# 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, :

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): n_i=Ce^{-\beta e_i),**

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

## References

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