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Equilibrium Statistical Mechanics

Describes systems in equilibrium, relating macroscopic properties to the microscopic behavior of particles through statistical averages and probability distributions.
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The statement of the theorem

Let H:R3N→R\mathcal{H}: \mathbb{R}^{3N} \to \mathbb{R} be the Hamiltonian of a system of NN particles. Define the inverse temperature β=1/(kBT)\beta = 1/(k_B T). The canonical ensemble probability density function ρ(q,p)\rho(\boldsymbol{q}, \boldsymbol{p}) for the phase space coordinates (q,p)∈R3N(\boldsymbol{q}, \boldsymbol{p}) \in \mathbb{R}^{3N} is given by: ρ(q,p)=e−βH(q,p)Z(β)\rho(\boldsymbol{q}, \boldsymbol{p}) = \frac{e^{-\beta \mathcal{H}(\boldsymbol{q}, \boldsymbol{p})}}{Z(\beta)}\nwhere the canonical partition function Z(β)Z(\beta) is the normalization constant: Z(β)=∫ ⁣ ⁣e−βH(q,p) d3Nqd3NpZ(\beta) = \int \!\! e^{-\beta \mathcal{H}(\boldsymbol{q}, \boldsymbol{p})} \, d^{3N} \boldsymbol{q} d^{3N} \boldsymbol{p}\nFor any observable O(q,p)\mathcal{O}(\boldsymbol{q}, \boldsymbol{p}), its ensemble average ⟨O⟩\langle \mathcal{O} \rangle is calculated as: ⟨O⟩=∫ ⁣ ⁣O(q,p)ρ(q,p) d3Nqd3Np\langle \mathcal{O} \rangle = \int \!\! \mathcal{O}(\boldsymbol{q}, \boldsymbol{p}) \rho(\boldsymbol{q}, \boldsymbol{p}) \, d^{3N} \boldsymbol{q} d^{3N} \boldsymbol{p}\nFurthermore, the Helmholtz free energy AA is related to Z(β)Z(\beta) by: A(β)=−kBTln⁡Z(β)A(\beta) = -k_B T \ln Z(\beta)\nThis framework establishes the link between the microscopic dynamics (via H\mathcal{H}) and the macroscopic thermodynamic potential AA.