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Kinetic Energy Equation

Relates the average kinetic energy of particles to the absolute temperature of the system, a fundamental equation in kinetic theory.
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The statement of the theorem

Let H(p,q)\mathcal{H}(\mathbf{p}, \mathbf{q}) be the Hamiltonian of a system of NN particles in a volume VV, where p\mathbf{p} and q\mathbf{q} are the canonical momentum and position vectors, respectively. The average kinetic energy ⟨K⟩\langle K \rangle is defined by the expectation value over the phase space Γ\Gamma: ⟨K⟩=1Z∫e−βH(p,q)(∑i=1Npi22m)dΓ\langle K \rangle = \frac{1}{Z} \int e^{-\beta \mathcal{H}(\mathbf{p}, \mathbf{q})} \left( \sum_{i=1}^{N} \frac{\mathbf{p}_i^2}{2m} \right) d\Gamma. For a system where the potential energy U(q)U(\mathbf{q}) is independent of momentum, the partition function ZZ is separable, and the average kinetic energy is given by ⟨K⟩=32NkBT\langle K \rangle = \frac{3}{2} N k_B T. Alternatively, using the equipartition theorem, for each quadratic degree of freedom xix_i, the average energy is ⟨12kBT∂2H∂xi2⟩=12kBT\langle \frac{1}{2} k_B T \frac{\partial^2 \mathcal{H}}{\partial x_i^2} \rangle = \frac{1}{2} k_B T. Thus, the total average kinetic energy is ⟨K⟩=12kBT∑i=13N∂2H∂xi21kBT⋅32NkBT\langle K \rangle = \frac{1}{2} k_B T \sum_{i=1}^{3N} \frac{\partial^2 \mathcal{H}}{\partial x_i^2} \frac{1}{k_B T} \cdot \frac{3}{2} N k_B T.
Source: Wikipedia