Canonical ensemble

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  • Number of Particles,  N
  • Volume,  V

Partition Function[edit]

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)


  •  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

See also[edit]