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 (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

$n_{i}=Ce^{-\beta e_{i}},$

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

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

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