Low temperature Fermi gas chemical potential
Posted by peeterjoot on April 24, 2013
Question: Low temperature Fermi gas chemical potential
 section 8.1 equation (33) provides an implicit function for
In class, we assumed that was quadratic in as a mechanism to invert this non-linear equation. Without making this quadratic assumption find the lowest order, non-constant approximation for .
To determine an approximate inversion, let’s start by multiplying eq. 1.0.2 by to non-dimensionalize things
If we are looking for an approximation in the neighborhood of , then the LHS factor is approximately one, whereas the fractional difference term is large (with a corresponding requirement for to be small. We must then have
This gives us the desired result
 RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.