Equipartition Theorem

Motivation

Consider a system in thermal equilibrium with the constraint

\sum_{i} p_{i} E_{i} = C,

the energy distribution is given by the Boltzmann distribution

p_{i} = \frac{e^{- β E_{i}}}{\sum_{j = 1}^{n} e^{- β E_{j}}}

Defining

Z (β) = \sum_{j = 1}^{n} e^{- β E_{j}},

then the expected energy is

⟨ E ⟩ (β) = \sum_{i = 1}^{n} \frac{E_{i} e^{- β E_{i}}}{Z (β)} = - \frac{\partial}{\partial β} \ln Z (β) .

You then determine $β$ by solving

- \frac{\partial}{\partial β} \ln Z (β) = C .

In most cases this equation

C = \frac{\sum_{i} E_{i} e^{- β E_{i}}}{\sum_{i} e^{- β E_{i}}}

has a unique solution $β = β (C)$ (because $⟨ E ⟩ (β)$ is a strictly decreasing function of $β$ ), so the temperature $T = 1 / (k β)$ is completely fixed by the desired $⟨ E ⟩ = C$ . Thus there is indeed a one‐to‐one correspondence between the coolness $β$ (hence the temperature) and the imposed mean energy $C$ .

A particularly simple correspondence happens for special cases:

Equipartition theorem

If a system at temperature $T$ has a Hamiltonian in which each independent degree of freedom appears only quadratically,

H = \sum_{i = 1}^{N} \frac{1}{2} a_{i} x_{i}^{2},

then at thermal equilibrium the average energy per quadratic degree of freedom is

⟨ \frac{1}{2} a_{i} x_{i}^{2} ⟩ = \frac{1}{2} k T,

and hence

⟨ E ⟩ = \frac{1}{2} N k T .

Clarification on degrees of freedom

Degree of freedom: one independent generalized coordinate $q_{i}$ needed to specify the system’s configuration. See configuration space.
Here we call quadratic degree of freedom to each independent squared variable in the energy (each appearance of a term like $q_{i}^{2}$ or $p_{i}^{2}$ ).
Example: A 1D harmonic oscillator has one degree of freedom $q$ , but its phase space is $R^{2}$ with coordinates $(q, p)$ . The Hamiltonian

H (q, p) = \frac{p^{2}}{2 m} + \frac{1}{2} k q^{2}

is a quadratic form on $R^{2}$ , so equipartition counts two quadratic terms and gives

⟨ H ⟩ = 2 \times \frac{1}{2} k T = k T .

Applicability

It holds for classical systems in the canonical ensemble.
A "degree of freedom" here must appear quadratically in the Hamiltonian (e.g. $\frac{1}{2} m v^{2}$ , $\frac{1}{2} k x^{2}$ ) for equipartition to apply.
Does not generally hold in the quantum regime, especially at low temperatures.

Examples

A 3D ideal gas particle has 3 degrees of freedom and 3 quadratic kinetic terms, giving mean energy $\frac{3}{2} k T$ . See gas in a box#The Physical Meaning of Temperature.
A classical harmonic oscillator (1D) has 1 configurational DoF but 2 quadratic terms (kinetic + potential), giving mean energy $k T$ .

Limitations

Fails when quantum effects are significant (e.g., low $T$ or discrete energy levels).
Does not apply to non-quadratic or constrained degrees of freedom.