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Condensed Matter Physics deals primarily with large collections of similar particles that are interacting in a strong way. These collections, called ensembles, contain many, many, many particles! For example, a single liter of water at standard temperature and pressure contains roughly 3x10²⁵ molecules!

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Statistical Mechanics

Phase Transitions

Now, if we were trying to describe the dynamics of the ensemble (the collection of the molecules in a lung-ful of air), we would write something called a Hamiltonian. A Hamiltonian is a mathematical statement that describes all the possible energy states of an ensemble. An energy state is a single possibility among several or many possible states. Basically, a single state describes the energy of the system in a particular configuration. For example, in our lung-ful of air, there is a state that describes the configuration: all nitrogen molecules on the left, all oxygen molecules on the right. That is one single, possible state of our ensemble. Granted, it's not very likely, but it is possible, if we wait for just slightly longer than the universe has already been in existence...

Remember, though, that there are 3x10²⁵ molecules in our liter of water! How many ways can so many things be arranged so that each arrangement is unique? One whole heck of a lot, that's how many! And yes, that is a technical term.

Okay, you get the idea. Our Hamiltonian would be ginormous because it would have to take into account each and every particle and how those particles interact with their neighbors, the walls of the container, etc. Now, if we pour a liter of water into Lake Michigan, which contains something like 5x10¹⁵ liters of water (1.5x10⁴¹ molecules), what chance do we have now of describing all the interactions??

Well, condensed matter physics has to deal with these sorts of problems, and it uses the tools of statistical mechanics to make an intractible problem into a solvable one.

In its essence, statistical mechanics (called "stat mech" by its students and practitioners) simplifies large systems by focusing not on the details of each and every particle, but on their collective, average behaviors. Despite the atmosphere having so many particles, we can look at a thermometer and get a value for the average energy of our local collection of atmosphere. Temperature is an average of the energies of all the particles! Other quantities also summarize, or average, various properties and states of a given ensemble.

For example, the pressure of a gas inside a box is the average force per unit area that the gas particles hit the walls of the box. Some hit the walls with great force, some with slight force. If we were able to record all the "hits" onto the walls of the box, and measure each force of impact, we could add them all up, divide by the number of hits, divide that by the area of the walls, and voila, we'd have the pressure! Or, we could stick a thermometer in the box, get the temperature, and then calculate the pressure in a single step! P = N*R*T / V, where N is the number of molecules, R is a constant, T is the temperature, and V is the volume of the box. Such is the power of statistical mechanics, for it allows us to find things out about an ensemble without going to all the difficulty of measuring something about each and every member of the ensemble. Thus, its name statistical mechanics.

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Ice melting into water... Frozen CO₂ sublimating into gas... Morning dew condesing on your lawn... All these things represent phase transitions, which are, well, transitions from one phase to another phase. Typically, we think of phase transitions as stuff melting or freezing or becoming vapor, but there are an awful lot of other interpretations, too.

For example, a commonly studied system in this area of physics is the ferromagnet (a material like iron that becomes magnetic when placed near another, real magnet). Ferromagnets undergo phase transitions in the sense that in one phase (an ordered phase), the magnetic moments of the individual atoms are all aligned with the external magnetic field. Each individual moment prefers to be in the same direction as its neighbors. As you raise the temperature of the ferromagnet, more and more of the magnetic moments flip at random rather than obey the external field. At some point, you reach a temperature that is high enough to cause the ferromagnet to be in a disordered phase. In other words, you may as well take away the external magnet, because the ferromagnet no longer cares that it's there! Its individual atoms have so much wiggling energy from the enhanced temperature, that they flip back and forth randomly.

So, there are experimental methods to locating phase transitions, and there are theoretical methods. Something they both have in common is the notion of critical points. In the above example, the temperature at which the ferromagnet changes from an ordered to a disordered phase is called the critical temperature. Other parameters of a system can also be critical points. For example, water has critical points in temperature and pressure that define its current phase (or, its coexisting phases if conditions are just right - in fact, there is a point called the tricritical point at which all three phases of water, liquid, solid, and gas all exist simultaneously!).

To define critical points theoretically, we usually look first to a particular function of our system called the Free Energy. Once we establish an equation that defines the free energy, then critical points can be predicted by finding locations where the free energy or its derivatives are discontinuous. Discontinuous is a fancy way of saying that the graph is not smooth, but has a kink or break in it, as in the picture below.

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