Equipartition Theorem - General Formulation of The Equipartition Theorem | General Formulation Equipartition Theorem

General Formulation of The Equipartition Theorem

See also: Generalized coordinates, Hamiltonian mechanics, Microcanonical ensemble, and Canonical ensemble

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_n, the following equipartition formula holds in thermal equilibrium for all indices m and n:

$\! \Bigl\langle x_{m} \frac{\partial H}{\partial x_{n}} \Bigr\rangle = \delta_{mn} k_{B} T.$

Here δ_mn is the Kronecker delta, which is equal to one if m = n and is zero otherwise. The averaging brackets is assumed to be an ensemble average over phase space or, under an assumption of ergodicity, a time average of a single system.

The general equipartition theorem holds in both the microcanonical ensemble, when the total energy of the system is constant, and also in the canonical ensemble, when the system is coupled to a heat bath with which it can exchange energy. Derivations of the general formula are given later in the article.

The general formula is equivalent to the following two:

If a degree of freedom x_n appears only as a quadratic term a_nx_n2 in the Hamiltonian H, then the first of these formulae implies that

$k_{B} T = \Bigl\langle x_{n} \frac{\partial H}{\partial x_{n}}\Bigr\rangle = 2\langle a_n x_n^2 \rangle,$

which is twice the contribution that this degree of freedom makes to the average energy . Thus the equipartition theorem for systems with quadratic energies follows easily from the general formula. A similar argument, with 2 replaced by s, applies to energies of the form a_nx_ns.

The degrees of freedom x_n are coordinates on the phase space of the system and are therefore commonly subdivided into generalized position coordinates q_k and generalized momentum coordinates p_k, where p_k is the conjugate momentum to q_k. In this situation, formula 1 means that for all k,

$\Bigl\langle p_{k} \frac{\partial H}{\partial p_{k}} \Bigr\rangle = \Bigl\langle q_{k} \frac{\partial H}{\partial q_{k}} \Bigr\rangle = k_{\rm B} T.$

Using the equations of Hamiltonian mechanics, these formulae may also be written

$\Bigl\langle p_{k} \frac{dq_{k}}{dt} \Bigr\rangle = -\Bigl\langle q_{k} \frac{dp_{k}}{dt} \Bigr\rangle = k_{\rm B} T.$

Similarly, one can show using formula 2 that

$\Bigl\langle q_{j} \frac{\partial H}{\partial p_{k}} \Bigr\rangle = \Bigl\langle p_{j} \frac{\partial H}{\partial q_{k}} \Bigr\rangle = 0 \quad \mbox{ for all } \, j,k$

and

$\Bigl\langle q_{j} \frac{\partial H}{\partial q_{k}} \Bigr\rangle = \Bigl\langle p_{j} \frac{\partial H}{\partial p_{k}} \Bigr\rangle = 0 \quad \mbox{ for all } \, j \neq k.$

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