Difference between revisions of "Inverse temperature"

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(cleaned a bit, added reference to quantum stat. mech.)
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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, such that one has a set of occupation numbers <math>n_i</math>. Introducing the  [[partition function]]:
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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>
 
:<math>\Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,</math>
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An additional constraint, which applies only to dilute gases, is:
 
An additional constraint, which applies only to dilute gases, is:
  
:<math>\sum_i n_i e_i=E</math>,
+
:<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>.
 
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
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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>,
 
  
 +
:<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
 
where the two Lagrange multipliers enforce the two conditions and permit the treatment of
 
the occupations as independent variables. The minimization leads to
 
the occupations as independent variables. The minimization leads to
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:<math>\beta := \frac{1}{k_BT} .</math>
 
:<math>\beta := \frac{1}{k_BT} .</math>
  
 
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Similar methods are used for [[quantum statistics]] of dilute gases (Ref. 1, pp. 179-185).
  
 
==References==
 
==References==
#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition, pp. 79-85  (1987)
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#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987)
 
[[category: Classical thermodynamics]]
 
[[category: Classical thermodynamics]]
 
[[category: statistical mechanics]]
 
[[category: statistical mechanics]]
 
[[category: Non-equilibrium thermodynamics]]
 
[[category: Non-equilibrium thermodynamics]]

Revision as of 13:15, 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 (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 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 entails 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

n_i=C e^{-\beta e_i},

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

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

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

References

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