Math 2.0

Statistical mechanics, structure of matter Kinetic theory Example

Kac ring

Builds on Loschmidt's paradoxPoincaré recurrence theorem

Example

Let N≥2N \ge 2 and place NN sites 0,1,…,N−10, 1, \dots, N - 1 on a circle. Each site holds one ball, black or white, and a set SS of mm of the NN edges {i,i+1}\set{i, i + 1} (indices modulo NN) is marked. At each time step every ball moves from its site ii to the site i+1i + 1, and it changes colour if the edge {i,i+1}\set{i, i + 1} is marked. This is the Kac ring, introduced by Mark Kac.

Check of reversibility and recurrence: the dynamics is a bijection on the 2N2^N colourings. Running it backward is the same kind of dynamics in the other direction. After NN steps every ball has gone once around the circle and has crossed each of the mm marked edges once. So every colour has changed mm times. Hence after 2N2N steps every ball is back at its site with its original colour, and the motion is periodic. For example, take N=3N = 3, S={{0,1}}S = \set{\set{0, 1}} and colours (black, white, white) at sites 0,1,20, 1, 2. The next states are (white, white, white), (white, black, white) and (white, black, black). The last one is the initial state with all colours switched.

Remarks

Let B(t)B(t) and W(t)W(t) be the numbers of black and white balls, Δ(t)=B(t)−W(t)\Delta(t) = B(t) - W(t) and μ=m/N\mu = m/N. In the spirit of molecular chaos, assume that a fraction μ\mu of the black balls, and of the white balls, sit just before a marked edge at every step. Then Δ(t+1)=(1−2μ)Δ(t)\Delta(t + 1) = (1 - 2\mu) \Delta(t), so Δ(t)=(1−2μ)tΔ(0)\Delta(t) = (1 - 2\mu)^t \Delta(0) decays to 00 monotonically or with alternating signs. This contradicts the exact periodicity Δ(2N)=Δ(0)\Delta(2N) = \Delta(0), a toy version of Loschmidt's paradox and of the objection based on the Poincaré recurrence theorem. The answer: if each edge is marked independently with probability μ\mu, then for every fixed tt and fixed initial colours, Δ(t)/N−(1−2μ)tΔ(0)/N\Delta(t)/N - (1 - 2\mu)^t \Delta(0)/N tends to 00 in probability as N→∞N \to \infty. So the decay law holds for typical markings and for times small compared with NN, but not for all times.

Reading path

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

Show the path as a graph
Almost everywhere, concept, no evidence levelAlmost everywhereconceptBoltzmann equation, concept, no evidence levelBoltzmann equationconceptFinite measure, concept, no evidence levelFinite measureconceptHamiltonian system, concept, no evidence levelHamiltonian systemconceptMeasure-preserving transformation, concept, no evidence levelMeasure-preserving tr…conceptLoschmidt's paradox, problem, no evidence levelLoschmidt's paradoxproblemPoincaré recurrence theorem, statement, no evidence levelPoincaré recurrence t…statementKac ring, example, no evidence levelKac ringexample
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. Almost everywhere concept
  2. Boltzmann equation concept
  3. Finite measure concept
  4. Hamiltonian system concept
  5. Measure-preserving transformation concept
  6. Loschmidt's paradox problem
  7. Poincaré recurrence theorem statement
  8. Kac ring example

Further reading

  1. Web https://en.wikipedia.org/wiki/Kac_ring
About this page
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Kindexample
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MSC82C03, 37A60
IllustratesLoschmidt's paradox
UsesLoschmidt's paradox, Poincaré recurrence theorem
Same as
Changed3 November 2026

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