Inverse temperature

It is often convenient to define a dimensionless inverse temperature, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 asigned to $K$ space-momentum cells, such that one has a set of occupation numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_i} . Introducing the partition function:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,}

one could maximize its logarithm (a monotonous function):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_i n_i=N}

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

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

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} is the total energy and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_i=p_i^2/2m} is the energy of cell Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} .

The method of Lagrange multipliers entails finding the extremum of the function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_i=C e^{-\beta e_i}, }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta := \frac{1}{k_BT} .}

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