## PHY452H1S Basic Statistical Mechanics. Lecture 18: Fermi gas thermodynamics. Taught by Prof. Arun Paramekanti

Posted by peeterjoot on March 26, 2013

# Disclaimer

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

**Review**

Last time we found that the low temperature behaviour or the chemical potential was quadratic as in fig. 1.1.

**Specific heat**

where

**Low temperature **

The only change in the distribution fig. 1.2, that is of interest is over the step portion of the distribution, and over this range of interest is approximately constant as in fig. 1.3.

so that

Here we’ve made a change of variables , so that we have near cancelation of the factor

Here we’ve extended the integration range to since this doesn’t change much. FIXME: justify this to myself? Taking derivatives with respect to temperature we have

With , we have for

Using eq. 1.1.4 at the Fermi energy and

we have

Giving

or

This is illustrated in fig. 1.4.

**Relativisitic gas**

- Relativisitic gas
- graphene
- massless Dirac Fermion
We can think of this state distribution in a condensed matter view, where we can have a hole to electron state transition by supplying energy to the system (i.e. shining light on the substrate). This can also be thought of in a relativisitic particle view where the same state transition can be thought of as a positron electron pair transition. A round trip transition will have to supply energy like as illustrated in fig. 1.6.

**Graphene**

Consider graphene, a 2D system. We want to determine the density of states ,

We’ll find a density of states distribution like fig. 1.7.

so that

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