Kac ring

Although artificial,[4] the model is notable as a mathematically transparent example of coarse-graining[5] and is used as a didactic tool[6] in non-equilibrium thermodynamics.

Whenever a ball leaves a marked site, it switches color from black to white and vice versa.

(If, however, the starting point is not marked, the ball completes its move without changing color.)

An imagined observer can only measure coarse-grained (or macroscopic) quantities: the ratio and the overall color where B, W denote the total number of black and white balls respectively.

Without the knowledge of detailed (microscopic) configuration, any distribution of M marks is considered equally likely.

denote the color of a ball at point k and time t with a convention The microscopic dynamics can be mathematically formulated as where and

Indeed, if balls would move clockwise (instead of counterclockwise) and marked points changed color upon entering them (instead of leaving), the motion would be equivalent, except going backward in time.

Considering the limit when N approaches infinity but t, i, and μ remain constant, the random variable X converges to the binomial distribution, i.e.:[5] Hence, the overall color after t steps will be Since

the overall color will, on average, converge monotonically and exponentially to 50% grey (a state that is analogical to thermodynamic equilibrium).

It is also possible to show that the variance approaches zero:[5] Therefore, when N is huge (of order 1023), the observer has to be extremely lucky (or patient) to detect any significant deviation from the ensemble averaged behavior.