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Partition Function

A central quantity in statistical mechanics, representing the sum of Boltzmann factors over all possible states of a system.
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The statement of the theorem

Let H\mathcal{H} be the Hilbert space describing the system's quantum states, and let H^:H→R\hat{H}: \mathcal{H} \to \mathbb{R} be the Hamiltonian operator. Define the inverse temperature β=1/(kBT)\beta = 1/(k_B T). The canonical partition function, ZZ, is rigorously defined as the trace of the thermal density operator ρ^=e−βH^\hat{\rho} = e^{-\beta \hat{H}}: \n\nZ=Tr(e−βH^)Z = \text{Tr}(e^{-\beta \hat{H}}) \n\nIf the system is classical and the phase space is Γ\Gamma, the partition function is given by the integral over the phase space, weighted by the Boltzmann factor and the phase space volume element dΓd\Gamma: \n\nZ=1h3NN!∫Γe−βH(q,p)dqdpZ = \frac{1}{h^{3N} N!} \int_{\Gamma} e^{-\beta \mathcal{H}(\mathbf{q}, \mathbf{p})} d\mathbf{q} d\mathbf{p} \n\nwhere H(q,p)\mathcal{H}(\mathbf{q}, \mathbf{p}) is the classical Hamiltonian, and hh is Planck's constant, ensuring proper normalization for quantum-to-classical correspondence.
Source: Wikipedia