Content deleted Content added
m Added = <E^2> - <E>^2 = to the variance. This makes the relationship to the second order partial derivative on the right a little easier to derive. |
|||
Line 108:
For a gas of <math> N </math> identical classical particles in three dimensions, the partition function is
<math display="block"> Z=\frac{1}{N!h^{3N}} \int \, \exp \left(-\beta \sum_{i=1}^N H(\textbf q_i, \textbf p_i) \right) \; \mathrm{d}^3 q_1 \cdots \mathrm{d}^3 q_N \, \mathrm{d}^3 p_1 \cdots \mathrm{d}^3 p_N = \frac{Z_{\text{single}}^N}{N!}</math>
where
* <math> h </math> is the [[Planck constant]];
Line 117:
* <math> p_i </math> is the [[Canonical coordinates|canonical momentum]] of the respective particle;
* <math> \mathrm{d}^3 </math> is shorthand notation to indicate that <math> q_i </math> and <math> p_i </math> are vectors in three-dimensional space.
* <math> Z_{\text{single}} </math> is the classical continuous partition function of a single particle as given in the previous section.
The reason for the [[factorial]] factor ''N''! is discussed [[#Partition functions of subsystems|below]]. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not [[dimensionless]]. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by ''h''<sup>3''N''</sup> (where ''h'' is usually taken to be Planck's constant).
| |||