Canonical ensemble
From SklogWiki
Variables:
- Number of Particles, \( N \)
- Volume, \( V \)
- Temperature, \( T \)
[edit] Partition Function
The partition function, \(Q\), for a system of \(N\) identical particles each of mass \(m\) is given by
\[Q_{NVT}=\frac{1}{N!h^{3N}}\iint d{\mathbf p}^N d{\mathbf r}^N \exp \left[ - \frac{H({\mathbf p}^N,{\mathbf r}^N)}{k_B T}\right]\]
where \(h\) is Planck's constant, \(T\) is the temperature, \(k_B\) is the Boltzmann constant and \(H(p^N, r^N)\) is the Hamiltonian corresponding to the total energy of the system. For a classical one-component system in a three-dimensional space, \( Q_{NVT} \), is given by:
\[ Q_{NVT} = \frac{V^N}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] ~~~~~~~~~~ \left( \frac{V}{N\Lambda^3} \gg 1 \right) \]
where:
- \( \Lambda \) is the de Broglie thermal wavelength (depends on the temperature)
- \( \beta := \frac{1}{k_B T} \), with \( k_B \) being the Boltzmann constant, and T the temperature.
- \( U \) is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
- \( \left( R^*\right)^{3N} \) represent the 3N position coordinates of the particles (reduced with the system size): i.e. \( \int d (R^*)^{3N} = 1 \)
