Difference between revisions of "Inverse temperature"

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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, \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, such that one has a set of occupation numbers n_i. 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. The maximization must be performed subject to the constraint:

\sum_i n_i=N

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

\sum_i n_i e_i=E,

where E is the total energy and e_i=p_i^2/2m is the energy of cell i.

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

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

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

\beta := \frac{1}{k_BT} .


References

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