Thermal and Statistical Physics

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Lecture 1: Accessible States Aug 21, 2006

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It's a new year, with new lectures. Today is an introduction to Thermal and Statistical Physics. We do a lightning-fast review of quantum mechanics, aided by the excellent program Atom in a Box available at www.daugerresearch.com. We also discuss the fundamental assumption of statistical mechanics. Although statistical mechanics is based on probabilities about processes on very small (atomic) scales, we can still make firm predictions about large-scale behavior. This works for the same reason that casinos make money: the statistics of large numbers becomes deterministic.
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New Course: Solid State Physics Jan 10, 2006

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Prof. Carlson is teaching a new course this semester, entitled Solid State Physics.
Here is a taste of the new course, which has a separate podcast in iTunes. Search podcasts for Carlson to find the new one, which hopefully will appear in the directory structure of iTunes soon.
Lecture 1 of Solid State Physics: The Failure of Reductionism
Reductionism is the idea that by breaking things into their smallest constituents, we will learn all about them. For example, we might want to learn about solids by breaking them into atoms, then learn about the atoms by breaking them into the constituent electrons and nuclei, and so on. But reductionism is merely a philosophy handed down to us by the Greeks -- is it really correct? New ideas in the field point toward the inadequacy of reductionism, and lead to "emergence" as a better paradigm for gaining knowledge in condensed matter/solid state physics. Emergence is the idea that when many particles get together, new phenomena appear which are not encoded in the microscopic laws, and in fact are independent of the microscopic laws. For example, all solids are hard, regardless of what atoms are in them. Likewise, if we changed the microscopic laws, by changing, say, the shape of the Coulomb potential, we'd still get solids that are hard. That means hardness, and many properties like it, is not caused by the underlying microscopic laws, but rather is caused by deeper physics. It turns out in fact that it's related to symmetry.
Also, find out the difference between correct and useful.
Solid State Physics Lecture Blog

Final Review 2 Dec 12, 2005

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This is a final review for the last 1/4 of the course. This is a very short lecture, because we had a field trip to go see the prestigious Bagwell Lecture given by Purdue's very own Prof. Albert Overhauser of the world-famous Overhauser Effect.
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Final Review 1 Dec 05, 2005

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This is a final review for the first 3/4 of the course.
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Lecture 24: Fluctuation-Dissipation Theorem Dec 01, 2005

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We finish two more examples of the Fluctuation-Dissipation Theorem. This is a theorem that pops up everywhere! It means that the very same microscopic processes responsible for establishing thermal equilibrium are the same microscopic processes that cause resistance in metals, drag in fluids, and other types of dissipation. We discuss thermal noise in resistors (also known as Johnson noise or Nyquist noise), and demonstrate the fluctuation-dissipation theorem in this system. We also derive the magnetic susceptibility of a collection of free spins in a magnetic field. It turns out (due to the fluctuation dissipation theorem, of course) that the higher the amount of thermal fluctuations in the system at thermal equilibrium, the easier it is to magnetize the system.
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Lecture23: Brownian Motion and Diffusion Nov 29, 2005

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Brownian motion was discovered by a botanist named Brown, when he looked at water under a microscope, and observed pollen grains "jiggling" about in it. Einstein eventually explained it as due to the random collisions the pollen grain experienced from the water molecules. We compare the pollen grain to a drunk person walking home, and calculate how far the pollen grain can get by this type of diffusion. We also introduce the fluctuation-dissipation theorem, a far-reaching principle in advanced statistical mechanics that says that the microscopic thermal fluctuations in a system are the same microscopic processes that are responsible for things like drag, viscosity, and electrical resistance. (Why is that so cool? Because it means you can predict nonequilibrium properties -- those in the presence of an applied field like voltage -- to equilibrium properties like thermal fluctuations.) We also derive Fick's law of diffusion -- particles diffuse away from high concentrations. Go figure!
Shown in class:
Nice movies on the web about colloid particles in milk executing Brownian motion.
There's a great applet on Brownian motion to play with here.
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Lecture 22: Nucleation in First Order (Abrupt) Phase Transitions Nov 16, 2005

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Supercooling Demonstration (thanks to special guest Prof. Ken Ritchie): Put filtered water in a plastic bottle in your freezer for, say, 4 hours. Now, carefully remove it from the freezer, and shake the bottle vigorously. We did this, and saw ice crystals begin to slowly form in the water, because the liquid water was supercooled, and the ice phase was technically more stable. (Some crystals even resembled snowflakes, and grew larger as they floated to the top.) You may have to experiment with how long you leave the bottle in the freezer. Too short a time, and nothing happens. If you freeze the bottle longer, a vigorous shake will turn the whole bottle white as crystals form everywhere. Too long, and it will all freeze in the freezer. Do try this at home!
Today we discuss nucleation in first order (abrupt) phase transitions. The ice crystals in our supercooled bottle of water formed through nucleation -- tiny ice crystals grew larger over time. The arctic cod is a supercooled fish, living in water too salty to freeze even though it's at -1.9 degrees Celsius! The reason the fish doesn't freeze solid is due to antifreeze glycoproteins, which inhibit the growth of nucleated ice crystals. We calculate the energy barriers to nucleation at the liquid-gas transiton, and find that a nucleated liquid bubble in the gas phase must be large enough before it will turn the whole substance liquid. If it's too small, the bubble is unstable and converts back into gas.
We also discuss: Slushy ice -- where is that on our phase diagram? Surface tension and faceting in crystals. Plant-eating bacteria which secrete enzymes that encourage ice nucleation on plants. And quite a bit about how snowflakes form.
Much of today is from Jim Sethna's statistical mechanics book, and the part about snowflakes and ice formation is from research at my alma mater, Caltech, as presented at www.snowcrystals.com.
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Lecture 21: Alloys, Mixing, and Phase Separation Nov 14, 2005

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Oil and water -- they don't mix. Or do they? Due to the entropy of mixing, any tiny amount of impurity is highly favored entropically. This means that in general, you can get a small amount of a substance to mix into another. But take that too far, and they no longer mix, but "phase separate" into 2 different concentrations. We discuss this from the following perspectives: energy, entropy, and free energy. Examples: binary alloy with interactions, and a mixture of He3 (fermions) and He4 (bosons).
Class discussion: Can you get oil and water to mix if you heat them in a pressure cooker?
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Lecture 20: Landau Theory of Phase Transitions; Oil, Water, and Alloys Nov 09, 2005

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Now that we know what order parameters are (see last lecture), we'll use the order parameter of a phase to construct the Landau free energy. The Landau free energy depends on the order parameter, and retains all the symmetries of the physical system. It's amazing how much you can get from symmetry, and we're able to see how it is that a ferromagnet can have what's called a continuous phase transition. That is, starting from zero temperature with a saturated magnetization, upon raising the temperature, the magnetization slowly decreases, until it has smoothly (continuously, in fact) gone to zero. This makes it a continous transition. We also show what a first order (discontinuous) phase transition would look like. First order phase transitions can exhibit supercooling and superheating. We also discuss the physics of alloys like bronze, and under what conditions two different materials will mix and form a binary mixture, and under what conditions there will be phase separation into 2 distinct concentrations, as happens with oil and water. A small concentration of impurities is always favorable according to entropy, and will always mix. But larger concentrations may "fall apart" and phase separate.
Class discussions: Lots about supercooling and superheating. More about nonequilibrium behavior like window glass.
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Lecture 19: Symmetries, Order Parameters, and the Failure of Reductionism Nov 07, 2005

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We finish the van der Waals equation of state, and use it to illustrate the liquid-gas phase transition. It turns out that at low pressure, the van der Waals equation of state has a wiggle where (dp/pV)>0. Since this would cause an explosion, the system instead undergoes phase separation so that part of the container has liquid, and part has gas in it.
More is different: We discuss the failure of reductionism. Reductionism is the idea that you will learn everything about an object by breaking it into its smallest bits -- like atoms, then electrons and protons, then quarks, then strings. But large collections of particles (like liquids, gases, and solids) have many properties which aren't really due to their constituents per se, but rather are due to larger organizing principles, and the symmetry of the associated phase. Example: All solids are hard, even though they're made out of different substances. So the property "hardness" is not actually caused by the particular form of the potentials for the particular atoms in that solid. Rather, it's due to the symmetry of the regular crystalline structure the atoms take, and is independent of the type of atom.
To illustrate, we discuss several phases of matter, and identify the corresponding "order parameter", which is a measurable quantity that captures the symmetry of the phase.
Visual Aids: Rotini pasta to demonstrate twisted nematic phases. Specimens from my rock collection: quartz, amethyst, hematite, and others to see how all crystals are similar, despite being made from different atoms. The "sameness" manifests itself in the basic property of a solid: being hard. The "differenc" manifests itself in color, and in the shape of the crystals, which reveal the underlying quantum mechanics of how the chemical bonds form from atom to atom. Plus, the return of the squishy crystal to illustrate phonons. 0
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