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Statistical mechanics, structure of matter Definition

Microcanonical ensemble

Builds on Statistical ensembleMicrostatePhase spaceLebesgue measure

Definition

Let Ω\Omega be the set of microstates of a system and let H ⁣:Ω→ℝH \colon \Omega \to \R be its energy.

  1. Discrete case: let Ω\Omega be countable and let EE be a value of HH with W(E)=#{ω∈Ω:H(ω)=E}W(E) = \#\set{\omega \in \Omega : H(\omega) = E} finite. The microcanonical ensemble at energy EE is the uniform distribution on {H=E}\set{H = E}: it gives each such ω\omega the probability 1/W(E)1 / W(E).
  2. Classical case: let Ω=Γ\Omega = \Gamma be a phase space ℝ2n\R^{2n} with Lebesgue measure (Liouville measure) dΓd\Gamma. Fix EE and a width δ>0\delta > 0, and let ΓE,δ={x∈Γ:E≤H(x)<E+δ}\Gamma_{E, \delta} = \set{x \in \Gamma : E \le H(x) < E + \delta} have finite positive measure. The microcanonical ensemble is the uniform probability measure on ΓE,δ\Gamma_{E, \delta}, with density 𝟏ΓE,δ/∣ΓE,δ∣\mathbf{1}_{\Gamma_{E, \delta}} / \abs{\Gamma_{E, \delta}}.

It is a statistical ensemble in which the energy, the volume and the number of particles are fixed.

Remarks

Example: take NN two-level units, each with energy 00 or ε>0\varepsilon > 0, and the total energy E=KεE = K \varepsilon with 0≤K≤N0 \le K \le N. Then W(E)=(NK)W(E) = \binom{N}{K}, and each configuration with exactly KK excited units has probability 1/(NK)1 / \binom{N}{K}. Non-example: the distribution that gives the configuration with KK excited units probability proportional to e−βKεe^{-\beta K \varepsilon} is not microcanonical, since it spreads over many energies. In the classical case one may also let δ→0\delta \to 0 and use the measure dσ/∣∇H∣d\sigma / \abs{\nabla H} on the surface H=EH = E, where dσd\sigma is surface measure. The microcanonical entropy is S(E)=kBln⁡W(E)S(E) = k_B \ln W(E), with kBk_B the Boltzmann constant.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptEuclidean space, object, no evidence levelEuclidean spaceobjectMicrostate, concept, no evidence levelMicrostateconceptOuter measure, concept, no evidence levelOuter measureconceptPhase space, concept, no evidence levelPhase spaceconceptStatistical ensemble, concept, no evidence levelStatistical ensembleconceptLebesgue measure, object, no evidence levelLebesgue measureobjectMicrocanonical ensemble, definition, no evidence levelMicrocanonical ensemb…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Euclidean space object
  3. Microstate concept
  4. Outer measure concept
  5. Phase space concept
  6. Statistical ensemble concept
  7. Lebesgue measure object
  8. Microcanonical ensemble definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Microcanonical_ensemble
About this page
TagZCMQGKHGSV
URIm2://ZCMQGKHGSV
Kinddefinition
WritingW0 stub
MSC82B03, 82B05
DefinesMicrocanonical ensemble
UsesStatistical ensemble, Microstate, Phase space, Lebesgue measure
Changed3 November 2026

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