equipartition theorem

E57415

physical law result in classical thermodynamics theorem in statistical mechanics

The equipartition theorem is a principle in classical statistical mechanics stating that, at thermal equilibrium, each independent quadratic degree of freedom of a system contributes an average energy of (1/2)kT.

Statements (51)

Predicate	Object
instanceOf	physical law → result in classical thermodynamics → theorem in statistical mechanics →
appliesTo	classical systems → harmonic oscillator → ideal gas → molecules in a gas → solids in the classical limit → systems in thermal equilibrium →
assumes	classical limit → continuous energy spectrum → thermal equilibrium → validity of classical mechanics →
concerns	average energy → distribution of energy among degrees of freedom → thermal equilibrium properties →
contrastsWith	quantum statistics →
energyPerDegreeOfFreedom	(1/2)kT →
field	classical mechanics → statistical mechanics → thermodynamics →
hasLimitation	does not hold when energy levels are quantized and not thermally accessible → fails at low temperatures due to quantum effects → overestimates heat capacities of diatomic gases at room temperature → overestimates heat capacity of solids at low temperature →
historicallyAttributedTo	James Clerk Maxwell → Ludwig Boltzmann →
implies	Dulong–Petit law for molar heat capacity of many solids at high temperature → average translational kinetic energy per particle is (3/2)kT in three dimensions → each quadratic term in the Hamiltonian contributes (1/2)kT to the mean energy → heat capacity of a monatomic ideal gas is (3/2)Nk → internal energy of a monatomic ideal gas is (3/2)NkT →
mathematicallyBasedOn	canonical ensemble average → classical Hamiltonian mechanics → phase space integrals →
relatedTo	Boltzmann distribution → Dulong–Petit law → Maxwell–Boltzmann statistics → canonical ensemble → kinetic theory of gases → partition function → specific heat of gases →
statedAs	each independent quadratic degree of freedom contributes an average energy of (1/2)kT →
usedFor	approximating high-temperature behavior of solids → deriving ideal gas internal energy → estimating heat capacities → understanding molecular motion →
usesQuantity	Boltzmann constant → absolute temperature → degree of freedom →
validWhen	kT is large compared to energy level spacings →

Referenced by (4)

Subject (surface form when different)	Predicate
Boltzmann constant →	appearsIn
Rayleigh–Jeans law at low frequencies →	derivedFrom
k_B →	playsRoleIn
Maxwell–Boltzmann statistics →	relatedTo