28 November, 2021


As the chlorophyll wanes, now is the heyday of the xanthophylls,
carotenoids and anthocyanins. These contain carbon rings and chains whose electrons become delocalized… their wavefunctions resonating at different frequencies, emitting photons of yellow, orange and red!

Yes, it’s fall. I’m enjoying it.

I wrote about two xanthophylls in my May 27, 2014 diary entry: I explained how they get their color from the resonance of delocalized electrons that spread all over a carbon chain with alternating single and double bonds:

I discussed chlorophyll, which also has such a chain, in my May 29th entry. I wrote about some carotenoids in my July 2, 2006 entry: these too have long chains of carbons with alternating single and double bonds.

I haven’t discussed anthocyanins yet! These have rings rather than chains of carbon, but the basic mechanism is similar: it’s the delocalization of electrons that makes them able to resonate at frequencies in the visual range. They are often blue or purple, but they contribute to the color of many red leaves:

Click on these two graphics for more details! I got them from a website called Science Notes, and it says:

Some leaves make flavonoids. Anthocyanins are flavonoids which vary in color depending on pH. Anthocyanins are not usually present in leaves during the growing season. Instead, plants produce them as temperatures drop. They acts as a natural sunscreen and protect against cold damage. Anthocyanins also deter some insects that like to overwinter on plants and discourage new seedlings from sprouting too close to the parent plant. Plants need energy from light to make anthocyanins. So, vivid red and purple fall colors only appear if there are several sunny autumn days in a row.

This raises a lot of questions, like: how do anthocyanins protect
leaves from cold, and why do some leaves make them only shortly before they die? Or are they there all along, hidden behind the chlorophyll Maybe this paper would help:

• D. Lee and K. Gould, Anthocyanins in leaves and other vegetative organs: an introduction, Advances in Botanical Research 37 (2002), 1–16.

Thinking about anthocyanins has led me to ponder the mystery of aromaticity. Roughly, a compound is aromatic if it contains one or more rings with pi electrons delocalized over the whole ring. But people fight over the exact definition.

I may write more about this if I ever solve some puzzles that are bothering me, like the mathematical origin of Hückel’s rule, which says a planar ring of carbon atoms is aromatic if it has 4n + 2 pi electrons. I want to know where the formula 4n + 2 comes from, and I’m getting close.

An early paper by Linus Pauling discusses the resonance of electrons in anthocyanins and other compounds with rings of carbon. This one is freely available, and it’s pretty easy to read:

• Linus Pauling, Recent work on the configuration and electronic structure of molecules; with some applications to natural products, in Fortschritte der Chemie Organischer Naturstoffe, 1939, Springer, Vienna, pp. 203–235.

Compositional Thermostatics

22 November, 2021

At the Topos Institute this summer, a group of folks started talking about thermodynamics and category theory. It probably started because Spencer Breiner and my former student Joe Moeller, both working at NIST, were talking about thermodynamics with some people there. But I’ve been interested in thermodynamics for quite a while now –and Owen Lynch, a grad student visiting from the University of Utrecht, wanted to do his master’s thesis on the subject. He’s now working with me. Sophie Libkind, David Spivak and David Jaz Myers also joined in: they’re especially interested in open systems and how they interact.

Prompted by these conversations, a subset of us eventually wrote a paper on the foundations of equilibrium thermodynamics:

• John Baez, Owen Lynch and Joe Moeller, Compositional thermostatics.

The idea here is to describe classical thermodynamics, classical statistical mechanics and quantum statistical mechanics in a unified framework based on entropy maximization. This framework can also handle ‘generalized probabilistic theories’ of the sort studied in quantum foundations—that is, theories like quantum mechanics, but more general.

To unify all these theories, we define a ‘thermostatic system’ to be any convex space X of ‘states’ together with a concave function

S \colon X \to [-\infty, \infty]

assigning to each state an ‘entropy’.

Whenever several such systems are combined and allowed to come to equilibrium, the new equilibrium state maximizes the total entropy subject to constraints. We explain how to express this idea using an operad. Intuitively speaking, the operad we construct has as operations all possible ways of combining thermostatic systems. For example, there is an operation that combines two gases in such a way that they can exchange energy and volume, but not particles—and another operation that lets them exchange only particles, and so on.

It is crucial to use a sufficiently general concept of ‘convex space’, which need not be a convex subset of a vector space. Luckily there has been a lot of work on this, so we can just grab a good definition off the shelf:

Definition. A convex space is a set X with an operation c_\lambda \colon X \times X \to X for each \lambda \in [0, 1] such that the following identities hold:

1) c_1(x, y) = x

2) c_\lambda(x, x) = x

3) c_\lambda(x, y) = c_{1-\lambda}(y, x)

4) c_\lambda(c_\mu(x, y) , z) = c_{\lambda'}(x, c_{\mu'}(y, z)) for all 0 \le \lambda, \mu, \lambda', \mu' \le 1 satisfying \lambda\mu = \lambda' and 1-\lambda = (1-\lambda')(1-\mu').

To understand these axioms, especially the last, you need to check that any vector space is a convex space with

c_\lambda(x, y) = \lambda x + (1-\lambda)y

So, these operations c_\lambda describe ‘convex linear combinations’.

Indeed, any subset of a vector space closed under convex linear combinations is a convex space! But there are other examples too.

In 1949, the famous mathematician Marshall Stone invented ‘barycentric algebras’. These are convex spaces satisfying one extra axiom: the cancellation axiom, which says that whenever \lambda \ne 0,

c_\lambda(x,y) = c_\lambda(x',y) \implies x = x'

He proved that any barycentric algebra is isomorphic to a convex subset of a vector space. Later Walter Neumann noted that a convex space, defined as above, is isomorphic to a convex subset of a vector space if and only if the cancellation axiom holds.

Dropping the cancellation axiom has convenient formal consequences, since the resulting more general convex spaces can then be defined as algebras of a finitary commutative monad, giving the category of convex spaces very good properties.

But dropping this axiom is no mere formal nicety. In our definition of ‘thermostatic system’, we need the set of possible values of entropy to be a convex space. One obvious candidate is the set [0,\infty). However, for a well-behaved formalism based on entropy maximization, we want the supremum of any set of entropies to be well-defined. This forces us to consider the larger set [0,\infty], which does not obey the cancellation axiom.

But even that is not good enough! In thermodynamics you often read about the ‘heat bath‘, an idealized system that can absorb or emit an arbitrarily large amount of energy while keeping a fixed temperature. We want to treat the ‘heat bath’ as a thermostatic system on an equal footing with any other. To do this, we need to allow consider negative entropies—not because the heat bath can have negative entropy, but because it acts as an infinite reservoir of entropy, and the change in entropy from its default state can be positive or negative.

This suggests letting entropies take values in the convex space \mathbb{R}. But then the requirement that any set of entropies have a supremum (including empty and unbounded sets) forces us to use the larger convex space [-\infty,\infty].

This does not obey the cancellation axiom, so there is no way to think of it as a convex subset of a vector space. In fact, it’s not even immediately obvious how to make it into a convex space at all! After all, what do you get when you take a nontrivial convex linear combination of \infty and -\infty? You’ll have to read our paper for the answer, and the justification.

We then define a thermostatic system to be a convex set X together with a concave function

S \colon X \to [-\infty, \infty]

where concave means that

S(c_\lambda(x,y)) \ge c_\lambda(S(x), S(y))

We give lots of examples from classical thermodynamics, classical and quantum statistical mechanics, and beyond—including our friend the ‘heat bath’.

For example, suppose X is the set of probability distributions on an n-element set, and suppose S \colon X \to [-\infty, \infty] is the Shannon entropy

\displaystyle{ S(p) = - \sum_{i = 1}^n p_i \log p_i }

Then given two probability distributions p and q, we have

S(\lambda p + (1-\lambda q)) \ge \lambda S(p) + (1-\lambda) S(q)

for all \lambda \in [0,1]. So this entropy function is convex, and S \colon X \to [-\infty, \infty] defines a thermostatic system. But in this example the entropy only takes nonnegative values, and is never infinite, so you need to look at other examples to see why we want to let entropy take values in [-\infty,\infty].

After looking at examples of thermostatic systems, we define an operad whose operations are convex-linear relations from a product of convex spaces to a single convex space. And then we prove that thermostatic systems give an algebra for this operad: that is, we can really stick together thermostatic systems in all these ways. The trick is computing the entropy function of the new composed system from the entropy functions of its parts: this is where entropy maximization comes in.

For a nice introduction to these ideas, check out Owen’s blog article:

• Owen Lynch, Compositional thermostatics, Topos Institute Blog, 9 September 2021.

And then comes the really interesting part: checking that this adequately captures many of the examples physicists have thought about!

The picture at the top of this post shows one that we discuss: two cylinders of ideal gas with a movable divider between them that’s permeable to heat. Yes, this is an operation in an operad—and if you tell us the entropy function of each cylinder of gas, our formalism will automatically compute the entropy function of the resulting combination of these two cylinders.

There are many other examples. Did you ever hear of the ‘canonical ensemble’, the ‘microcanonical ensemble’, or the ‘grand canonical ensemble’? Those are famous things in statistical mechanics. We show how our formalism recovers these.

I’m sure there’s much more to be done. But I feel happy to see modern math being put to good use: making the foundations of thermodynamics more precise. Once Vladimir Arnol’d wrote:

Every mathematician knows that it is impossible to understand any elementary course in thermodynamics.

I’m not sure our work will help with that—and indeed, it’s possible that once the mathematicians finally understand thermodynamics, physicists won’t understand what the mathematicians are talking about! But at least we’re clearly seeing some more of the mathematical structures that are hinted at, but not fully spelled out, in such an elementary course.

I expect that our work will interact nicely with Simon Willerton’s work on the Legendre transform. The Legendre transform of a concave (or convex) function is widely used in thermostatics, and Simon describes this for functions valued in [-\infty,\infty] using enriched profunctors:

• Simon Willerton, Enrichment and the Legendre–Fenchel transform I, The n-Category Café, April 16, 2014.

• Simon Willerton, Enrichment and the Legendre–Fenchel transform II, The n-Category Café, May 22, 2014.

He also has a paper on this, and you can see him talk about it on YouTube.

The Kuramoto–Sivashinsky Equation (Part 7)

3 November, 2021

I have a lot of catching up to do. I want to share a bunch of work by Steve Huntsman. I’ll start with some older material. A bit of this may be ‘outdated’ by his later work, but I figure it’s all worth recording.

One goal here is to define ‘stripes’ for the Kuramoto–Sivashinky equation in a way that lets us count them, their births, and their mergers, and so on. We need a good definition to test the conjectures I made in Part 1.

While I originally formulated my conjectures for the ‘integral form’ of the
Kuramoto–Sivashinky equation:

h_t + h_{xx} + h_{xxxx} + \frac{1}{2} (h_x)^2 = 0

Steve has mostly been working with the derivative form:

u_t + u_{xx} + u_{xxxx} + u u_x = 0

so you can assume that unless I say otherwise. He’s using periodic boundary conditions such that

u(t,x) = u(t,x+L)

for some length L. The length depends on the particular experiment he’s doing.

First, a plot of stripes. It looks like L = 100 here:

Births and deaths are shown as green and red dots, respectively. But to see them, you may need to click on the picture to enlarge it!

According to my conjecture there should be no red dots. The red dots at the top and the bottom of the image don’t count: they mostly arise because this program doesn’t take the periodic boundary conditions into account. There are two other red dots, which are worth thinking about.

Nice! But how are stripes being defined here? He describes how:

The stripe definition is mostly pretty simple and not image processy at all, and the trick to improve it is limited to removing little blobs and is easily explained.

Let u(t,x) be the solution to the KSE. Then let

v(t,x) := u(t,x)- u(t,x+a)

where a is the average integer offset (maybe I’m missing a minus sign a la -a) that maximizes the cross-correlation between u(t,x) and -u(t,x+a). Now anywhere v exceeds its median is part of a stripe.

The image processing trick is that I delete little stripes (and I use what image processors would call 4-connectivity to define simply connected regions—this is the conservative idea that a pixel should have a neighbor to the north, south, east, or west to be connected to that neighbor, instead of the aggressive 8-connectivity that allows NE, NW, SE, SW too) whose area is less than 1000 grid points. So it uses lots of image processing machinery to actually do its job, but the definition is simple and easily explained mathematically.

An obvious fix that removes the two nontrivial deaths in the picture I sent is to require a death to be sufficiently far away from another stripe: here I am guessing that the characteristic radius of a stripe will work just fine.

Learn Applied Category Theory!

27 October, 2021

Do you like the idea of learning applied category theory by working on a project, as part of a team led by an expert? If you’re an early career researcher you can apply to do that now!

Mathematical Research Community: Applied Category Theory, meeting 2022 May 29–June 4. Details on how to apply: here. Deadline to apply: Tuesday 2022 February 15 at 11:59 Eastern Time.

After working with your team online, you’ll take an all-expenses-paid trip to a conference center in upstate New York for a week in the summer. There will be a pool, bocci, lakes with canoes, woods to hike around in, campfires at night… and also whiteboards, meeting rooms, and coffee available 24 hours a day to power your research!

Later you’ll get invited to the 2023 Joint Mathematics Meetings in Boston.

There will be three projects to choose from:

Valeria de Paiva (Topos Institute) will lead a study in the context of computer science that investigates indexed containers and partial compilers using lenses and Dialectica categories.

Nina Otter (Queen Mary University of London) will lead a study of social networks using simplicial complexes.

John Baez (University of California, Riverside) will lead a study of chemical reaction networks using category theoretic methods such as structured cospans.

The whole thing is being organized by Daniel Cicala of the University of New Haven:

and Simon Cho of Two Six Technologies:

I should add that this is just one of four ‘Mathematical Research Communities’ run by the American Mathematical Society in 2022, and you may prefer another. The applied category theory session will be held at the same time and place as one on data science! Then there are two more:

• Week 1a: Applied Category Theory

Organizers: John Baez, University of California, Riverside; Simon Cho, Two Six Technologies; Daniel Cicala, University of New Haven; Nina Otter, Queen Mary University of London; Valeria de Paiva, Topos Institute.

• Week 1b: Data Science at the Crossroads of Analysis, Geometry, and Topology

Organizers: Marina Meila, University of Washington; Facundo Mémoli, The Ohio State University; Jose Perea, Northeastern University; Nicolas Garcia Trillos, University of Wisconsin-Madison; Soledad Villar, Johns Hopkins University.

• Week 2a: Models and Methods for Sparse (Hyper)Network Science

Organizers: Sinan G. Aksoy, Pacific Northwest National Laboratory; Aric Hagberg, Los Alamos National Laboratory; Cliff Joslyn, Pacific Northwest National Laboratory; Bill Kay, Oak Ridge National Laboratory; Emilie Purvine, Pacific Northwest National Laboratory; Stephen J. Young, Pacific Northwest National Laboratory; Jennifer Webster, Pacific Northwest National Laboratory.

• Week 2b: Trees in Many Contexts

Organizers: Miklós Bóna, University of Florida; Éva Czabarka, University of South Carolina; Heather Smith Blake, Davidson College; Stephan Wagner, Uppsala University; Hua Wang, Georgia Southern University.

Applicants should be ready to engage in collaborative research and should be “early career”—either expecting to earn a PhD within two years or having completed a PhD within five years of the date of the summer conference. Exceptions to this limit on the career stage of an applicant may be made on a case-by-case basis. The Mathematical Research Community (MRC) program is open to individuals who are US citizens as well as to those who are affiliated with US institutions and companies/organizations. A few international participants may be accepted. Depending on space and other factors, a small number of self-funded participants may be admitted. Individuals who have once previously been an MRC participant will be considered for admission, and their applications must include a rationale for repeating. Please note that individuals cannot participate in the MRC program more than twice: applications from individuals who have twice been MRC participants will not be considered.

We seek individuals who will both contribute to and benefit from the MRC experience, and the goal is to create a collaborative research community that is vibrant, productive, and diverse. We welcome applicants from academic institutions of all types, as well as from private industry and government laboratories and agencies. Women and under-represented minorities are especially encouraged to apply.

All participants are expected to be active in the full array of MRC activities—the summer conference, special sessions at the Joint Mathematics Meetings, and follow-up collaborations.

The Kuramoto–Sivashinsky Equation (Part 6)

25 October, 2021

guest post by Theodore Kolokolnikov

I coded up a simple dynamical system with the following rules (loosely motivated by theory of motion of spikes in reaction-diffusion systems, see e.g. appendix of this paper, as well as this paper):

• insert a particle if inter-particle distance is more than some maxdist
• merge any two particles that collide
• otherwise evolve particles according to the ODE

\displaystyle{ x'_k(t) = \sum_{j=1}^N G_x(x_k, x_j) }

Here, G is a Green’s function that satisfies

G_{xx}-\lambda^2 G = -\delta(x,y)

inside the interval [-L,L] with Neumann boundary conditions G_x(\pm L, y)=0. Explicitly,

\displaystyle{ G(x,y) = \frac{\cosh((x+y)\lambda)+\cosh((2L-|x-y|) \lambda )}{2 \lambda \sinh(2L \lambda ) }  }


\displaystyle{ G_x(x,y)= \frac{\sinh((x+y) \lambda )+ \sinh((|x-y|-2L) \lambda ) \mbox{sign}(x-y) } {2 \sinh(2L\lambda)} }

where sign(0) is taken to be zero so that

\displaystyle{ G_x(y,y) := \frac{G_x(y^+,y)+ G_x(y^-,y)}{2} }

In particular, for large \lambda, one has

\displaystyle{ G(x,y)\sim\frac{e^{-\lambda | x-y|}}{2\lambda} }


\displaystyle{ G_x(x,y)\sim-\frac{e^{-\lambda | x-y|}} {2} \mbox{sign}(x-y), ~~\lambda \gg 1 }

Here are some of the resulting simulations, with different \lambda (including complex \lambda). This is mainly just for fun but there is a wide range of behaviours. In particular, I think the large-lambda limit should be able to capture analogous dynamics in the Keller-Siegel model with logistic growth (Hillen et. al.), see e.g. figures in this paper.

The Kuramoto–Sivashinsky Equation (Part 5)

24 October, 2021

In Parts 3 and 4, I showed some work of Cheyne Weis on the ‘derivative form’ of the Kuramoto–Sivashinksy equation, namely

u_t + u_{xx} + u_{xxxx} + u u_x = 0

Steve Huntsman’s picture of a solution above gives you a good feel for how this works.

Now let’s turn to the ‘integral form’, namely

h_t + h_{xx} + h_{xxxx} + \frac{1}{2} (h_x)^2 = 0

This has rather different behavior, though it’s closely related, since if h is any solution of the integral form then

u = h_x

is a solution of the derivative form.

Cheyne drew a solution of the integral form:

You’ll immediately see the most prominent feature: it slopes down! I’ll show later that the average of h over space can never increase with time, and it decreases unless h is constant as a function of space. By contrast, we saw in Part 2 that the average of u over space never changes with time.

However, we can subtract off the average of h over space to eliminate this dramatic but rather boring effect. The result looks like this:

Now it’s very easy to see the ‘stripes’ I’m so obsessed with: they are the ridges in these pictures. You can see how as time increases from left to right these stripes are born and merge, but never die or split.

But how can we mathematically define these stripes, to make it possible to state precise conjectures about them? We could try defining them to be points where u is locally maximized of as a function of x at any time t. With this definition, Cheyne gets stripes like this:

The previous picture shows up in the lower right hand corner of this one.

These stripes look pretty good, but you’ll see some gaps where they momentarily disappear and then reappear. I don’t think these invalidate my conjecture that stripes never ‘die’. I just think this definition of stripe is not quite right. (Of course I would think that, wouldn’t I? I want the conjecture to be true!)

Cheyne thought that maybe overlaying maxima in time would help:

This fills in some gaps, but there are still stripes that momentarily die, only to be shortly reborn. It might be good to define stripes to be points where this function— u minus its average over space—exceeds a certain cutoff.

Let’s conclude by proving that the average of h over space can never increase with time. To prove this, just take the time derivative of the integral of h over space, and show it’s \le 0. Remember that we’re assuming h(t,x) is periodic in x with period L, so ‘space’ is the interval [0,L] with its endpoints identified to form a circle. So, we get

\begin{array}{ccl}  \displaystyle{ \frac{d}{d t} \int_0^L h(t,x) \, dx } &=& \displaystyle{ \int_0^L h_t(t,x) \, dx } \\ \\  &=& \displaystyle{ -\int_0^L \left( h_{xx} + h_{xxxx} + \frac{1}{2} (h_x)^2 \right) \, dx } \\ \\  &=& \displaystyle{  -\left( h_x + h_{xxx} \right) \Big|_0^L -\int_0^L (h_x)^2 \, dx } \\ \\  &=& \displaystyle{ -\int_0^L (h_x)^2 \, dx }  \end{array}

This is \le 0, as desired. Moreover, it’s zero iff h is constant as a function on space!

The Kuramoto–Sivashinsky Equation (Part 4)

23 October, 2021

Here is some more work by Cheyne Weis. Last time I explained that Cheyne and Steve Huntsman were solving the ‘derivative form’ of the Kuramoto–Sivashinsky equation, namely this:

u_t + u_{xx} + u_{xxxx} + u u_x = 0

Above is one of Steve’s pictures of a typical solution with its characteristic ‘stripes’. Cheyne started trying to identify these stripes as locations where du/dx \le c for some cutoff c, for example c = -0.7. He made some nice 3d views of du/dx illustrating the problems with this. As he explains:

So there is the tradeoff that if I make the cutoff too high, the sections that look greenish are getting identified as stripes. If I make it too low, the points near where two stripes merge disappear. I attached some 3D plots showing the landscape of du/dx reiterating this point.

The disappearance of the stripes as they merge is unavoidable to a certain extent. The plots in the PowerPoint show how even when there is no cutoff, there is some gap between the merging stripes. There may be some characteristic length scale needed to qualify them as merging.

You can click on these pictures to make them bigger.

The Kuramoto–Sivashinsky Equation (Part 3)

23 October, 2021

I’ve been getting a lot of help from Steve Huntsman and also Cheyne Weis, who is a physics grad student at the University of Chicago. You can see a lot, but far from all, of Steve’s work as comments on part 1. Here are some things Cheyne has been doing.

Cheyne started out working with the ‘derivative form’ of the Kuramoto–Sivashinsky equation, meaning this:

u_t + u_{xx} + u_{xxxx} + u u_x = 0

and he soon noticed what Steve made clear in the image above: the ‘stripes’ in solutions of this equation aren’t ‘bumps’ (regions where u is large) but regions where the solution is rapidly changing from positive to negative. This suggests a way to define stripes: look for where du/dx < c for some negative c. It seems c = -0.7 is a pretty good choice.

I thought maybe it would be better to use the derivative of the PDE’s solution (du/dx) to define the stripes. You can find an image of this in the attached PowerPoint.

The second slide has another image where the lines represent the minima of du/dx (as a function of x) that are below a certain threshold c. You can see these lines appearing and combining as apparent in Thien An’s animation. Hopefully this is some progress on the definition of a “bump”. If you agree, I could use this to test some of your other conjectures.

Here are the result for a range of alternative choices of c. The problem, if we’re seeking a definition of ‘stripe’ where stripes never die as time passes, is the presence of short ‘ministripes’ that die shortly after they appear. What’s really going on, I believe, is that when small stripes merge with larger ones, the derivative du/dx becomes smaller in absolutely value, thus going above the cutoff c. In short, merging is being misinterpreted as death.

The Kuramoto–Sivashinsky Equation (Part 2)

22 October, 2021

I love the Kuramoto–Sivashinsky equation, beautifully depicted here by Thien An, because it’s one of the simplest partial differential equations that displays both chaos and a visible ‘arrow of time’. Last time I made some conjectures about it. Most notably, I conjecture that the ‘stripes’ you see above can appear or merge as time passes, but never disappear or split.

But I was quite confused for a while, because the Kuramoto–Sivashinsky equation comes in two forms. The integral form, which I spent most of my time discussing, is this:

h_t + h_{xx} + h_{xxxx} + \frac{1}{2} (h_x)^2 = 0

where h(t,x) is a real function of two real variables, ‘time’ t and ‘space’ x. The derivative form is this:

u_t + u_{xx} + u_{xxxx} + u u_x = 0

where u(t,x) is again a real function of two real variables. If h is a solution of the integral form,

u = h_x

is a solution of the derivative form. This is easy to see: just take the x derivative of the equation with h in it and see what you get.

By the way, beware of this: Thien An’s wonderful movie above shows the integral form of the equation, but she calls the function u.

Note that if h has a local maximum as a function of x, then u will go positive and then negative. This is important because it affects the definition of ‘stripe’. Here’s what stripes look in the derivative form of the Kuramoto–Sivashinsky equation, as computed by Steve Huntsman:

As you can see, the function u goes negative and then positive as we increase x moving through a stripe. This means that h would have a local minimum.

Another thing puzzling me was that Steve Huntsman found solutions of the derivative form where the stripes in u mostly move in the same direction as time passes:

This suggests that there’s some way to take a solution where the stripes aren’t moving much, and switch to a moving frame of reference to get a solution with moving stripes. And indeed I’ll show that’s true! But I’ll also show it’s not true for the integral form of the Kuramoto–Sivashinsky equation—at least not with periodic boundary conditions, which is what we’re using here.

Galilean transformations

Galileo was the guy who first emphasized that the laws of physics look the same in a moving frame of reference, but later Maxwell explained the idea very poetically:

Our whole progress up to this point may be described as a gradual development of the doctrine of relativity of all physical phenomena. Position we must evidently acknowledge to be relative, for we cannot describe the position of a body in any terms which do not express relation… There are no landmarks in space; one portion of space is exactly like every other portion, so that we cannot tell where we are. We are, as it were, on an unruffled sea, without stars, compass, sounding, wind or tide, and we cannot tell in which direction we are going. We have no log which we can case out to take a dead reckoning by; we may compute our rate of motion with respect to the neighboring bodies, but we do not know how these bodies may be moving in space.

Before Einstein came along, a transformation into a moving frame of reference worked like this:

t \mapsto t
x \mapsto x + vt

where v is a constant, the velocity.

These are called ‘Galilean transformations’ And here’s something cool: the derivative form of the Kuramoto–Sivashinsky equation has Galilean transformations as symmetries! If u(t,x) is a solution, so is the boosted function

u'(t,x) = u(t,x+vt) - v

The prime here does not mean derivative: u' is the name of our new boosted solution. To get this boosted solution, we do the obvious coordinate transformation into a moving frame of reference, but then a bit more: we must subtract the constant v.

Let’s see how this works! Suppose u is a solution of the derivative form of the Kuramoto–Sivashinsky equation:

u_t + u_{xx} + u_{xxxx} + u u_x = 0

Then defining

u'(t,x) = u(t, x+vt) - v

we see the t derivative of u' has an extra vu_x term:

u'_t(t,x) = u_t(t,x+vt) + vu_x(t,x+vt)

while its x derivatives are simple:

u'_x(t,x) = u_x(t,x+vt)

and so on for the higher x derivatives.

This lets us check that the boosted function u' is also a solution:

\begin{array}{ccl}  u'_t + u'_{xx} + u'_{xxxx} + u' u'_x &=& u_t(t,x+vt) + vu_x(t,x+vt)  \\ \\  &&  + u_{xx}(t,x+vt) + u_{xxxx}(t,x+tv)  \\ \\  && + (u(t,x+vt) - v) u_x(t,x+vt) \\ \\  &=& 0  \end{array}

Note how subtracting the constant v when we transform u exactly cancels the extra term vu_x term we get when we transform the time derivative of u.

I got this idea from here:

• Uriel Frisch, Zhensu She and Olivier Thual, Viscoelastic behaviour of cellular solutions to the Kuramoto–Sivashinsky model, Journal of Fluid Mechanics 168 (1986), 221–240.

thanks to someone on Twitter.

Does the integral form of the Kuramoto-Sivashinsky equation also have Galilean transformations as symmetries? Yes, but there’s a certain problem with them. Let’s see how it goes.

If h(t,x) is a solution, then so is this boosted function:

h'(t,x) = h(t,x+vt) - vx - \frac{1}{2}t v^2

But now the ‘fudge factors’ in the boosted function are more sneaky! We don’t just subtract a constant, as we did with the derivative form. We subtract a function that depends on both the space and time coordinates.

Let’s check. The first time derivative of h' works like this:

h'_t(t,x) = h_t(t,x+vt) + vh_t(t,x+vt) - \frac{1}{2} v^2

The first space derivative works like this:

h'_x(t,x) = h_x(t,x+vt) - v

The second space derivative is simpler:

h'_{xx}(t,x) = h_{xx}(t,x+vt)

and the higher space derivatives work the same way. Now suppose h was a solution of the integral form of the Kuramoto-Sivashinsky equation:

h_t + h_{xx} + h_{xxxx} + \frac{1}{2} (h_x)^2 = 0

We can use our formulas to check that h' is a solution:

\begin{array}{ccl}  h'_t \! +\! h'_{xx}\! +\! h'_{xxxx} \!+ \! \frac{1}{2} (h'_x)^2 \!  &=& h_t(t,x+vt) + vh_t(t,x+vt) \! - \!\frac{1}{2} v^2 \\ \\  && + h_{xx}(t,x+vt)   \\ \\  && + h_{xxxx}(t,x+vt) \\ \\  && + \frac{1}{2}(h_x(t,x+vt) - v)^2 \\ \\  &=& h_x(t,x+vt) + h_{xx}(t,x+vt)  \\ \\  && + h_{xxxx}(t,x+vt) + \frac{1}{2} h_x(t,x+vt)^2  \\ \\  &=& 0  \end{array}

The cancellations in the second step rely crucially on those sneaky fudge factors in the definition of the boosted solution h'. I chose those fudge factors precisely to make these cancellations happen.

Periodic boundary conditions

A lot of really interesting results about the Kuramoto-Sivashinsky equation involve looking at solutions that are periodic in space. For the derivative form this means

u(t,x+L) = u(t,x)

It’s easy to see that if u obeys this equation, so does its boosted version:

u'(t,x) = u(t,x+vt) - v

After all, a quick calculation shows

\begin{array}{ccl}  u'(t,x+L) &=& u(t,x+L+vt) - v \\ \\  &=& u(t,x+vt) - v \\ \\  &=& u'(t,x)   \end{array}

So for any spatially periodic solution with some stripes that are basically standing still, there’s a spatially periodic boosted version where they’re moving at velocity v. You can think of a spatially periodic solution as a function on the cylinder. In the boosted version, the stripes spiral around this cylinder like the stripes on a barber pole!

But for the integral form this doesn’t work! Suppose h is a solution of the integral form that is periodic in space:

h(t,x+L) = h(t,x)

Now its boosted version is defined by

h'(t,x) = h(t,x+vt) - vx - \frac{1}{2}t v^2

and this is not periodic in space:

\begin{array}{ccl}  h'(t,x+L) &=& h(t,x+L+vt) - v(x+L) - \frac{1}{2}t v^2  \\ \\  &=& h(t,x+vt) - v(x+L) - \frac{1}{2}t v^2 \\ \\  &\ne& h(t,x+vt) = h'(t,x)  \end{array}

A conserved quantity

The derivative form of the Kuramoto–Shivashinsky equation has an unexpected symmetry under Galilean transformations, or boosts, even for periodic solutions. It also has an interesting conserved quantity, namely

\displaystyle{ \int_0^L u(t,x) \, dx }

To see this, note that

\begin{array}{ccl}  \displaystyle{ \frac{d}{dt} \int_0^L u(t,x) \, dx } &=& \displaystyle{ \int_0^L u_t(t,x) \, dx }  \\ \\  &=& \displaystyle{ - \int_0^L \left( u_{xx} + u_{xxxx} + u u_x \right) \, dx } \\ \\  &=& \displaystyle{ \left. - \left( u_x + u_{xxx} + \frac{1}{2} u^2 \right)   \right|_0^L  } \\ \\  &=& 0  \end{array}

Note that if we boost a solution, replacing u(t,x) by u(t,x+vt) - v, we also subtract a constant from this conserved quantity, namely v L. So, it seems that the conserved quantity

\displaystyle{ \int_0^L u(t,x) \, dx }

is a way of measuring ‘how much the stripes are moving, on average’.

I’ll venture a guess that when

\displaystyle{ \int_0^L u(t,x) \, dx = 0 }

the stripes are stationary, on average. One reason I’m willing to guess this is that this equation has another meaning, too.

I mentioned that we can get solutions u of the derivative form from solutions h of the integral form by differentiating them, like this:

u = h_x

Does every solution of the derivative form arise this way from a solution of the integral form? Given u we can define h by

\displaystyle{ h(t,x) = \int_0^x u(t,y) \, dy }

and we can check it obeys the integral form of the Kuramoto–Sivashinsky equation. I’ve been showing you lots of calculations and we’re probably both getting tired of them, so I won’t actually go through this check. But here’s the interesting part. Suppose we restrict attention to spatially periodic solutions! If u is spatially periodic, h will be so iff

\displaystyle{ \int_0^L u(t,x) \, dx = 0}

since we have

\begin{array}{ccl}  h(t,x+L) &=& \displaystyle{ \int_0^{x+L} u(t,y) \, dy } \\ \\  &=&  \displaystyle{ \int_0^x u(t,y) \, dy + \int_x^{x+L} u(t,y) \, dy } \\ \\  &=& h(t,x)  \end{array}

where in the last step, the first integral is h(t,x) by definition and the second is zero because

\displaystyle{ \int_0^L u(t,x) \, dx = 0 }

and u is periodic with period L.


Limiting ourselves to spatially periodic solutions, we see:

• The derivative form of the Kuramoto–Sivashinsky equation has symmetry under boosts.

• The integral form does not.

• Solutions of the integral form correspond precisely to solutions of the derivative form with

\displaystyle{ \int_0^L u(t,x) \, dx = 0 }

• Boosting a solution of the derivative form with

\displaystyle{ \int_0^L u(t,x) \, dx = 0 }

gives a solution where this quantity is not zero.

Given all this, I think the good theorems about spatially periodic solutions of the Kuramoto–Sivashinsky equation will focus on solutions of the integral form, or equivalently solutions of the differential form with

\displaystyle{ \int_0^L u(t,x) \, dx = 0 }

Epilogue: symmetries of the heat equation

At first I was shocked that the Kuramoto–Sivashinsky equation had Galilean transformations as symmetries, because it’s a relative of the heat equation, and the heat equation obviously doesn’t have Galilean transformations as symmetries: you can’t get a solution of the heat equation where a wave of heat moves along at constant velocity as it spreads out.

But then I remembered that the heat equation

u_t = u_{xx}

is very similar to Schrödinger’s equation:

u_t = i u_{xx}

And Schrödinger’s equation does have Galilean transformations as symmetries, since it describes a free particle! Physicists know how they work. You don’t just replace u(t,x) with u(t,x+vt), you also multiply it by a suitable complex phase that depends on t and x. That is, you multiply u(t,x+vt) by the exponential of some imaginary-valued function of t and x.

So then I realized that the heat equation does have Galilean symmetries! They work just as for Schrödinger’s equation, but now you have to multiply u(t,x+vt) by the exponential of some real-valued function of t and x.

This grows exponentially as you move in one direction in space, so people thinking about heat often don’t think about such solutions. Yes, you can get a ‘wave of heat that moves along at constant velocity as it spreads out’, but it’s like a huge tsunami wave rolling in from infinity!

I was very proud of myself for having discovered this weird Galilean invariance of the heat equation, and I posed it as a puzzle on Twitter. But then Tom Price pointed out a paper about it:

• U. Niederer, Schrödinger invariant generalized heat equations, Helvetica Physica Acta 51 (1978), 220–239.

It turns out the Galilean invariance of the heat equation, and much more, was discovered by Sophus Lie in 1882!

The so-called ‘Schrödinger group’ also includes certain fractional linear transformations of the plane. So this raises the question of whether the Kuramoto–Sivashinsky equation has even more symmetries than those in the Galilei group (translations, spatial reflections and Galilean transformations).

But it seems this too has been studied:

• Mehdi Nadjafikhah and Fatemeh Ahangari, Lie symmetry analysis of the two-dimensional generalized Kuramoto–Sivashinsky equation, Mathematical Sciences 6 (2012).

This discusses the analogue of the Kuramoto–Sivashinsky equation in two dimensions of space and one of time. It uses some techniques to compute its symmetry group, but I don’t see any sign of invariance under a big group like the ‘Schrödinger group’.

Topos Institute Postdoc

19 October, 2021

The Topos Institute is trying to hire a postdoc to work on polynomial functors! Here is the ad, written by David Spivak.

Dear all,

I’m happy to announce that Topos Institute is hiring a postdoctoral research fellow to study polynomial functors and their applications. A mathjobs listing is here:

and information on the precise project can be found here:

Applicants should be strong in mathematics or computer science and know category theory. Strong programming experience or strong math research experience is required, and a combination of the two is preferred. The position is for 1 year, but a renewal may be possible, depending on funding.

Topos is committed to building a team with diverse perspectives and life experiences, so those with underrepresented personal or professional backgrounds are highly encouraged to apply. Unfortunately, Topos is unable to sponsor visas for international candidates at this time.

We are looking to hire as soon as possible. Please feel free to contact me if you have any questions.

Best regards,