Inverse temperature

It is often convenient to define a dimensionless inverse temperature, $\beta$:

$\beta :={\frac {1}{k_{B}T}}$

This notation likely comes from its origin as a Lagrangian multiplier, for which Greek letters are customarily written.

Indeed, it shown in Ref. 1 that this is the way it enters. The task is to maximize number of ways $N$ particles may be asigned to $K$ space-momentum cells. Introducing the partition function:

$\Omega \propto {\frac {N!}{n_{1}!n_{2}!\ldots n_{K}!}},$

one could maximize its logarithm (a monotonous function):

$\log \Omega \approx \log N-N-\sum _{i}(\log n_{i}+n_{i})+\mathrm {consts} ,$

where Stirling's approximation for large numbers has been used.

References

1. Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition, pp. 79-85 (1987)