# Inverse temperature

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

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

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 ${\displaystyle N}$ particles may be assigned to ${\displaystyle K}$ space-momentum cells, such that one has a set of occupation numbers ${\displaystyle n_{i}}$. Introducing the partition function:

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

one could maximize its logarithm (a monotonous function):

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

where Stirling's approximation for large numbers has been used. The maximization must be performed subject to the constraint:

${\displaystyle \sum _{i}n_{i}=N}$

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

${\displaystyle \sum _{i}n_{i}e_{i}=E,}$

where ${\displaystyle E}$ is the total energy and ${\displaystyle e_{i}=p_{i}^{2}/2m}$ is the energy of cell ${\displaystyle i}$. The method of Lagrange multipliers entails finding the extremum of the function

${\displaystyle L=\log \Omega -\alpha (\sum _{i}n_{i}-N)-\beta (\sum _{i}n_{i}e_{i}-E),}$

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

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

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

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

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