In Chapter 12 we will discuss in more detail some of the key ideas involved in coming to this conclusion. But the basic point is that the phenomenon of entropy increase implied by the Second Law is a more or less direct consequence of the phenomenon discovered in this book that even with simple initial conditions many systems can produce complex and seemingly random behavior.

One aspect of the generation of randomness that we have noted several times in earlier chapters is that once significant randomness has been produced in a system, the overall properties of that system tend to become largely independent of the details of its initial conditions.

In any system that is reversible it must always be the case that different initial conditions lead to at least slightly different states—otherwise there would be no unique way of going backwards. But the point is that even though the outcomes from different initial conditions differ in detail, their overall properties can still be very much the same.

The pictures on the facing page show an example of what can happen. Every individual picture has different initial conditions. But whenever randomness is produced the overall patterns that are obtained look in the end almost indistinguishable.

The reversibility of the underlying rules implies that at some level it must be possible to recognize outcomes from different kinds of initial conditions. But the point is that to do so would require a computation far more sophisticated than any that could meaningfully be done as part of a practical measurement process.

So this means that if a system generates sufficient randomness, one can think of it as evolving towards a unique equilibrium whose properties are for practical purposes independent of its initial conditions.

This fact turns out in a sense to be implicit in many everyday applications of physics. For it is what allows us to characterize all sorts of physical systems by just specifying a few parameters such as temperature and chemical composition—and avoids us always having to know the details of the initial conditions and history of each system.

The existence of a unique equilibrium to which any particular system tends to evolve is also a common statement of the Second Law of

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From Stephen Wolfram: A New Kind of Science [citation]