Appendix Origins of Irreversibility—The Kac Ring Model

While no one debates the validity of Boltzman s equation (equation 9.32) - one good reason for this is that there has been no experimental evidence suggesting that the theory is wrong - it is in a very important sense laced with a fundamental 

Consider a circular ring with N equally spaced fixed sites, m of which are marked in some fashion say, with a cross x (see figure 8.27). 

Before exploring the behavior of this system, we mention its two most obvious properties, consistent with what we expect of any Newtonian mechanical system  

Perhaps our first instinct is to take the most direct approach. Namely, to simply write down and solve a set of Newtonian cquations-of-motion describing the scattering dynamics in terms of basic system valuables. While this naive, intuitive approach might be attractive, we show below that it turns out to be both right and wrong . 

We also have two constraints. We require that the total number of balls and number of marked sites both remain fixed for all time  

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