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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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This is a final review for the first 3/4 of the course.
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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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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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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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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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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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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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We derive the shape of the phase boundary for solid to gas transitions (sublimation), examples being dry ice (CO2) or ice at low pressure. We derive the van der Waals equation of state, which is an improvement on the ideal gas equation pV=nRT. The ideal gas equation is based on two assumptions: 1. Particles occupy zero volume, and 2. Particles do not interact. Allowing for particles to have a finite size, and also allowing for the fact that at close range, gas particles feel van der Waals attractions, we get the new improved van der Waals equation of state for a gas made of sticky but hard molecules. Van der Waals attractions work because at close range, atoms and molecules notice each other's dipole moments. The dipole moments are due to the fact that at any given instant, the electron cloud is not quite centered on the nucleus of the atom (although it will be centered on average). This instantaneous dipole moment causes atoms in the vicinity to arrange their instantaneous dipoles so as to lower their energy, which causes attraction.
It turns out that geckos can cling to walls and ceilings because of van der Waals attractions. Gecko feet have tiny hairs that split many times to make many very fine tips, giving the hairs a very large total surface area. The fine hairs are able to form many contacts with any surface, and the surface-to-hair contact is adhesive due to van der Waals forces. One gecko foot can support the weight of an entire human.
Video: Sticky gecko feet, and their van der Waals adhesive properties.
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We finish discussing chemical reactions, including how fast they progress, and what a catalyst can do for you. Then we begin a new topic: phases of matter and phase transitions between them. You've heard of solid, liquid, and gas, but did you know about the other phases of matter? Other phases include liquid crystals (of which there are many types). Also, electrons inside of a solid have their own phase transitions.
For example, metals carry current when the electrons inside flow -- that's a liquid phase of electrons. Refrigerator magnets are in a different electronic phase -- there, electrons execute tiny current loops around individual atoms, forming nanosize magnets. When they all align, the phase is called a "ferromagnet", and can be used to post notes to your refrigerator. We also discuss how you can go from liquid to gas and never encounter a phase transition! This is because liquid and gas aren't all that different to begin with. Class discussions: liquid crystal screens, melting snow, and what you really see when water boils.
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We define the Gibbs Free Energy, which is the right energy function to use when you can control temperature, pressure, and particle number. This means chemists like it, because chemical reactions in a lab often take place under these conditions.
We use this to derive the Law of Mass Action, which shows that the relative concentration of reactants depends only on temperature, and apply this to dissociation of the Hydrogen molecule, water, and hydrochloric acid.
We also return to last lecture's discussion of how superconductors repel magnetic fields. Demo: We use liquid nitrogen to cool the high temperature superconductor YBCO
below its superconducting transition temperature, so that it is in the superconducting state, and able to levitate magnets. Class discussions: How not to use a refrigerator to cool your apartment; High temperature superconductors and a small part of what's known about them.
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How refrigerators work. Why you can't cool your apartment by leaving the refrigerator door open. How heat and work depend on which path is taken. How to do completely meaningless work, the kind that's turned entirely into heat. We prove why the free energy is a useful concept: it tells you the maximum amount of work you can expect to extract from a system. The free energy is about the useful energy. We show that chemical potentials drive chemical work. How to levitate Tosanumi the sumo wrestler with superconductors and magnets, and how much work the superconductor does in the process.
Class discussions: The heat death of the universe. (Don't worry -- the sun will die out far before that.) More about high temperature superconductors, and possible applications.
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We're having a midterm exam Wednesday, and today is a review of everything in chapters 1-7 in the text, Kittel and Kroemer's Thermal Physics. Topics include: Fundamental assumption of statistical mechanics, Laws of Thermodynamics, Probabilities and the Partition Function, Entropy and Temperature, Heat Capacity and Energy, Thermodynamic Identity, Helmholtz free energy, Free energy and the partition function, Maxwell Relations, Planck Distribution Function and blackbody radiation, Chemical potential, Grand partition function, Density of States, Fermions and Bosons, Ideal Gas, Ideal Gas Processes, and a few equations to memorize for the exam.
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Storytime with Thursday Next (Jasper Fforde), and her Uncle Mycroft's entropy-detecting entroposcope. Why are large-scale systems capable of producing irreversible processes (like glass breaking, or red and blue Kool-aid mixing), even though the microscopic processes are reversible? We finish the electronic heat capacity of metals, first with an easy estimate to see that C~T, then with the full calculation. Using ideal gas processes (isothermal expansion, isentropic expansion), we build a Carnot engine and discuss its efficiency. You can't beat Sadi Carnot. Class Exercise: calculate the work done in one cycle. Class discussions: the chemical potential has a slight temperature dependence in three dimensions, but not in two. Why you should never hook lead pipes to aluminum pipes in your house. A little bit about melting. Can you convert heat entirely into work, or work entirely into heat?
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More about Bose condensates. They're really weird -- at the lowest temperature, all bosons flock to the lowest available state, producing a "Bose condensate".
Due to quantum mechanics, this is a remarkably stable state of matter, and is very hard to disturb. In fact, because the chemical potential becomes negative, it costs negative energy to add a new particle to the condensate. (Yes, bosons are "sticky" due to their statistics.) We also show why Bose condensates give rise to superfluidity (and superconductivity if the bosons are charged.) Class demonstration: The Wave (Just like the one in a baseball stadium.) The point is that many-body excitations often have very different character from the constituents. That is, "The Wave" in a crowd is an excitation of the crowd that doesn't look anything like the constituents (individual persons). Class discussions: What are superfluids and superconductors good for? What about the cuprate high temperature superconductors? Since they're ceramics, can you ever make them into wires? Are there higher temperature superconductors? How would room temperature superconductors make your life better?
We also discuss the heat capacity of metals at the end of class. Some of the electrons in a metal are free to flow, and are in a fluid phase of matter that allows us to use the Fermi ideal gas to describe some of their behavior.
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Now that we've derived absolutely everything about the ideal gas from scratch,
it's time to do something useful with it! We'd like to eventually learn how to use this stuff to build engines and refrigerators. Today we discuss the basic processes (reversible expansions) that are the building blocks of engines and refrigerators.
We also cover Bose condensation at the end of class, and learn why their statistics makes bosons sticky.
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Review of Fermions and Bosons. Review of Fermi Gas. All about the Bose gas, and its ditsrubution function. In the classical limit, the Fermi-Dirac distribution function and the Bose-Einstein distribution function approach the same form, and we recover ideal gas physics. We derive many properties about the ideal gas, and extend it to the case of internal degrees of freedom. More detail about the equipartition theorem, and how as temperature is raised, the heat capacity jumps up every time a new degree of freedom becomes excited. Example: Diatomic molecule. (Visual aids: many diatomic molecule models made from balls and springs.) Example: Experimental verification of the ideal gas law through the Sackur-Tetrode equation for entropy.
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Why no two pieces of matter may occupy the same space at the same time. Fermions are antisocial; bosons are social. Bosonic examples: lasers and superfluid helium. All about Fermions. Fermions obey the Pauli exclusion principle, and each state may have either 0 or 1 fermions in it, and no more. Class Discussions: more about aluminum, what about positrons, why gecko feet are sticky. Simulation Demo: Fermi distrubution function at various temperatures.
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When the system and reservoir can trade particles, you can't use the Boltzmann factor and the partition function anymore. Instead, use the Gibbs factor, and the grand partition function (or Gibbs sum). We introduce these new things, and then apply them to semiconductors, aluminum soft drink cans, and blood.
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Introducing a new thermodynamically conjugate pair of variables: number of particles and chemical potential. Internal and external chemical potential. Voltmeters measure the total chemical potential. Great class brainstorm on internal voltages in your life. How to get a theory named after yourself. Spins in a magnetic field. Why atmospheric pressure falls off with height, hiking in high altitude, and how to solve that deuterated Kool-Aid problem we talked about in Lecture 6. Lead-Acid batteries and your car.
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