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Equipartition Theorem

States that energy is equally distributed among all available degrees of freedom of a system in thermal equilibrium, a cornerstone of kinetic theory.
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The statement of the theorem

Let H(θ)\mathcal{H}(\boldsymbol{\theta}) be the Hamiltonian of a system with generalized coordinates θ=(θ1,…,θN)\boldsymbol{\theta} = (\theta_1, \dots, \theta_N) in thermal equilibrium at temperature TT. Assume the contribution of the ii-th degree of freedom to the Hamiltonian is quadratic, Hi=12kiθi2H_i = \frac{1}{2} k_i \theta_i^2. The canonical partition function is Z=∫e−H(θ)/kBTdθZ = \int e^{-\mathcal{H}(\boldsymbol{\theta})}/k_B T \text{d}\boldsymbol{\theta}. The average energy ⟨Ei⟩\langle E_i \rangle associated with this degree of freedom is given by the expectation value: ⟨Ei⟩=1Z∫e−H(θ)/kBT(∂H∂θiθi)dθ\langle E_i \rangle = \frac{1}{Z} \int e^{-\mathcal{H}(\boldsymbol{\theta})}/k_B T \left( \frac{\partial \mathcal{H}}{\partial \theta_i} \theta_i \right) \text{d}\boldsymbol{\theta}. Under the assumption that the system is ergodic and the Hamiltonian is separable into quadratic terms, the theorem states that for every such degree of freedom ii: ⟨Ei⟩=12kBT \langle E_i \rangle = \frac{1}{2} k_B T
Source: Wikipedia