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− | It is often convenient to define a dimensionless | + | 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 | + | |

+ | 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> | ||

+ | 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, | ||

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:<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). | ||

+ | |||

==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 |