MathLiveEquipartition theorem
The equipartition theorem gives the average energy carried by each quadratic degree of freedom of a classical system in thermal equilibrium, which is one half k_B T per quadratic term. The form shown relates averages of phase-space coordinates and derivatives of the Hamiltonian to temperature; it breaks down for quantum systems at low temperatures, where some degrees of freedom freeze out.
\left\langle x_m \frac{\partial H}{\partial x_n} \right\rangle = \delta_{mn} k_{\mathrm{B}} TWhat the symbols mean
| ensemble or time average over the system in thermal equilibrium | |
| generalized phase-space coordinates (positions or momenta) | |
| Hamiltonian (total energy) of the system, in joules | |
| Kronecker delta, equal to 1 if m = n and 0 otherwise | |
| Boltzmann constant, in joules per kelvin | |
| absolute temperature, in kelvin |
About this formula
- FieldPhysics
- TopicStatistical mechanics
- LevelUniversity
- Also calledlaw of equipartition of energy, equipartition of energy, equipartition, equipartition theorem thermodynamics, average energy per degree of freedom
- SourceWikidata Q255996
Write it in Obsidian
MathLive is a free visual math editor for Obsidian. Run Find a formula, type “equipartition theorem”, and it goes into your note as editable LaTeX. You can also scan formulas from screenshots, PDFs and handwriting.