Kac ring
In statistical mechanics, the Kac ring is a toy model introduced by Mark Kac in 1956 to explain how the second law of thermodynamics emerges from time-symmetric interactions between molecules. Although artificial, the model is notable as a mathematically transparent example of coarse-graining and is used as a didactic tool in non-equilibrium thermodynamics.
Formulation
The Kac ring consists of equidistant points in a circle. Some of these points are marked. The number of marked points is, where . Each point represents a site occupied by a ball, which is black or white. After a unit of time, each ball moves to a neighboring point counterclockwise. Whenever a ball leaves a marked site, it switches color from black to white and vice versa.An imagined observer can only measure coarse-grained quantities: the ratio
and the overall color
where, denote the total number of black and white balls respectively. Without the knowledge of detailed configuration, any distribution of marks is considered equally likely. This assumption of equiprobability is comparable to Stosszahlansatz, which leads to Boltzmann equation.
Detailed evolution
Let denote the color of a ball at point and time with a conventionThe microscopic dynamics can be mathematically formulated as
where
and is taken modulo. In analogy to molecular motion, the system is time-reversible. Indeed, if balls would move clockwise and marked points changed color upon entering them, the motion would be equivalent, except going backward in time. Moreover, the evolution of is periodic, where the period is at most. Periodicity of the Kac ring is a manifestation of more general Poincaré recurrence.
Coarse-graining
Assuming that all balls are initially white,where is the number of times the ball will leave a marked point during its journey. When marked locations are unknown, becomes a random variable. Considering the limit when approaches infinity but,, and remain constant, the random variable converges to the binomial distribution, i.e.:
Hence, the overall color after steps will be
Since the overall color will, on average, converge monotonically and exponentially to 50% grey. An identical result is obtained for a ring rotating clockwise. Consequently, the coarse-grained evolution of the Kac ring is irreversible.
It is also possible to show that the variance approaches zero:
Therefore, when is huge, the observer has to be extremely lucky to detect any significant deviation from the ensemble averaged behavior.