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:

$\mathrm{\Omega }\propto \frac{N!}{n_1!n_2!\dots n_K!},$

one could maximize its logarithm (a monotonous function):

$\log \mathrm{\Omega }\approx \log N-N-{\sum }_i(\log n_i+n_i)+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{s},$

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

$L=\log \mathrm{\Omega }-\alpha ({\sum }_i{n}_i-N)-\beta ({\sum }_i{n}_i{e}_i-E),$

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,

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

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