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