PHZ3400-09 Emergence in Condensed Matter

“ The whole is more than the sum of its parts ”

Do we have to watch every atom?

Are we concerned about the macroscopic or the microscopic? This is a question of granularity.

This is a question of granularity and the answer to it depends on the context of the problem. If one is looking at the motion of a soccer ball as a whole then the answer is no. In this case, we are comprising our system of large components, mainly the soccer ball. The particles of air inside the ball move randomly, but there is no point in keeping track of every air particle in the ball, if all we care about is the overall motion of the ball. This is an example of coarse-graining, where only large particles comprise the system of interest. In condensed matter physics, scientists often begin with a microscopic system and use coarse graining to achieve a description of a macroscopic system.

However, if one wanted to know the air pressure inside the ball, knowing properties of the individual particles is probably more convenient. This is an example of fine-grained system.

When deciding on using a coarse-grained or fine-grained system, one should decide if the property desired is macroscopic or microscopic.

Defining Macroscopic Bodies

A macroscopic body is made up of many microscopic bodies usually on the order of at least one mole. The amount of particles in one mole is given by Avogadro's number, on the order of $\mathbb {N} _{A}=10^{23}$. Were every particle to be watched, there would be something on the order of $e^{10^{23}}$ states to solve for. A significantly high number capable of being too much for even the best computers.

Rather than doing that, most of the time only the center of mass is needed to draw a complete picture about the object. The Center of Mass is found using the following expression:

$m_{CM}={\dfrac {\sum _{i}^{N}m_{i}R_{i}}{\sum _{i}^{N}m_{i}}}$

Center of mass is an example of a collective co-ordinate in which all of the particles are treated as one.

Many phases of matter

Phase diagram used in a chocolate factory (do you believe it?).

Snow Flake.

Carbon Phase Diagram.

Carbon_phase_diagram.jpg

Matter comes in many forms macroscopically. Why does it matter if it's macroscopic (large) or not? Water comes in solid (like the snowflake shown), liquid, and gaseous forms. What governs these transitions for water is temperature. To compound this phenomenon further, compounds, such as carbon chains, can have different types of the same state, which for solid carbon includes graphite, coal, and diamonds. These different forms of the same state are known as allotropes. The pressure-temperature graph to the right shows that these different types of solid carbon are formed at different pressures. Another method of phase shifting can be seen in the vanilla-chocolate phase diagram. As one can see the percent of chocolate in the vanilla-chocolate mixture also can cause a phase shift. Though this example appears simplistic, compounds do exhibit property changes as the proportions of the constituent elements change.

How important is Quantum Mechanics?

The cause of these phase changes is an interesting question. Can these phase changes be found by classical or quantum mechanical means?

For example, why don't we suffer the fate of Rumpelstinskin (drop through the floor)? This is because objects are solid. But is solidity a quantum or classical phenomena? Solidity is most (are we sure or it's a guess??) likely a macroscopic property as solids contain many particles. But classical physics gives no reason for objects to be solid other than molecular vibrations from the temperature of the solid. So ideally one could manipulate the temperature and affect the solidity of the floor. Except for the extreme case where the temperature is brought to the level where the floor dissociates, the temperature does not affect solidity. However, quantum mechanics can solve this problem. Heisenberg's uncertainty principle forces the electrons of the floor to be discrete values. This also means that the orbitals are incompressible unless the electrons can orbit in the same energy level (that is, possess non distinct quantum numbers). The Pauli Exclusion principle and Fermi-Dirac statistics show that this is also not a possibility, because electrons are fermions and therefore cannot occupy the same quantum state simultaneously. So despite solidity being macroscopically observed, it is quantum mechanics that keeps solids from passing through each other.

Phase diagram of high temperature superconductors: chemically doped ceramics (magnetic insulators) to introduce mobile charge carriers.