approaches to thermodynamics are equivalent given some reasonable assumptions, and all the formulas and relations can be derived using the Legendre transformation trick I explained last time. So, I won’t repeat what we did in Example 3 for all these other cases!

Example 5. In classical statics, consider a particle on a manifold This particle will seek to minimize its potential energy which we’ll assume is some smooth function of its position We then have

where are local coordinates on and

is called the force in the ith direction.

Conjugate variables

So, the partial derivatives of the quantity we’re trying
to minimize or maximize are very important! As a result, we often want to give them more of an equal status as independent quantities in their own right. Then we call them ‘conjugate variables’.

To make this precise, consider the cotangent bundle which has local coordinates (coming from the coordinates on ) and (the corresponding coordinates on each cotangent space). We then call the conjugate variable of the coordinate

Given a smooth function

the 1-form can be seen as a section of the cotangent bundle. The graph of this section is defined by the equation

and this equation ties together two intuitions about ‘conjugate variables’: as coordinates on the cotangent bundle, and as partial derivatives of the quantity we’re trying to minimize or maximize.

The tautological 1-form

There is a lot to say here, especially about Legendre transformations, but I want to hasten on to a bit of symplectic geometry. And for this we need the ‘tautological 1-form’ on

We can think of as a map

sending each point to the point where is defined by the equation we just saw:

Using this map, we can pull back any 1-form on to get a 1-form on

What 1-form on might we like to get? Why, of course!

Amazingly, there’s a 1-form on such that when we pull it back using the map we get the 1-form —no matter what smooth function we started with!

Thanks to this wonderfully tautological property, is called the tautological 1-form on You should check that it’s given by the formula

If you get stuck, try this.

So, if we want to see how much changes as we move along a path in we can do this in three equivalent ways:

• Evaluate at the endpoint of the path and subtract off at the starting-point.

• Integrate the 1-form along the path.

• Use to map the path over to and then integrate over this path in

The last method is equivalent thanks to the ‘tautological’ property of It may seem overly convoluted, but it shows that if we work in where the conjugate variables are accorded equal status, everything we want to know about the change in is contained in the 1-form no matter which function we decide to use!

So, in this sense, knows everything there is to know about the change in Hamilton’s principal function in classical mechanics, or the change in entropy in thermodynamics… and so on!

But this means it must know about things like Hamilton’s equations, and the Maxwell relations.

The symplectic structure

We saw last time that the fundamental equations of classical mechanics and thermodynamics—Hamilton’s equations and the Maxwell relations—are mathematically just the same. They both say simply that partial derivatives commute:

where is the function we’re trying to minimize or maximize.

I also mentioned that this fact—the commuting of partial derivatives—can be stated in an elegant coordinate-free way:

Perhaps I should remind you of the proof:

but

changes sign when we switch and while

does not, so It’s just a wee bit more work to show that conversely, starting from it follows that the mixed partials must commute.

How can we state this fact using the tautological 1-form ? I said that using the map

we can pull back to and get . But pulling back commutes with the operator! So, if we pull back we get But So, has the magical property that when we pull it back to we always get zero, no matter what we choose!

This magical property captures Hamilton’s equations, the Maxwell relations and so on—for all choices of at once. So it shouldn’t be surprising that the 2-form

is colossally important: it’s the famous symplectic structure on the so-called phase space

Well, actually, most people prefer to work with

It seems this whole subject is a monument of austere beauty… covered with minus signs, like bird droppings.

Example 6. In classical mechanics, let

as in Example 1. If has local coordinates then has these along with the conjugate variables as coordinates. As we explained, it causes little trouble to call these conjugate variables by the same names we used for the partial derivatives of namely, and So, we have

and thus

Example 7. In thermodynamics, let

as in Example 3. If has coordinates then the conjugate variables deserve to be called So, we have

and

You’ll see that in these formulas for variables get paired with their conjugate variables. That’s nice.

But let me expand on what we just saw, since it’s important. And let me talk about without tossing in that extra sign.

What we saw is that the 2-form is a ‘measure of noncommutativity’. When we pull back to we get zero. This says that partial derivatives commute—and this gives Hamilton’s equations, the Maxwell relations, and all that. But up in is not zero. And this suggests that there’s some built-in noncommutativity hiding in phase space!

Indeed, we can make this very precise. Consider a little parallelogram up in :

Suppose we integrate the 1-form up the left edge and across the top. Do we get the same answer if integrate it across the bottom edge and then up the right?

No, not necessarily! The difference is the same as the integral of all the way around the parallelogram. By Stokes’ theorem, this is the same as integrating over the parallelogram. And there’s no reason that should give zero.

However, suppose we got our parallelogram in by taking a parallelogram in and applying the map

Then the integral of around our parallelogram would be zero, since it would equal the integral of around a parallelogram in … and that’s the change in as we go around a loop from some point to… itself!

And indeed, the fact that a function doesn’t change when we go around a parallelogram is precisely what makes

So the story all fits together quite nicely.

The big picture

I’ve tried to show you that the symplectic structure on the phase spaces of classical mechanics, and the lesser-known but utterly analogous one on the phase spaces of thermodynamics, is a natural outgrowth of utterly trivial reflections on the process of minimizing or maximizing a function on a manifold .

The first derivative test tells us to look for points with

while the commutativity of partial derivatives says that

everywhere—and this gives Hamilton’s equations and the Maxwell relations. The 1-form is the pullback of the tautologous 1-form on and similarly is the pullback of the symplectic structure The fact that

says that holds noncommutative delights, almost like a world where partial derivatives no longer commute! But of course we still have

everywhere, and this becomes part of the official definition of a symplectic structure.

All very simple. I hope, however, the experts note that to see this unified picture, we had to avoid the most common approaches to classical mechanics, which start with either a ‘Hamiltonian’

or a ‘Lagrangian’

Instead, we started with Hamilton’s principal function

where is not the usual configuration space describing possible positions for a particle, but the ‘extended’ configuration space, which also includes time. Only this way do Hamilton’s equations, like the Maxwell relations, become a trivial consequence of the fact that partial derivatives commute.

But what about those ‘noncommutative delights’? First, there’s a noncommutative Poisson bracket operation on functions on . This makes the functions into a so-called Poisson algebra. In classical mechanics of a point particle on the line, for example, it’s well-known that we have

In thermodynamics, the analogous relations

seem sadly little-known. But you can see them here, for example:

• M. J. Peterson, Analogy between thermodynamics and mechanics, American Journal of Physics 47 (1979), 488–490.

at least up to one of those pesky minus signs! We can use these Poisson brackets to study how one thermodynamic variable changes as we slowly change another, staying close to equilibrium all along.

Second, we can go further and ‘quantize’ the functions on . This means coming up with an associative but noncommutative product of these function that mimics the Poisson bracket to some extent. In the case of a particle on a line, we’d get commutation relations like

where is Planck’s constant. Now we can represent these quantities as operators on a Hilbert space, the uncertainty principle kicks in, and life gets really interesting.

In thermodynamics, the analogous relations would be

The math works just the same, but what does it mean physically? Are we now thinking of temperature, entropy and the like as ‘quantum observables’—for example, operators on a Hilbert space? Are we just quantizing thermodynamics?

That’s one possible interpretation, but I’ve never heard anyone discuss it. Here’s one good reason: as Blake Stacey pointed out below, these equations don’t pass the test of dimensional analysis! The quantities at left have units of energy, while Plank’s constant has units of action. So maybe we need to introduce a quantity with units of time at right, or maybe there’s some other interpretation, where we don’t interpret the parameter as the good old-fashioned Planck’s constant, but something else instead.

And if you’ve really been paying attention, you may wonder how quantropy fits into this game! I showed that at least in a toy model, the path integral formulation of quantum mechanics arises, not exactly from maximizing or minimizing something, but from finding its critical points: that is, points where its first derivative vanishes. This something is a complex-valued quantity analogous to entropy, which I called ‘quantropy’.

Now, while I keep throwing around words like ‘minimize’ and ‘maximize’, most everything I’m doing works just fine for critical points. So, it seems that the apparatus of symplectic geometry may apply to the path-integral formulation of quantum mechanics.

But that would be weirdly interesting! In particular, what would happen when we go ahead and quantize the path-integral formulation of quantum mechanics?

If you’re a physicist, there’s a guess that will come tripping off your tongue at this point, without you even needing to think. Me too. But I don’t know if that guess is right.

Less mind-blowingly, there is also the question of how symplectic geometry enters into classical statics via the idea of Example 4.

But there’s a lot of fun to be had in this game already with thermodynamics.

Appendix

I should admit, just so you don’t think I failed to notice, that only rather esoteric physicists study the approach to quantum mechanics where time is an operator that doesn’t commute with the Hamiltonian In this approach commutes with the momentum and position operators. I didn’t write down those commutation equations, for fear you’d think I was a crackpot and stop reading! It is however a perfectly respectable approach, which can be reconciled with the usual one. And this issue is not only quantum-mechanical: it’s also important in classical mechanics.

Namely, there’s a way to start with the so-called extended phase space for a point particle on a manifold :

with coordinates and and get back to the usual phase space:

with just and as coordinates. The idea is to impose a constraint of the form

to knock off one degree of freedom, and use a standard trick called ‘symplectic reduction’ to knock off another.

Similarly, in quantum mechanics we can start with a big Hilbert space

on which and are all operators, then impose a constraint expressing in terms of and and then use that constraint to pick out states lying in a smaller Hilbert space. This smaller Hilbert space is naturally identified with the usual Hilbert space for a point particle:

Here is called the configuration space for our particle; its cotangent bundle is the usual phase space. We call the extended configuration space for a particle on the line; its cotangent bundle is the extended phase space.

I’m having some trouble remembering where I first learned about these ideas, but here are some good places to start:

• Toby Bartels, Abstract Hamiltonian mechanics.

• Nikola Buric and Slobodan Prvanovic, Space of events and the time observable.

• Piret Kuusk and Madis Koiv, Measurement of time in nonrelativistic quantum and classical mechanics, Proceedings of the Estonian Academy of Sciences, Physics and Mathematics 50 (2001), 195–213.