# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

# Fermions summary

We’ve considered a momentum sphere as in fig. 1.1, and performed various appromations of the occupation sums fig. 1.2.

The physics of Fermi gases has an extremely wide range of applicability. Illustrating some of this range, here are some examples of Fermi temperatures (from )

- Electrons in copper:
- Neutrons in neutron star:
- Ultracold atomic gases:

# Bosons

We’d like to work with a fixed number of particles, but the calculations are hard, so we move to the grand canonical ensemble

Again, we’ll consider free particles with energy as in fig. 1.3, or

Again introducing fugacity , we have

We’ll consider systems for which

Observe that at large energies we have

For small energies

Observe that we require (or ) so that the number distribution is strictly positive for all energies. This tells us that the fugacity is a function of temperature, but there will be a point at which it must saturate. This is illustrated in fig. 1.4.

Let’s calculate this density (assumed fixed for all temperatures)

With the substitution

we find

This implicitly defines a relationship for the fugacity as a function of temperature .

It can be shown that

As we end up with a zeta function, for which we can look up the value

where the *Riemann zeta function* is defined as

At high temperatures we have

(as does down, goes up)

Looking at leads to

**How do I satisfy number conservation?**

We have a problem here since as the term in above drops to zero, yet cannot keep increasing without bounds to compensate and keep the density fixed. The way to deal with this was worked out by

- Bose (1924) for photons (examining statistics for symmetric wave functions).
- Einstein (1925) for conserved particles.

To deal with this issue, we (somewhat arbitrarily, because we need to) introduce a non-zero density for . This is an adjustment of the approximation so that we have

as in fig. 1.5, so that

Given this, we have

We can illustrate this as in fig. 1.6.

At we have , whereas at we must introduce a non-zero density if we want to be able to keep a constant density constraint.