- #1
John Baez
Also available at http://math.ucr.edu/home/baez/week231.html
May 9, 2006
This Week's Finds in Mathematical Physics (Week 230)
John Baez
Enceladus is a moon of Saturn with a cracked icy surface, twisted
and buckled by tidal forces, hinting at mysteries beneath:
1) NASA, Enceladus the storyteller,
http://www.nasa.gov/mission_pages/cassini/multimedia/pia07800.html
Recently the NASA space probe Cassini has been getting a good look at
Enceladus. In March, Cassini discovered that it has geysers among the
cracks near its south pole - geysers that spray water right out into space!
2) NASA's Cassini discovers potential liquid water on Enceladus,
http://saturn.jpl.nasa.gov/news/press-release-details.cfm?newsID=639
3) Special issue on Enceladus, Science 311 (March 10th 2006).
The water freezes in microscopic crystals, which replenish Saturn's
E ring - a diffuse bluish ring that was previously a mystery.
The currently popular theory for the geysers looks like this:
4) NASA, Enceladus "cold geyser" model,
http://www.nasa.gov/mission_pages/cassini/multimedia/pia07799.html
Enceladus is now the the only place besides Earth where liquid water
has been seen - though people believe Jupiter's moon Europa has oceans
under a layer of ice, and maybe Ganymede and Callisto do too.
While we tend to take it for granted, water is a very strange chemical:
5) Martin Chaplin, Forty-one anomalies of water,
http://www.lsbu.ac.uk/water/anmlies.html
As you probably know, the specific heat of water is unusually high,
which stabilizes the Earth's temperature. And no other simple compound
exhibits so many different forms. There at least 18 forms of ice!
You can tour them here:
6) Martin Chaplin, The phase diagram of water,
http://www.lsbu.ac.uk/water/phase.html
The hexagonal form of ice we find here on Earth is called ice Ih.
There's also a slightly denser cubic phase, ice Ic, which forms when
water vapor is condensed on a cold substrate. Below -130 Celsius,
a low-density amorphous solid form called LDA is possible. By
compressing ordinary ice Ih to high pressures, you get a different
higher-density amorphous form, called HDA. And there's an even denser
amorphous form called VHDA.
(It's unusual for a crystal to become amorphous when you compress it
or cool it, but ordinary ice is unusually light: it floats on liquid
water! That's because the powerful hydrogen bonds of water allow it
to maintain a very sparse hexagonal crystal structure - so sparse you
could even fit extra water molecules in the gaps. When you crush this,
it becomes amorphous.)
There are also crystal forms called ice II through ice XIV, in order
of discovery. It would take a few weeks to discuss all these, but
luckily Chaplin's website has a separate page on each kind, with nice
explanations and pictures of the crystal structures.
Kurt Vonnegut wrote a novel called "Cat's Cradle" staring a substance
called ice IX, which was supposedly more stable than liquid water at
ordinary temperatures and pressures. When it got loose, it destroyed
the world. Luckily the actual ice IX isn't like that, and it couldn't be:
the most stable form of water already prevails.
But enough about ice IX. I want to talk about ice X!
This is one of the most extreme forms of ice known. It's only stable
at pressures of about 50 gigapascals - in other words, roughly 50,000
atmospheres.
Hmm. Do those quantities mean as little to you as they do to me?
A "pascal" is a unit of pressure, or force per area, equal to one
Newton per square meter. An "atmosphere" is another unit of pressure,
basically the average air pressure at sea level here on Earth. This
has the annoying value of 101,325 pascals. Personally I have some
trouble getting a feel for how much pressure this is, since a Newton
per square meter isn't much, but 101,325 of them sounds like a lot.
So for me, being an American, it's helpful to know that an atmosphere
equals 2116 pounds per square foot. If you're a metric sort of person,
that's about the weight of 1 kilogram pushing down on each square
centimeter. That's a lot of pressure we're under! No wonder we feel
stressed sometimes.
(Yes, I know a kilogram is not a unit of weight. I mean the weight
corresponding to a mass of a kilogram in the Earth's gravitational field
at sea level. Sheesh!)
But I digress. I was saying that ice X only forms at a pressure
of about 50 gigapascals. But I've actually read figures ranging
from 44 to 80 gigapascals. This raises the question: how do people
know these things? Do they actually know, or just guess?
Well, some overgrown kids get paid to study these issues by actually
squashing water to enormous pressures using "diamond anvil cells".
Not many substances can withstand such huge pressures, but diamonds
can: as you know, they're really hard! They're also transparent,
so you can see what's going on while you're squashing something.
You basically just stick something between two carefully carved
diamonds, surrounded by a metal foil gasket, and squash the heck
out of it:
7) Diamond anvil cell, Wikipedia,
http://en.wikipedia.org/wiki/Diamond_Anvil_Cell
Apparently they can get pressures of up to 360 gigapascals this way,
which is the pressure at the center of the Earth.
Another method, which sounds even more fun, is to use a "light gas gun".
Here you explode a few kilograms of gunpowder to shoot a piston down
a tube. As it shoots forwards, the piston pushes some gas down the tube.
The tube narrows to a tiny tip at the end, so the gas is going really fast
by the time it shoots out. It shoots out into a much narrower tube,
where it pushes a projectile. You can then fire the projectile into
something, to generate very high pressures for a very short time.
8) Light gas gun, Wikipedia, http://en.wikipedia.org/wiki/Light_Gas_Gun
It's not called a "light" gas gun because it's wimpy - in fact they're
huge, and everyone evacuates the lab when they run the one at NASA!
It's called that because the speed of the projectile is limited only
by the speed of sound in the gas, which is higher for a light gas like
helium - or even better, hydrogen. Even better, that is, you don't
mind exploding gunpowder near highly flammable hydrogen! But, as you
can imagine, people who do this stuff are precisely the sort who don't
mind. You may enjoy reading how folks at Lawrence Livermore National
Laboratory used a light gas gun to compress hydrogen to pressures of
up to 200 gigapascals, enough to convert it into a metal:
9) Robert C. Cauble, Putting more pressure on hydrogen,
http://www.llnl.gov/str/Cauble.html
This supports the theory that the hydrogen at Jupiter's core is in
metallic form, which would explain its enormous magnetic field.
They know their hydrogen became a metal because they fired a laser
at it and saw it was shiny! In fact, they fired three lasers at
it simultaneously, just for kicks.
(By the way, this article erroneously says a "megabar" is 100 pascals.
It's a million atmospheres, or 100 gigapascals.)
But I'm digressing again. I was saying ice X forms at a pressure
of around 50 gigapascals. It's pretty far-out stuff. It's a cubic
crystal with density 2.5 times that of ordinary liquid water.
It's so compressed that separate water molecules no longer exist!
Instead, the oxygen atoms form a body-centered cubic. This means
they lie at the corners of a lattice of cubes, but with one at the
center of each cube too, like the red dots in this picture by
Cavazzoni:
10) Carlo Cavazzoni, Large scale first-principles simulations of water and
ammonia at high pressure and temperature, Ph.D. thesis, Scuola
Internazionale Superiore di Studi Avanzati, October 1998.
Figure 4.10: symmetric and super-ionic ice X structures, p. 57.
Available at http://sirio.cineca.it/~acv0/thesis.html
Hydrogen ions - in other words, protons - sit at the midpoints of half
the edges connecting cube corners to cube centers. There are two ways
they can do this. They can form a right-side-up tetrahedron, or an
upside-down tetrahedron.
Each oxygen has 4 hydrogens next to it. If you compress water a bit
less than enough to make ice X, you get ice VII. This is almost the
same, but two of those hydrogens are closer to the oxygen than the
other two, so there are still separate water molecules! It's completely
random which two hydrogens are closer than the other two. But if you
cool down ice VII, you get ice VIII, where it's *not* random.
So, Nature explores all the options.
Recently people have gotten interested in ice at even higher pressures -
and also higher temperatures, to understand the interiors of planets
like Neptune and Uranus. Here pressures range from 20 to 800 gigapascals,
and temperatures from 2000 to 8000 kelvin. In "week160" I mentioned
that on Neptune it may rain diamonds, formed by methane in the atmosphere.
But what happens to the water, and the ammonia? If they became good
electrical conductors, that might explain the magnetic fields of these
planets.
People have done computer simulations to study this:
12) C. Cavazzoni, G. L. Chiarotti, S. Scandolo, E. Tosatti, M. Bernasconi
and M. Parrinello, Superionic and metallic states of water and ammonia
at giant planet conditions, Science 283 (January 1999), 44-46.
Also available at http://www.sciencemag.org/cgi/content/full/283/5398/44
Phase diagram at http://math.ucr.edu/home/baez/cavazzoni_ice_phases.jpg
It seems that when you heat up ice X, it goes into a "superionic"
state where the little tetrahedra of hydrogen ions in each cube are
constantly randomizing themselves, instead of remaining fixed.
It's a curious hybrid of a solid and a liquid, since the hydrogens
are moving around, while the oxygens stay in their body-centered
cubic crystal.
But if you heat it even more, the oxygen melts too! As you can see
from the phase diagram above, it then becomes an ionic fluid.
As you heat it even more, you enter the region labelled "gap closure",
where the water starts to act like a metallic plasma. Then it's a
really good conductor of electricity.
The curve labelled "Neptune isentrope" describes the pressures and
temperatures you'd experience if you unwisely jumped into Neptune!
As you fell in, it would keep getting hotter and the pressure would
keep rising until you entered this chart, at a temperature of about
2000 kelvin. At this point you'd see molecular fluid water - I say
this because at temperatures above 650 kelvin (the critical point
for water), there's no sharp difference between liquid and gas.
Then the fluid would become ionic... and then you'd start drifting
towards gap closure and the metallic plasma phase. Down deep,
metallic plasmas of water and ammonia might explain the magnetic
field of this planet.
Recently people have done some experiments with water at extremely
high pressures, checking what theorists like Cavazzoni and company
predict. For example, this paper says that using "extremely large
lasers", people have studied water at pressures near a terapascal -
1000 gigapascals:
13) P. M. Celliers et al, Electronic conduction in shock-compressed
water, Plasmas 11 (2004), L41-L48.
They also mention that "a single datum at 1.4 terapascals from an
underground nuclear experiment has never been repeated." Some people
just don't know when to stop in the quest for higher pressures.
While I'm at it, I should mention a few more interesting articles
on weird forms of ice. There's a lot of research on this subject!
Here's a quick overview:
14) Nancy McGuire, The many phases of water, American Chemical Society,
http://www.chemistry.org/portal/a/c/s/1/feature_pro.html?id=c373e9fbed0a01c78f6a4fd8fe800100
Here's a webpage with some nice pictures and an interesting story:
15) J. L. Finney, The phase diagram of water and a new metastable form of
ice, http://www.cmmp.ucl.ac.uk/people/finney/soi.html
And finally, there's a paper that talks about how ordinary ice Ih
but also silica and ice XI become amorphous when you squeeze them
enough:
16) Koichiro Umemoto, Renata M. Wentzcovitch, Stefano Baroni and Stefano
de Cironcoli, Anomalous pressure-induced transition(s) in ice XI, Physical
Review Letters 92 (2004), 105502-1. Also available at
http://www.cems.umn.edu/research/wentzcovitch/papers/Phys._Rev._Lett._92_105502_(2004).pdf
There's some interesting math in here, because they do computer
simulations of the transition from a crystal to an amorphous
substance, which is interesting to study using Fourier analysis.
The idea is that certain vibrational modes of the crystal
"go soft", so they get easily excited. When a bunch of modes
go soft that have wavelengths not equal to the crystal lattice
spacing, the crystal structure becomes unstable, and there can
be a transition to an amorphous state.
There's also interesting math lurking in Cavazzoni et al's
models of ice X! If you think particle physics is hard, just
wait until you try understanding something complicated, like water.
I've been sort of obsessed with ice lately. If you like it too,
I recommend this book for general information:
16) Mariana Gosnell, Ice: The Nature, the History, and the Uses of
an Astonishing Substance, Alfred A. Knopf, New York, 2005.
but I bought this one, because it tells an interesting history of
the science of climate change as seen from icy peaks:
17) Mark Bowen, Thin Ice: Unlocking the Secrets of Climate in the
World's Highest Mountains, Henry Holt & Co., 2005.
Now for some math. Last week I said a bit about quivers, the McKay
correspondence, and string theory. I want to dig deeper into the
relation between these subjects, because Urs Schreiber has some
interesting ideas about them, which he's mentioned here:
18) Urs Schreiber, A note on RCFT and quiver reps,
http://golem.ph.utexas.edu/string/archives/000794.html
But, I'm not feeling sufficiently energetic to explain these ideas
right now, especially since he already has! For some more clues,
try this:
19) Paul Aspinwall, D-branes on Calabi-Yau manifolds, section
7.3.1, The McKay correspondence, p. 101 and following. Available
as hep-th/0403166.
For more on the representation theory of quivers, see the references
in the "Addenda" to "week230", and also this excellent book:
20) David J. Benson, Representations and Cohomology I,
Cambridge U. Press, Cambridge 1991.
You'll see how the non-simply-laced Dynkin diagrams get into the act!
A more thorough treatment, fascinating but somewhat quirky, can be
found here:
21) P. Gabriel and A. V. Roiter, Representations of Finite-Dimensional
Algebras, Enc. of Math. Sci., 73, Algebra VIII, Springer, Berlin 1992.
If you like category theory you may enjoy this book, because
it's all about representations of categories, i.e. functors
F: C -> Vect
where C is a category. It's full of nontrivial theorems about
these, starting with Gabriel's classification of quivers into those
of finite representation type (see "week230"), the tame quivers
(which have a countable set of indecomposable representations),
and the wild ones. But, you may be puzzled when you read about
"svelte" categories, or functors that "preserve heteromorphisms"!
I might as well say what those are. A category is "svelte" if
its isomorphism classes of objects form a mere set instead of
a proper class, like the category of finite-dimensional vector
spaces. Most people would say such a category is "essentially small".
And, a functor "preserves heteromorphisms" if it maps heteromorphisms
to heteromorphisms. Well, duh! But what's a "heteromorphism"?
It's their term for a morphism that's not an isomorphism. Most
people would say such a functor "reflects isomorphisms".
You may also be interested in what a "locular" category is,
or a "spectroid"... but I won't tell you! Read the book.
Speaking of category theory, this is my last week in Chicago, which
is really sad, because Steve Lack is just starting to give us a
crash course on "Australian category theory". Australia, you see,
is the center of macho category theory, where they're heavy on the
calculus of mates, doctrinal adjunctions are a dime a dozen, and
everything should be V-enriched if not W-enriched. But Chicago is
starting to get macho too: tomorrow Nick Gurski defends his Ph.D.
thesis on "Algebraic Tricategories"! So, the Chicago gang wants
to learn some Australian tricks. But next Monday I'm off to the
Perimeter Institute, to indulge the physics side of my personality...
-----------------------------------------------------------------------
Quote of the Week:
"That the glass would melt in heat,
That the water would freeze in cold,
Shows that this object is merely a state,
One of many, between two poles." - Wallace Stevens
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
May 9, 2006
This Week's Finds in Mathematical Physics (Week 230)
John Baez
Enceladus is a moon of Saturn with a cracked icy surface, twisted
and buckled by tidal forces, hinting at mysteries beneath:
1) NASA, Enceladus the storyteller,
http://www.nasa.gov/mission_pages/cassini/multimedia/pia07800.html
Recently the NASA space probe Cassini has been getting a good look at
Enceladus. In March, Cassini discovered that it has geysers among the
cracks near its south pole - geysers that spray water right out into space!
2) NASA's Cassini discovers potential liquid water on Enceladus,
http://saturn.jpl.nasa.gov/news/press-release-details.cfm?newsID=639
3) Special issue on Enceladus, Science 311 (March 10th 2006).
The water freezes in microscopic crystals, which replenish Saturn's
E ring - a diffuse bluish ring that was previously a mystery.
The currently popular theory for the geysers looks like this:
4) NASA, Enceladus "cold geyser" model,
http://www.nasa.gov/mission_pages/cassini/multimedia/pia07799.html
Enceladus is now the the only place besides Earth where liquid water
has been seen - though people believe Jupiter's moon Europa has oceans
under a layer of ice, and maybe Ganymede and Callisto do too.
While we tend to take it for granted, water is a very strange chemical:
5) Martin Chaplin, Forty-one anomalies of water,
http://www.lsbu.ac.uk/water/anmlies.html
As you probably know, the specific heat of water is unusually high,
which stabilizes the Earth's temperature. And no other simple compound
exhibits so many different forms. There at least 18 forms of ice!
You can tour them here:
6) Martin Chaplin, The phase diagram of water,
http://www.lsbu.ac.uk/water/phase.html
The hexagonal form of ice we find here on Earth is called ice Ih.
There's also a slightly denser cubic phase, ice Ic, which forms when
water vapor is condensed on a cold substrate. Below -130 Celsius,
a low-density amorphous solid form called LDA is possible. By
compressing ordinary ice Ih to high pressures, you get a different
higher-density amorphous form, called HDA. And there's an even denser
amorphous form called VHDA.
(It's unusual for a crystal to become amorphous when you compress it
or cool it, but ordinary ice is unusually light: it floats on liquid
water! That's because the powerful hydrogen bonds of water allow it
to maintain a very sparse hexagonal crystal structure - so sparse you
could even fit extra water molecules in the gaps. When you crush this,
it becomes amorphous.)
There are also crystal forms called ice II through ice XIV, in order
of discovery. It would take a few weeks to discuss all these, but
luckily Chaplin's website has a separate page on each kind, with nice
explanations and pictures of the crystal structures.
Kurt Vonnegut wrote a novel called "Cat's Cradle" staring a substance
called ice IX, which was supposedly more stable than liquid water at
ordinary temperatures and pressures. When it got loose, it destroyed
the world. Luckily the actual ice IX isn't like that, and it couldn't be:
the most stable form of water already prevails.
But enough about ice IX. I want to talk about ice X!
This is one of the most extreme forms of ice known. It's only stable
at pressures of about 50 gigapascals - in other words, roughly 50,000
atmospheres.
Hmm. Do those quantities mean as little to you as they do to me?
A "pascal" is a unit of pressure, or force per area, equal to one
Newton per square meter. An "atmosphere" is another unit of pressure,
basically the average air pressure at sea level here on Earth. This
has the annoying value of 101,325 pascals. Personally I have some
trouble getting a feel for how much pressure this is, since a Newton
per square meter isn't much, but 101,325 of them sounds like a lot.
So for me, being an American, it's helpful to know that an atmosphere
equals 2116 pounds per square foot. If you're a metric sort of person,
that's about the weight of 1 kilogram pushing down on each square
centimeter. That's a lot of pressure we're under! No wonder we feel
stressed sometimes.
(Yes, I know a kilogram is not a unit of weight. I mean the weight
corresponding to a mass of a kilogram in the Earth's gravitational field
at sea level. Sheesh!)
But I digress. I was saying that ice X only forms at a pressure
of about 50 gigapascals. But I've actually read figures ranging
from 44 to 80 gigapascals. This raises the question: how do people
know these things? Do they actually know, or just guess?
Well, some overgrown kids get paid to study these issues by actually
squashing water to enormous pressures using "diamond anvil cells".
Not many substances can withstand such huge pressures, but diamonds
can: as you know, they're really hard! They're also transparent,
so you can see what's going on while you're squashing something.
You basically just stick something between two carefully carved
diamonds, surrounded by a metal foil gasket, and squash the heck
out of it:
7) Diamond anvil cell, Wikipedia,
http://en.wikipedia.org/wiki/Diamond_Anvil_Cell
Apparently they can get pressures of up to 360 gigapascals this way,
which is the pressure at the center of the Earth.
Another method, which sounds even more fun, is to use a "light gas gun".
Here you explode a few kilograms of gunpowder to shoot a piston down
a tube. As it shoots forwards, the piston pushes some gas down the tube.
The tube narrows to a tiny tip at the end, so the gas is going really fast
by the time it shoots out. It shoots out into a much narrower tube,
where it pushes a projectile. You can then fire the projectile into
something, to generate very high pressures for a very short time.
8) Light gas gun, Wikipedia, http://en.wikipedia.org/wiki/Light_Gas_Gun
It's not called a "light" gas gun because it's wimpy - in fact they're
huge, and everyone evacuates the lab when they run the one at NASA!
It's called that because the speed of the projectile is limited only
by the speed of sound in the gas, which is higher for a light gas like
helium - or even better, hydrogen. Even better, that is, you don't
mind exploding gunpowder near highly flammable hydrogen! But, as you
can imagine, people who do this stuff are precisely the sort who don't
mind. You may enjoy reading how folks at Lawrence Livermore National
Laboratory used a light gas gun to compress hydrogen to pressures of
up to 200 gigapascals, enough to convert it into a metal:
9) Robert C. Cauble, Putting more pressure on hydrogen,
http://www.llnl.gov/str/Cauble.html
This supports the theory that the hydrogen at Jupiter's core is in
metallic form, which would explain its enormous magnetic field.
They know their hydrogen became a metal because they fired a laser
at it and saw it was shiny! In fact, they fired three lasers at
it simultaneously, just for kicks.
(By the way, this article erroneously says a "megabar" is 100 pascals.
It's a million atmospheres, or 100 gigapascals.)
But I'm digressing again. I was saying ice X forms at a pressure
of around 50 gigapascals. It's pretty far-out stuff. It's a cubic
crystal with density 2.5 times that of ordinary liquid water.
It's so compressed that separate water molecules no longer exist!
Instead, the oxygen atoms form a body-centered cubic. This means
they lie at the corners of a lattice of cubes, but with one at the
center of each cube too, like the red dots in this picture by
Cavazzoni:
10) Carlo Cavazzoni, Large scale first-principles simulations of water and
ammonia at high pressure and temperature, Ph.D. thesis, Scuola
Internazionale Superiore di Studi Avanzati, October 1998.
Figure 4.10: symmetric and super-ionic ice X structures, p. 57.
Available at http://sirio.cineca.it/~acv0/thesis.html
Hydrogen ions - in other words, protons - sit at the midpoints of half
the edges connecting cube corners to cube centers. There are two ways
they can do this. They can form a right-side-up tetrahedron, or an
upside-down tetrahedron.
Each oxygen has 4 hydrogens next to it. If you compress water a bit
less than enough to make ice X, you get ice VII. This is almost the
same, but two of those hydrogens are closer to the oxygen than the
other two, so there are still separate water molecules! It's completely
random which two hydrogens are closer than the other two. But if you
cool down ice VII, you get ice VIII, where it's *not* random.
So, Nature explores all the options.
Recently people have gotten interested in ice at even higher pressures -
and also higher temperatures, to understand the interiors of planets
like Neptune and Uranus. Here pressures range from 20 to 800 gigapascals,
and temperatures from 2000 to 8000 kelvin. In "week160" I mentioned
that on Neptune it may rain diamonds, formed by methane in the atmosphere.
But what happens to the water, and the ammonia? If they became good
electrical conductors, that might explain the magnetic fields of these
planets.
People have done computer simulations to study this:
12) C. Cavazzoni, G. L. Chiarotti, S. Scandolo, E. Tosatti, M. Bernasconi
and M. Parrinello, Superionic and metallic states of water and ammonia
at giant planet conditions, Science 283 (January 1999), 44-46.
Also available at http://www.sciencemag.org/cgi/content/full/283/5398/44
Phase diagram at http://math.ucr.edu/home/baez/cavazzoni_ice_phases.jpg
It seems that when you heat up ice X, it goes into a "superionic"
state where the little tetrahedra of hydrogen ions in each cube are
constantly randomizing themselves, instead of remaining fixed.
It's a curious hybrid of a solid and a liquid, since the hydrogens
are moving around, while the oxygens stay in their body-centered
cubic crystal.
But if you heat it even more, the oxygen melts too! As you can see
from the phase diagram above, it then becomes an ionic fluid.
As you heat it even more, you enter the region labelled "gap closure",
where the water starts to act like a metallic plasma. Then it's a
really good conductor of electricity.
The curve labelled "Neptune isentrope" describes the pressures and
temperatures you'd experience if you unwisely jumped into Neptune!
As you fell in, it would keep getting hotter and the pressure would
keep rising until you entered this chart, at a temperature of about
2000 kelvin. At this point you'd see molecular fluid water - I say
this because at temperatures above 650 kelvin (the critical point
for water), there's no sharp difference between liquid and gas.
Then the fluid would become ionic... and then you'd start drifting
towards gap closure and the metallic plasma phase. Down deep,
metallic plasmas of water and ammonia might explain the magnetic
field of this planet.
Recently people have done some experiments with water at extremely
high pressures, checking what theorists like Cavazzoni and company
predict. For example, this paper says that using "extremely large
lasers", people have studied water at pressures near a terapascal -
1000 gigapascals:
13) P. M. Celliers et al, Electronic conduction in shock-compressed
water, Plasmas 11 (2004), L41-L48.
They also mention that "a single datum at 1.4 terapascals from an
underground nuclear experiment has never been repeated." Some people
just don't know when to stop in the quest for higher pressures.
While I'm at it, I should mention a few more interesting articles
on weird forms of ice. There's a lot of research on this subject!
Here's a quick overview:
14) Nancy McGuire, The many phases of water, American Chemical Society,
http://www.chemistry.org/portal/a/c/s/1/feature_pro.html?id=c373e9fbed0a01c78f6a4fd8fe800100
Here's a webpage with some nice pictures and an interesting story:
15) J. L. Finney, The phase diagram of water and a new metastable form of
ice, http://www.cmmp.ucl.ac.uk/people/finney/soi.html
And finally, there's a paper that talks about how ordinary ice Ih
but also silica and ice XI become amorphous when you squeeze them
enough:
16) Koichiro Umemoto, Renata M. Wentzcovitch, Stefano Baroni and Stefano
de Cironcoli, Anomalous pressure-induced transition(s) in ice XI, Physical
Review Letters 92 (2004), 105502-1. Also available at
http://www.cems.umn.edu/research/wentzcovitch/papers/Phys._Rev._Lett._92_105502_(2004).pdf
There's some interesting math in here, because they do computer
simulations of the transition from a crystal to an amorphous
substance, which is interesting to study using Fourier analysis.
The idea is that certain vibrational modes of the crystal
"go soft", so they get easily excited. When a bunch of modes
go soft that have wavelengths not equal to the crystal lattice
spacing, the crystal structure becomes unstable, and there can
be a transition to an amorphous state.
There's also interesting math lurking in Cavazzoni et al's
models of ice X! If you think particle physics is hard, just
wait until you try understanding something complicated, like water.
I've been sort of obsessed with ice lately. If you like it too,
I recommend this book for general information:
16) Mariana Gosnell, Ice: The Nature, the History, and the Uses of
an Astonishing Substance, Alfred A. Knopf, New York, 2005.
but I bought this one, because it tells an interesting history of
the science of climate change as seen from icy peaks:
17) Mark Bowen, Thin Ice: Unlocking the Secrets of Climate in the
World's Highest Mountains, Henry Holt & Co., 2005.
Now for some math. Last week I said a bit about quivers, the McKay
correspondence, and string theory. I want to dig deeper into the
relation between these subjects, because Urs Schreiber has some
interesting ideas about them, which he's mentioned here:
18) Urs Schreiber, A note on RCFT and quiver reps,
http://golem.ph.utexas.edu/string/archives/000794.html
But, I'm not feeling sufficiently energetic to explain these ideas
right now, especially since he already has! For some more clues,
try this:
19) Paul Aspinwall, D-branes on Calabi-Yau manifolds, section
7.3.1, The McKay correspondence, p. 101 and following. Available
as hep-th/0403166.
For more on the representation theory of quivers, see the references
in the "Addenda" to "week230", and also this excellent book:
20) David J. Benson, Representations and Cohomology I,
Cambridge U. Press, Cambridge 1991.
You'll see how the non-simply-laced Dynkin diagrams get into the act!
A more thorough treatment, fascinating but somewhat quirky, can be
found here:
21) P. Gabriel and A. V. Roiter, Representations of Finite-Dimensional
Algebras, Enc. of Math. Sci., 73, Algebra VIII, Springer, Berlin 1992.
If you like category theory you may enjoy this book, because
it's all about representations of categories, i.e. functors
F: C -> Vect
where C is a category. It's full of nontrivial theorems about
these, starting with Gabriel's classification of quivers into those
of finite representation type (see "week230"), the tame quivers
(which have a countable set of indecomposable representations),
and the wild ones. But, you may be puzzled when you read about
"svelte" categories, or functors that "preserve heteromorphisms"!
I might as well say what those are. A category is "svelte" if
its isomorphism classes of objects form a mere set instead of
a proper class, like the category of finite-dimensional vector
spaces. Most people would say such a category is "essentially small".
And, a functor "preserves heteromorphisms" if it maps heteromorphisms
to heteromorphisms. Well, duh! But what's a "heteromorphism"?
It's their term for a morphism that's not an isomorphism. Most
people would say such a functor "reflects isomorphisms".
You may also be interested in what a "locular" category is,
or a "spectroid"... but I won't tell you! Read the book.
Speaking of category theory, this is my last week in Chicago, which
is really sad, because Steve Lack is just starting to give us a
crash course on "Australian category theory". Australia, you see,
is the center of macho category theory, where they're heavy on the
calculus of mates, doctrinal adjunctions are a dime a dozen, and
everything should be V-enriched if not W-enriched. But Chicago is
starting to get macho too: tomorrow Nick Gurski defends his Ph.D.
thesis on "Algebraic Tricategories"! So, the Chicago gang wants
to learn some Australian tricks. But next Monday I'm off to the
Perimeter Institute, to indulge the physics side of my personality...
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Quote of the Week:
"That the glass would melt in heat,
That the water would freeze in cold,
Shows that this object is merely a state,
One of many, between two poles." - Wallace Stevens
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Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
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