Inverse temperature: Difference between revisions
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It is often convenient to define a dimensionless ''inverse'' | 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]]: | |||
Indeed, it shown in Ref. 1 that this is the way it enters. The task is to maximize number of ways | |||
:<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]] 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 | ||
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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). | |||
==References== | ==References== | ||
#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition | #Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987) ISBN 978-0-471-81518-1 | ||
[[category: Classical thermodynamics]] | [[category: Classical thermodynamics]] | ||
[[category: statistical mechanics]] | [[category: statistical mechanics]] | ||
[[category: Non-equilibrium thermodynamics]] | [[category: Non-equilibrium thermodynamics]] | ||
Latest revision as of 14:39, 5 March 2010
It is often convenient to define a dimensionless inverse temperature, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} :
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} particles may be assigned to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} space-momentum cells, such that one has a set of occupation numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_i} . Introducing the partition function:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,}
one could maximize its logarithm (a monotonous function):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_i n_i=N}
An additional constraint, which applies only to dilute gases, is:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_i n_i e_i=E, }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} is the total energy and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_i=p_i^2/2m} is the energy of cell Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} . The method of Lagrange multipliers entails finding the extremum of the function
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_i=C e^{-\beta e_i}, }
and an application to the case of an ideal gas reveals the connection with the temperature,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta := \frac{1}{k_BT} .}
Similar methods are used for quantum statistics of dilute gases (Ref. 1, pp. 179-185).
References[edit]
- Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987) ISBN 978-0-471-81518-1