Editing Inverse temperature
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 1: | Line 1: | ||
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 | |||
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]]: | |||
:<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> | ||
Line 12: | Line 13: | ||
:<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. | ||
==References== | ==References== | ||
#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987) | #Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition, pp. 79-85 (1987) | ||
[[category: Classical thermodynamics]] | [[category: Classical thermodynamics]] | ||
[[category: statistical mechanics]] | [[category: statistical mechanics]] | ||
[[category: Non-equilibrium thermodynamics]] | [[category: Non-equilibrium thermodynamics]] |