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 asigned 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):

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sum _{i}n_{i}=N}

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n_{i}=Ce^{-\beta e_{i}},}

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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