The generalized partition function

The generalized partition function

A partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables such as the temperature and volume. Each partition function is constructed to represent a particulr statistical ensemble. We can derive a generalized form for the partition function by requiring that, if each of the microstates in phase space has some set of properties $\{b^{(j)}\}_i$, (a volume, energy, number of atoms, etc) the ensemble average for each of these properties $\langle B^{(j)} \rangle$ should be finite. In other words: \[ \sum_i p_i b^{(j)}_i = \langle B^{(j)} \rangle \] We can then use Lagrange's method of undetermined multipliers to derive the probability of being in a microstate as: \[ p_i = \frac{e^{-\sum_j \lambda^{(j)} b^{(j)}_i}}{Z} \] where $Z$ is the partition function, which is given by: \[ Z = = \sum_i e^{-\sum_j \lambda^{(j)} b^{(j)}_i} \]

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