## Low temperature Fermi gas chemical potential

Posted by peeterjoot on April 24, 2013

## Question: Low temperature Fermi gas chemical potential

[1] section 8.1 equation (33) provides an implicit function for

or

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 .

## Answer

To determine an approximate inversion, let’s start by multiplying eq. 1.0.2 by to non-dimensionalize things

or

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

or

This gives us the desired result

# References

[1] RK Pathria. *Statistical mechanics*. Butterworth Heinemann, Oxford, UK, 1996.

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