Editing Inverse temperature

It is often convenient to define a dimensionless ''inverse'' temperature, <math>\beta</math>:

:<math>\beta := \frac{1}{k_BT}</math>

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 <math>n_i</math>. Introducing the  [[partition function]]:

:<math>\Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,</math>

one could maximize its logarithm (a monotonous function):

:<math>\log \Omega \approx \log N -N - \sum_ i ( \log n_i + n_i) + \mathrm{consts} ,</math>

where [[Stirling's approximation]] for large numbers has been used. The maximization must be performed subject to the constraint:

:<math>\sum_i n_i=N</math>

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

:<math>\sum_i n_i e_i=E, </math>
where <math>E</math> is the total energy and <math>e_i=p_i^2/2m</math> is the energy of cell <math>i</math>.

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

:<math>L=\log\Omega - \alpha (\sum_i n_i - N ) - \beta ( \sum_i n_i e_i - E  ),</math>
where the two Lagrange multipliers enforce the two conditions and permit the treatment of
the occupations as independent variables. The minimization leads to

:<math>n_i=C e^{-\beta e_i}, </math>

and an application to the case of an ideal gas reveals the connection with the temperature,
:<math>\beta := \frac{1}{k_BT} .</math>

Similar methods are used for [[quantum statistics]] of dilute gases (Ref. 1, pp. 179-185).

==References==
#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987)
[[category: Classical thermodynamics]]
[[category: statistical mechanics]]
[[category: Non-equilibrium thermodynamics]]