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 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. 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.
References
- Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition, pp. 79-85 (1987)