Inverse temperature: Difference between revisions

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This notation likely comes from its origin as a Lagrangian multiplier, for which Greek letters are customarily written.
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]]:
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, 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>
:<math>\Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,</math>
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:<math>\log \Omega \approx \log N -N - \sum_ i ( \log n_i + n_i) + \mathrm{consts} ,</math>
:<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.
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]] entail 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=Ce^{-\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>
 
 


==References==
==References==

Revision as of 12:04, 4 March 2010

It is often convenient to define a dimensionless inverse temperature, :

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, such that one has a set of occupation numbers . Introducing the partition function:

one could maximize its logarithm (a monotonous function):

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

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

,

where is the total energy and is the energy of cell .

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

,

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 (syntax error): {\displaystyle n_i=Ce^{-\beta e_i), }

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


References

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