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General formulation of the equipartition theorem

The most general form of the equipartition theorem states that under suitable assumptions (discussed below), for a physical system with Hamiltonian energy function H and degrees of freedom x, the following equipartition formula holds in thermal equilibrium for all indices m and n: ⟨ x m ∂ H ∂ x n ⟩ = δ m n k B T. {\displaystyle \left\langle x_{m}{\frac {\partial H}{\partial x_{n}}}\right\rangle =\delta _{mn}k_{\text{B}}T.} Here δ is the Kronecker delta, which is equal to one if m = n and is zero otherwise. The averaging brackets ⟨ … ⟩ {\displaystyle \left\langle \ldots \right\rangle } is assumed to be an ensemble average over phase space or, under an assumption of ergodicity, a time average of a single system.