View source for Kac ring

{{Short description|Toy model in statistical physics.}}

In [[statistical mechanics]], the '''Kac ring''' is a [[toy model]]<ref>{{cite journal |last1=Luczak |first1=Joshua |title=Talk about toy models |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |date=2017 |volume=57 |pages=1–7|doi=10.1016/j.shpsb.2016.11.002 |bibcode=2017SHPMP..57....1L |s2cid=125757671 |url=https://ink.library.smu.edu.sg/soss_research/3290 }}</ref> introduced by [[Mark Kac]] in 1956<ref>{{cite journal |last1=Kac |first1=Mark |title=Some remarks on the use of probability in classical statistical mechanics |journal=Bulletins de l'Académie Royale de Belgique |date=1956 |volume=42 |pages=356–361}}</ref><ref>{{cite journal |last1=Thompson |first1=Colin J |title=The contributions of Mark Kac to mathematical physics |journal=The Annals of Probability |date=1986 |volume=14 |issue=4 |pages=1129–1138|doi=10.1214/aop/1176992357 |doi-access=free }}</ref> to explain how the [[second law of thermodynamics]] emerges from [[T-symmetry|time-symmetric]] interactions between molecules (see [[reversibility paradox]]). Although artificial,<ref name="Kac">{{cite book |last1=Kac |first1=Mark |title=Probability and related topics in physical sciences |date=1959 |publisher=American Mathematical Soc.}}</ref> the model is notable as a mathematically transparent example of [[coarse-grained modeling|coarse-graining]]<ref name="Gottwald&Oliver">{{cite journal |last1=Gottwald and Oliver |title=Boltzmann's Dilemma: An Introduction to Statistical Mechanics via the Kac Ring |journal=SIAM Review |date=2009 |volume=51 |issue=3 |pages=613–635|doi=10.1137/070705799 |bibcode=2009SIAMR..51..613G }}</ref> and is used as a didactic tool<ref name="Dorfman">{{cite book |last1=Dorfman |first1=Jay Robert |title=An Introduction to Chaos in Nonequilibrium Statistical Mechanics |date=1999 |publisher=Cambridge University Press |pages=34–39}}</ref> in [[non-equilibrium thermodynamics]].

== Formulation ==
[[File:Kac ring animation.gif|thumb|The Kac ring with <math>N = 20</math> and <math>M = 4</math>. The marked points are indicated by red gates placed on edges which connect them to their counterclockwise neighbor.]]
The Kac ring consists of {{mvar|N}} equidistant points in a circle. Some of these points are ''marked''. The number of marked points is {{mvar|M}}, where  <math>0 < 2M < N</math>. 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. (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 scale|macroscopic]]) quantities: the ratio
: <math>\mu = \frac{M}{N} < 0.5 </math>
and the overall color
: <math>\delta = \frac{W-B}{N},</math>
where {{mvar|B}}, {{mvar|W}} denote the total number of black and white balls respectively. Without the knowledge of detailed ([[Microscopic scale|microscopic]]) configuration, any distribution of {{mvar|M}} marks is considered equally likely. This assumption of equiprobability is comparable to [[Molecular chaos|''Stosszahlansatz'']], which leads to [[Boltzmann equation]].<ref>{{cite journal |last1=Jebeile |first1=Julie |title=The Kac Ring or the Art of Making Idealisations |journal=Foundations of Physics |date=2020 |volume=50 |issue=10 |pages=1152–1170|doi=10.1007/s10701-020-00373-1 |bibcode=2020FoPh...50.1152J |s2cid=225318291 |doi-access=free }}</ref>

== Detailed evolution ==
Let <math>\eta_k(t)</math> denote the color of a ball at point {{mvar|k}} and time {{mvar|t}} with a convention
: <math> \eta_k = \begin{cases}
+1 & \text{ball is white}\\
-1 & \text{ball is black}
\end{cases}. </math>
The microscopic dynamics can be mathematically formulated as
: <math> \eta_k(t) = \epsilon_{k-1} \eta_{k-1}(t-1), </math>
where 
: <math> \epsilon_k = \begin{cases}
+1 & \text{unmarked site}\\
-1 & \text{marked site}
\end{cases} </math>
and <math>k-1</math> is taken modulo {{mvar|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 evolution of <math>\eta_k(t)</math> is [[Periodic function|periodic]], where the period is at most <math>2N</math>. (After {{mvar|N}} steps, each ball visits all {{mvar|M}} marked points and changes color by a factor <math>(-1)^M</math>.) Periodicity of the Kac ring is a manifestation of more general [[Poincaré recurrence theorem|Poincaré recurrence]].<ref name="Dorfman"/>

== Coarse-graining == 
[[File:Kac Ring (Graph).pdf|thumb|Kac ring evolution for <math>N = 1000</math> and <math>M = 20</math> with logarithmic time scale. Blue line is the approximate mean behavior given by macroscopic model, indicating exponential relaxation to equilibrium. Orange line is an example of evolution given by microscopic description, which features Poincaré recurrence. Orange area is a confidence interval from 10% to 90% quantile (estimated numerically).]]
Assuming that all balls are initially white,
:<math> \eta_k(t) = \epsilon_{k-1} \epsilon_{k-2} \cdots \epsilon_{k-t} = (-1)^{X}, </math>
where <math>X = X(k,t)</math> 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), {{mvar|X}} becomes a random variable. Considering the limit when {{mvar|N}} approaches infinity but {{mvar|t}}, {{mvar|i}}, and {{mvar|μ}} remain constant, the random variable {{mvar|X}} converges to the [[binomial distribution]], i.e.:<ref name="Gottwald&Oliver"/> 
:<math> \lim_{N \to \infty} \text{Pr}(X = i) = 
\mu^i (1-\mu)^{t-i}
\begin{pmatrix}
t \\
i
\end{pmatrix}
,</math>
Hence, the overall color after {{mvar|t}} steps will be
: <math>
\begin{align}
\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{align} .</math>
Since <math> 0<1-2\mu<1 </math> the overall color will, on average, converge monotonically and exponentially to 50% grey (a state that is analogical to [[thermodynamic equilibrium]]). 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:<ref name="Gottwald&Oliver"/>
: <math>
\lim_{N \to \infty} \text{Var}(\delta(t)) = 0 .
</math>
Therefore, when {{mvar|N}} is huge (of order [[Avogadro constant|10<sup>23</sup>]]), the observer has to be extremely lucky (or patient) to detect any significant deviation from the ensemble averaged behavior.

== See also ==
* [[Ehrenfest model]]

== References ==
{{reflist}}

[[Category:Statistical mechanics]]