Math 2.0

Statistical mechanics, structure of matter Definition

Statistical mechanics

Builds on MicrostateStatistical ensemblePhase spaceHamiltonian mechanicsExpected value

Definition

Statistical mechanics is the part of mathematical physics that studies systems of many interacting components through probability. A model of statistical mechanics consists of the following data.

  1. A set Ω\Omega of microstates, such as the phase space of NN particles or the set of spin configurations on a finite graph.
  2. An energy function, the Hamiltonian H ⁣:Ω→ℝH \colon \Omega \to \R, as in Hamiltonian mechanics.
  3. A statistical ensemble, that is, a probability measure on Ω\Omega built from HH and from macroscopic parameters such as temperature, volume and particle number.

A macroscopic observable is a function A ⁣:Ω→ℝA \colon \Omega \to \R, and its predicted value is its expected value under the ensemble. Equilibrium statistical mechanics studies ensembles that do not change in time; nonequilibrium statistical mechanics studies how distributions evolve.

Remarks

Example: for NN independent spins σi∈{−1,+1}\sigma_i \in \set{-1, +1} in a magnetic field hh, take Ω={−1,+1}N\Omega = \set{-1, +1}^N and H(σ)=−h∑iσiH(\sigma) = -h \sum_i \sigma_i. The canonical ensemble at inverse temperature β\beta gives the mean magnetization per spin tanh⁡(βh)\tanh(\beta h), since each spin is +1+1 with probability eβh/(eβh+e−βh)e^{\beta h} / (e^{\beta h} + e^{-\beta h}). Non-example: a single planet moving around the sun is a mechanical system. It carries no probability distribution over states, so it is not a model of statistical mechanics in this sense. In quantum statistical mechanics the set Ω\Omega is replaced by a Hilbert space and the probability measure by a density operator.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Expected value, concept, no evidence levelExpected valueconceptHamiltonian mechanics, concept, no evidence levelHamiltonian mechanicsconceptMicrostate, concept, no evidence levelMicrostateconceptPhase space, concept, no evidence levelPhase spaceconceptStatistical ensemble, concept, no evidence levelStatistical ensembleconceptStatistical mechanics, definition, no evidence levelStatistical mechanicsdefinition
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. Expected value concept
  2. Hamiltonian mechanics concept
  3. Microstate concept
  4. Phase space concept
  5. Statistical ensemble concept
  6. Statistical mechanics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Statistical_mechanics
About this page
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Kinddefinition
WritingW0 stub
MSC82B03, 82B05, 82C03
DefinesStatistical mechanics
UsesMicrostate, Statistical ensemble, Phase space, Hamiltonian mechanics, Expected value
Changed3 November 2026

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