Statistical Mechanics/Quantum Gases

As per the spin-statistics theorem, there are really only two types of particles: fermions and bosons. Thus if we can study each one separately fairly well, then we know quite a bit!

We will be using the grand partition function 'Infrastructure' developed earlier for these models (in order to make it even more general). Specifically, we can use the grand partition function to create distribution functions, and use those to study specific properties of fermi/bose gases.

The Distribution Functions

Fermi Gas

Using our earlier definition:

f(ε) := <N(ε)>

Now, for a given orbital (i.e., for a given ε), there can only be two options with a fermion, either there's one particle or none. Therefore, we can use our previous defined method to calculate an average quantity in order to calculate <N(ε)> (we will do the sum by brute force, it's not too hard, only two terms at most):

f(ε)=1/(exp((ε-μ)/kT) + 1)

Bose Gas

The process is the same, only instead of none or one, we have 0 to infinite. If we use some sum relations we come up with:

f(ε)=1/(exp((ε-μ)/kT) - 1)

Which is surprisingly similar to the fermi gas case, but the minus sign causes EXTREME differences in behavior compared to the fermi gas.

The Classical Limit

Despite overall behavior of the fermi and bose gases, in a certain limit, we see that the two distributions are nearly similar. We call this limit 'The Classical Limit'. In particular, this limit is:

exp((ε-μ)/kT) >> 1

The only importance of this limit is to determine when we don't have to worry about the two gases distinct behaviors.

Fermi Energy

Because of the 0 or 1 limit, at very cold temperatures (note that cold is a fairly comparative term, for the free electrons on a sheet of metals, this can be up to hundred of degrees farenheight! Which shows that this odd quantum stuff does affect your daily life) fermi gases can only get 'locked down' to so many orbitals. They can't all fall into the ground state. The energy at which the gas is 'locked in' is known as the fermi energy.

In terms of the distribution function, this means at this 'cold' temperature, the function drops off sharply at one point.

Bose-Einstein Condensation

Unlike the fermi energy occurrence with a fermi gas, bosons can condense all they want, this is where the minus sign comes into play. As the energy level is lower and lower, eventually they might pass the chemical potential, which causes the exponential in the bose distribution function to be less than one, and thus the function to be negative, i.e., an impossible value. In fact, as ε approaches μ, the function approaches a singularity. Physically, this means that all the bosons can 'condense' into the same low energy spot at low temperatures: a phenomena known as boson condensation.