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It is often convenient to define a dimensionless '''inverse temperature''', <math>\beta</math>:
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It is often convenient to define a dimensionless ''inverse'' temperature, <math>\beta</math>:
  
 
:<math>\beta := \frac{1}{k_BT}</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.
 
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 <math>N</math> particles may be assigned to <math>K</math> 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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:<math>\sum_i n_i e_i=E, </math>
 
:<math>\sum_i n_i e_i=E, </math>
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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]] entails finding the extremum of the function
 
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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and an application to the case of an ideal gas reveals the connection with the temperature,
 
and an application to the case of an ideal gas reveals the connection with the temperature,
 
 
:<math>\beta := \frac{1}{k_BT} .</math>
 
:<math>\beta := \frac{1}{k_BT} .</math>
  
 
Similar methods are used for [[quantum statistics]] of dilute gases (Ref. 1, pp. 179-185).
 
Similar methods are used for [[quantum statistics]] of dilute gases (Ref. 1, pp. 179-185).
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==References==
 
==References==
 
#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987) ISBN 978-0-471-81518-1
 
#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987) ISBN 978-0-471-81518-1

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