User:Ondrej Kincl/sandbox

In statistical physics, the Kac ring is a toy model introduced by Mark Kac to explain how the second law of thermodynamics emerges from time-symmetric interactions between molecules (see reversibility paradox). Although artificial, the model is notable as a mathematically transparent example of coarse-graining and is used as a didactic tool.

Formulation

The Kac ring with $N=20$ and $M=4$. The marked points are represented by red gates placed on edges which connect them to their counterclockwise neighbor.

The Kac ring consists of N equidistant points in a circle. Some of these points are marked. The number of marked points is M, where $0<2M<N$. 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. (When its starting point is not marked, it completes the move without changing color.)

An imagined observer can only measure macroscopic quantities: the ratio

$\mu ={\frac {M}{N}}<0.5$

and the overall color

$\delta ={\frac {W-B}{N}},$

where B, W denote the total number of black and white balls respectively. Without the knowledge of detailed configuration, any distribution of M marks is considered equally likely.

Properties

Kac ring evolution for $N=1000$ and $M=20$ with logarithmic time scale. Blue line is the estimated behavior, showing exponential relaxation. Orange line is an example of detailed evolution, which features Poincaré recurrence. Orange area is a confidence interval from 10% to 90% quantile (estimated numerically).

Let $\eta _{k}(t)$ denote the color of a ball at point k and time t with a convention

$\eta _{k}={\begin{cases}+1&{\text{ball is white}}\\-1&{\text{ball is black}}\end{cases}}.$

The microscopic dynamics can be mathematically formulated as

$\eta _{k}(t)=\epsilon _{k-1}\eta _{k-1}(t-1),$

where

$\epsilon _{k}={\begin{cases}+1&{\text{unmarked site}}\\-1&{\text{marked site}}\end{cases}}$

and $k-1$ is taken modulo N. In analogy to molecular motion, the system is time-reversible. 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. Moreover, the system is periodic, where the period is at most $2N$. (After N steps, each ball visits all M marked points and changes color by a factor $(-1)^{M}$.) This fact is a manifestation of discrete Poincaré recurrence.

Assuming that all balls are initially white,

$\eta _{k}(t)=\epsilon _{k-1}\epsilon _{k-2}\cdots \epsilon _{k-t}=(-1)^{X},$

where $X=X(k,t)$ is the number of times the ball will leave a marked point during its journey. When marked locations are unknown (and all possibilities equally likely), X becomes a random variable. Assuming $t<N$, then X has hypergeometric distribution, i.e.:

${\text{Pr}}(X=i)={\frac {{\begin{pmatrix}N-t\\M-i\end{pmatrix}}{\begin{pmatrix}t\\i\end{pmatrix}}}{\begin{pmatrix}N\\M\end{pmatrix}}}.$

Considering the limit when N approaches infinity but t, i, and μ remain constant, then, using Stirling's approximation:

$\lim _{N\to \infty }{\text{Pr}}(X=i)=\mu ^{i}(1-\mu )^{t-i}{\begin{pmatrix}t\\i\end{pmatrix}},$

which we can identify as the binomial distribution. Hence, the overall color after t steps will be

${\begin{aligned}\lim _{N\to \infty }\langle \delta (t)\rangle &=\lim _{N\to \infty }{\frac {1}{N}}\sum _{k}\langle \eta _{k}(t)\rangle \\&=\lim _{N\to \infty }\langle \eta _{1}(t)\rangle \\&=\sum _{i=0}^{t}(-1)^{i}\mu ^{i}(1-\mu )^{t-i}{\begin{pmatrix}t\\i\end{pmatrix}}\\&=(1-2\mu )^{t}\end{aligned}}$

Since $0<1-2\mu <1$ the overall color will, on average, converge monotonically and exponentially to 50% grey (which is analogical to thermodynamic equilibrium). An identical result is obtained for a ring rotating clockwise. Consequently, the macroscopic behavior of the Kac ring is irreversible.

It is also possible to show that the variance approaches zero:

$\lim _{N\to \infty }{\text{Var}}(\delta (t))=0$

Therefore, when N is huge (of order $10^{23}$), the observer has to be extremely lucky (or patient) to detect any significant deviation from the mean behavior.

See also

Ehrenfest model