The Kac ring

A model of how reversible motion can produce an apparent approach to equilibrium in statistical mechanics.

01 Microscopic state
Particles on the Kac ringEach particle moves one site clockwise at every step. Crossing a fixed scatterer reverses its sign.
+1 particle −1 particle Scatterer

All particles move one site per step. A scatterer flips the sign of each particle that crosses it.

02 Colour imbalance m(t)+1.000
Exact dynamics
03 Boltzmann entropy S/Smax0.000

The entropy counts colour configurations with the same imbalance.

All particles begin in the +1 state.

Experiment parameters

Edit initial state or export data

Particle states over time

Each column is a fixed site and each row is a time step. Use the time slider or CSV export for exact values.
+1 −1

Reversibility, relaxation and return

A deterministic microscopic rule

The model places N particles on a ring, one at each site. Each has a sign, +1 or −1. At every time step all particles move one site clockwise. A fixed set of M scatterers reverses the signs of particles that cross them.

σi+1(t + 1) = εi σi(t)

Here εi = −1 if the outgoing edge at site i contains a scatterer, and +1 otherwise. Indices wrap around the ring. Reverse applies the inverse rule: move anticlockwise and cross the same scatterers again.

The molecular-chaos assumption

The colour imbalance is m = (N+ − N−)/N. Molecular chaos treats a particle’s sign as independent of whether its next edge contains a scatterer. Applying this assumption at each step gives an approximate law, with scatterer density ρ = M/N:

mmc(t) = m(0)(1 − 2ρ)t.

This is an additional statistical assumption, not a consequence of the reversible dynamics. It can describe early relaxation in typical large, randomly prepared rings. Initial independence need not persist: the motion develops correlations that the approximation neglects. The dashed curve is the prediction under this assumption; it is not an exact average over fixed-count scatterer configurations.

Boltzmann entropy

The Boltzmann entropy measures how many colour configurations have the same imbalance:

SB=kBln⁡(N!N+!N−!)

The plot shows SB/SB,max, where the maximum occurs at equal populations. This coarse-grained entropy can fall as well as rise. The microscopic dynamics remain invertible, and the fine-grained Gibbs entropy of a probability distribution is conserved.

Recurrence

After one circuit, every particle has crossed every scatterer. Its sign has therefore changed by the factor (−1)M.

σi(t + N) = (−1)M σi(t).

For even M, the initial colour state returns after N steps; for odd M, it returns after 2N. These are guaranteed return times; special colour patterns can recur sooner.

References

Randomness enters only when the initial configuration is prepared.

  1. G. A. Gottwald & M. Oliver, “Boltzmann’s dilemma: an introduction to statistical mechanics via the Kac ring,” SIAM Review 51 (2009), 613–635. Published article · Author manuscript.