So cute! This small lizard, called the 'thorny devil' or Moloch horridus, lives in the deserts and scrub lands of Australia.
It may look fierce, but it's not dangerous. It only eats ants. It's spiny so it doesn't get eaten. It can change color, for camouflage! And it has a "false head" on the back of its neck, which it shows to potential predators by dipping its real head. I'm not sure why.
It's also called a 'thorny dragon':
I thank Rasha Kamel for introducing me to this beast. She pointed out this article:
Scientists have recently figured out more about how this lizard gets water:
Researchers discovered long ago that because its mouth has evolved to eat ants, it cannot sip or even lick water from a source — instead it has to rely on other means. Prior research had found that the lizard has tiny folds on its skin that overlap, creating tube-like structures capable of carrying water — the tubes all lead to the back of the mouth. It was noted that setting the lizard in a small bucket of water caused the tubes to fill and the lizard to start swallowing. But what has remained a mystery is how such a technique could work in the desert, where there are rarely puddles to stand in. To solve the mystery, the researchers captured some specimens and took them back to their lab for study.In the lab, the researchers first tried allowing the lizards to stand on sand that had been wetted — this resulted in some water being drawn into the tubes, but not enough to get the lizard to start swallowing, which meant it wasn't enough. The answer, it turned out, was the lizard's habit of pushing sand onto its back — this caused any water from recent rains or even from dew to move slowly downward, due to gravity. Eventually, it would reach the skin, where it would be sucked into the tubes like a child with a straw. At some point, the tubes would fill and the lizard would swallow it.
The researchers note that such a drinking technique is likely merely supplementary — most of the water they get comes from the ants they eat; thus, using skin for drinking would likely only occur during extreme draught conditions.
I got this picture from a website with a lot of great photos of thorny devils:
These are Pristerognathus, very ancient mammal-like reptiles. They lived in the middle Permian, around 260 million years ago. That's long before the dinosaurs!
These animals were roughly dog-sized, and had long, narrow skulls and large canine teeth. They probably lived in woodlands, and preyed on smaller animals.
There were many kinds of mammal-liked reptiles back then. In general they are called therapsids. Some of them evolved to become mammals — like you and me! Fur has been found in the fossilized poop of some of these animals. So, at least some of them had hair.
These particular guys are called Pristerognathus vanderbyli. This picture is from Wikicommons.
The first dinosaurs showed up around 240 million years ago — and they
only became common after the great Triassic-Jurassic extinction, 200
million years ago. Therapsids started around 275 million years ago.
Some of them evolved into mammals 225 million years ago, and all the
non-mammalian ones went extinct by the early Cretaceous, 100 million
years ago. Most dinosaurs went extinct at the end of the Cretaceous,
65 million years ago. Some, however, are still sold as food at
grocery stores.
November 15, 2016
Nice animation by Phillipe Roux! Take some balls moving in the same direction and let them bounce around in this shape: a rectangle with ends rounded into semicircles. They will soon start moving in dramatically different ways. To keep things simple we don't let the balls collide — they pass right through each other. In a while they will be close to evenly spread over the whole billiard table.
This is an example of chaos: slightly different initial conditions lead to dramatically different trajectories.
It's also an example of ergodicity: for almost every choice of initial conditions, the trajectory of a ball will have an equal chance of visiting each tiny little region. That is, if we take a random choice of initial conditions, there is a 100% probability that the trajectory will have this property.
The Bunimovich stadium is a rectangle capped by semicircles in which a point particle moves at constant speed along straight lines, reflecting off the boundary in a way that the angle of incidence equals the angle of reflection. The animation shows a collection of such particles initially moving in the same direction. With each bounce their trajectories diverge, and after a while they are distributed almost evenly through the whole stadium, though for a while one can still see a density wave moving back and forth.
The Bunimovich stadium appears in the 1979 work of Leonid Bunimovich:
• Leonid A. Bunimovich, On the ergodic properties of nowhere dispersing billiards, Commun. Math. Phys. 65 (1979), 295–312.
He showed that the motion of a billiard in this stadium is ergodic. This is a way of making precise the intuition that given a billiard with randomly chosen initial position and velocity, over time its position almost surely becomes uniformly distributed over the whole stadium.
More precisely, we can define the phase space \(\Omega\) for the Bunimovich stadium to be the space of position-velocity pairs where the velocity is a unit vector. (Since the speed of the billiard does not change, we may assume it is normalized to 1.) There is a probability measure on \(\Omega\) for which time evolution defines measure-preserving dynamical system:
$$ T_t \colon \Omega \to \Omega , \qquad \qquad t \in \mathbb{R}. $$
Given a measure-preserving dynamical system, we say a measurable subset \(A \subseteq \Omega\) is 'invariant' if for all \(t \in \mathbb{R}\) the sets \(T_t(A)\) and \(A\) differ only by a null set, meaning that the symmetric difference \(T_t(A) \triangle A\) has measure zero. A measure-preserving dynamical system is ergodic if the only invariant measurable subsets \(A \subseteq \Omega\) are null sets and the complements of null sets.
The meaning of this is clarified by 'ergodic theorem'. Suppose \(T_t : \Omega \to \Omega\) is a measure-preserving dynamical system on a probability measure space \(\Omega,\mu\), and suppose \(f \colon \Omega \to \mathbb{R}\) is an integrable function. Then we can define two averages of \(f\), the 'time average' and 'phase space average'.
Time average: This is the following average (if it exists):
$$ \widehat{f}(x) = \lim_{t\rightarrow\infty}\; \frac{1}{t} \int_0^t f(T_s x) \, ds .$$
Phase space average: This is the integral of \(f\) over the phase space:
$$ \bar{f} = \int_\Omega f \, d \mu(x). $$
In general the time average and phase space average may be difference, and the time average may not exist. But if \(T_t\) is ergodic, Birkhoff's ergodic theorem says that
$$ \widehat{f}(x) = \bar{f} $$
for almost every \(x \in \Omega\).
Proving that a measure-preserving dynamical system is ergodic can be difficult. Bunimovich's thesis advisor, Yakov G. Sinai, showed that a billiard moving on a square table with a reflecting disk inside is ergodic.
Sinai Billiard - George Stamatiou
The curvature of the disk tends to amplify the angle between slightly different trajectories. The Bunimovich stadium is subtler because it lacks this feature: since its rounded ends are convex, they tend to focus billiards that bounce off them. The rectangular portion of the table counteracts this focusing effect, and over long enough times there tend to be an exponentially growing distance between initially nearby trajectories.
Bunimovich Stadium Trajectories - Jakob Scholbach
As Buminovich writes:
Moreover, a closer analysis of these billiards revealed a new mechanism of chaotic behavior of conservative dynamical systems, which is called a mechanism of defocusing. The key observation is that a narrow parallel beam of rays, after focusing because of reflection from a focusing boundary, may pass a focusing (in linear approximation) point and become divergent provided that a free path between two consecutive reflections from the boundary is long enough. The mechanism of defocusing works under condition that divergence prevails over convergence.
This is from:
However, this analysis is not sufficient to understand the ergodicity of the Bunimovich stadium, because in 1973 Lazutkin showed that a convex billiard table with infinitely differentiable boundary cannot be ergodic. In fact he showed this for a convex table whose boundary has 553 continuous derivatives! In 1982 Douady showed 6 continuous derivatives is enough — and he conjectured that 4 is enough. For references, see:
This explained an interesting question which was addressed by later work:
Also try Carlos Scheidegger's great webpage that lets you play around with billiards on the Bunimovich stadium as well as elliptical table, where their motion is completely integrable:
Check out more of Phillipe Roux's animations here:
George Stamatiou put his picture of the Sinai billiard on Wikicommons under a Creative Commons Attribution 2.5 Generic license. Jakob Scholbach put his picture of billiard trajectories in the Bunimovich stadium on Wikicommons under a Attribution-ShareAlike 3.0 Unported license.
November 16, 2016
Check out Carlos Scheidegger's great webpage that lets you play around with billiards on two tables:
The table here is elliptical, and you'll see that the billiards trace out nice patterns — not at all random. Often there's a region of the table that they never enter! Not in this particular example, but try others and you'll see what I mean.
Puzzle 1. What shape is this 'forbidden region', and why?
It will be easier to answer if you experiment a bit.
The other table is a rectangle with rounded ends, called the 'Bunimovitch stadium'. For that one the billiards move chaotically. After a while they seem randomized.
This illustrates two very different kinds of dynamical systems. The 'completely integrable' systems, like the elliptical billiards, do very predictable things. The 'ergodic' ones seem random.
With some math, we can make these ideas precise. I'll be quick: a system whose motion is described by Hamiltonian mechanics is completely integrable if it has the maximum number of conserved quantities. It's ergodic if it has the minimum number. All sorts of in-between cases are also possible!
For a particle moving around in \(n\) dimensions the maximum number of conserved quantities is \(n\). More precisely, we can write every conservated quantity as a function of \(n\) such quantities. The minimum number is 1, since energy is always conserved.
So, for a billiard ball, the maximum number is 2, and that's what we have for the elliptical billiard ball table. One of them is the energy, or if you prefer, the speed of the billiard ball.
Puzzle 2. What's the other?
This is related to Puzzle 1, since it's this extra conserved quantity that sometimes forbids the billiard ball from entering certain regions in the ellipse.
For more on complete integrability versus ergodicity, try these:
For some very nice answers to the puzzles, see the comments on
my G+ post.
November 17, 2016
This photo by Kei Nomiyama shows fireflies just above the ground in a bamboo forest.
Photographing fireflies is popular in Japan, and this article shows some other nice examples:
She writes:
Japan is a beautiful country full of breathtaking buildings, landscapes, and scenery any time of year. In the height of summer, however, something particularly magical happens. Throughout the countryside, twinkling fireflies take to the evening skies in search of a mate. This natural phenomenon creates a beautifully ethereal glow through trees and leaves that is nothing short of breathtaking.Of course, in this phenomenon, Japanese and visiting photographers have found a gorgeous source of inspiration. Capturing the lights of the fireflies, however, can be extremely difficult. Fireflies are very sensitive to other sources of light besides themselves, meaning that camera flashes, cell phones, flashlights, and other things that photographers often need to get their equipment set up can drive the little creatures away.
The difficulty of capturing photos of the fireflies, however, hasn't deterred the most dedicated photographers. They've simply adapted their strategy to account for the habits of the fireflies. Photographers often scout an area out days in advance to see where the fireflies congregate and then return very early on the day they want to shoot, setting up in daylight before the twinkling lights begin and lying in still, silent wait for hours.
You can see more of Kei Nomiyama's firefly photos here:
What puzzles me is this: the glowing fireflies in these photos seem more orange than what I see in the eastern US. I'm used to firefly light being yellow-green. So:
Puzzle 1. Are fireflies in Japan from a different species than US fireflies?
and more importantly:
Puzzle 2. Do they use a different chemical mechanism to make light?
or more generally:
Puzzle 3. How do fireflies make light, and how do they turn the chemical reaction on and off?
For some attempts at answers, see the comments on my G+ post.
November 27, 2016
Sure, there were a few absolutely fundamental results like the second law, which says that entropy cannot decrease as we carry a system from one equilibrium state to another. But the complications you see when you boil a pot of water... those were largely out of bounds.
This has changed in the last 50 years. One example is the Jarzynski equality, discovered by Christopher Jarzynski in 1997.
The second law implies that the change in 'free energy' of a system is less than or equal to the amount of work done on it. But the Jarzynski equality gives a precise equation relating these two concepts, which implies that inequality. I won't explain it here, but it's terse and beautiful.
Last week at the Santa Fe Institute, Jarzynski gave an incredibly clear hour-long tutorial on thermodynamics, starting with the basics and zipping forward to modern work. With his permission, you can see his slides here:
along with links to an explanation of the Jarzynski equality, and a proof.
I had a great time in Santa Fe, and this was one of the high points.
