sensitive dependence on initial conditions, and with the chaos phenomenon that we discussed in Chapter 4.

But while there are certainly mathematical equations that exhibit this phenomenon, none of those typically investigated have any close connection to realistic descriptions of fluid flow.

And in the model on the facing page it turns out that there is essentially no sensitive dependence on initial conditions, at least at the level of overall fluid behavior. If one looks at individual particles, then changing the position of even one particle will typically have an effect that spreads rapidly. But if one looks instead at the average behavior of many particles, such effects get completely washed out. And indeed when it comes to large-scale fluid behavior, it seems to be true that in almost all cases there is no discernible difference between what happens with different detailed initial conditions.

So is there ever sensitive dependence on initial conditions?

Presumably there do exist situations in which there is some kind of delicate balance—say of whether the first eddy is shed at the top or bottom of an object—and in which small changes in initial conditions can have a substantial effect. But such situations appear to be very much the exception rather than the rule. And in the vast majority of cases, small changes instead seem to damp out rapidly—just as one might expect from everyday experience with viscosity in fluids.

So what this means is that the randomness we observe in fluid flow cannot simply be a reflection of randomness that is inserted through the details of initial conditions. And as it turns out, in the pictures on the facing page, the initial conditions were specifically set up to be very simple. Yet despite this, there is still apparent randomness in the overall behavior that is seen.

And so, once again, just as for many other systems that we have studied in this book, there is little choice but to conclude that in a turbulent fluid most of the randomness we see is not in any way inserted from outside but is instead intrinsically generated inside the system itself. In the pictures on page 378 considerable randomness was already evident at the level of individual particles. But since changes in the configurations of such particles do not seem to have any discernible

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