Inverse temperature

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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 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:

\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.


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