# Inverse temperature

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

$\beta := \frac{1}{k_BT}$

This notation likely comes from its origin as a Lagrangian multiplier, for which Greek letters are customarily written. Indeed, it shown in Ref. 1 (pp. 79-85) that this is the way it enters. The task is to maximize number of ways $N$ particles may be assigned to $K$ space-momentum cells, such that one has a set of occupation numbers $n_i$. 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. The maximization must be performed subject to the constraint:

$\sum_i n_i=N$

An additional constraint, which applies only to dilute gases, is:

$\sum_i n_i e_i=E,$

where $E$ is the total energy and $e_i=p_i^2/2m$ is the energy of cell $i$. The method of Lagrange multipliers entails finding the extremum of the function

$L=\log\Omega - \alpha (\sum_i n_i - N ) - \beta ( \sum_i n_i e_i - E ),$

where the two Lagrange multipliers enforce the two conditions and permit the treatment of the occupations as independent variables. The minimization leads to

$n_i=C e^{-\beta e_i},$

and an application to the case of an ideal gas reveals the connection with the temperature,

$\beta := \frac{1}{k_BT} .$

Similar methods are used for quantum statistics of dilute gases (Ref. 1, pp. 179-185).

## References

1. Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987) ISBN 978-0-471-81518-1