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Canonical Ensemble

Applies to systems in thermal equilibrium with a heat bath, where energy, volume, and number of particles are fixed.
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The statement of the theorem

Let H:Phase Space→R\mathcal{H}: \text{Phase Space} \to \mathbb{R} be the Hamiltonian of the system. Define the inverse temperature β=1/(kBT)\beta = 1/(k_B T). The Canonical Partition Function ZZ for a system with fixed volume VV and particle number NN is given by the integral over the phase space Γ\Gamma: Z(β,V,N)=∫Γe−βH(p,q)dpdqZ(\beta, V, N) = \int_{\Gamma} e^{-\beta \mathcal{H}(\mathbf{p}, \mathbf{q})} d\boldsymbol{p} d\boldsymbol{q} where dpdqd\boldsymbol{p} d\boldsymbol{q} is the measure on the phase space. The probability density ρ(p,q)\rho(\mathbf{p}, \mathbf{q}) of finding the system in a microstate (p,q)(\mathbf{p}, \mathbf{q}) is the Boltzmann distribution: ρ(p,q)=e−βH(p,q)Z(β,V,N)\rho(\mathbf{p}, \mathbf{q}) = \frac{e^{-\beta \mathcal{H}(\mathbf{p}, \mathbf{q})}}{Z(\beta, V, N)} Furthermore, the Helmholtz Free Energy AA is derived from the partition function via the Legendre transform relationship: A(β,V,N)=−kBTln⁡Z(β,V,N)A(\beta, V, N) = -k_B T \ln Z(\beta, V, N) This ensemble characterizes the statistical mechanical description of systems governed by the fixed parameters (β,V,N)(\beta, V, N).
Source: Wikipedia