Chain Reactions

I am a member of a church supper group. Dinners are scheduled once a month, in which a bunch of couples come to this one couple's house who hosts them and prepares a dinner for them. This allows us to get to know each other better. I thought of what happens if not enough couples are assigned to the dinners. If only 4 couples are assigned at a time, then if one or two can't make it (this can easily happen), then the hosts have to reschedule the dinner so that they all can make it. So there would be a large number of reschedulings; these would conflict with other dinners, church, and other events and cause more reschedulings. This reminds me of something I saw back in the 1950s in shows, perhaps on Walt Disney's Tomorrowland, of a demonstration of a nuclear fission chain reaction.

In such a chain reaction, a neutron hits an atom of Uranium 235 or Plutonium 239. This causes the atom to split, ejecting energy according to E=mc², two smaller atoms such as those of barium and krypton, and one to three more neutrons. These in turn would hit other atoms, which in turn would break up into two smaller atoms and produce more neutrons. The number of free neutrons grows exponentially until the entire chunk of uranium or plutonium is consumed. The resulting energy is tremendous, and produces the familiar mushroom cloud of a nuclear explosion.

In the 1950s they demonstrated the concept by covering an empty room's floor with mousetraps with ping pong balls on them. Then someone would throw a ping pong ball into the room. The result is dramatic. Almost instantly, the air would be filled with balls pinging and ponging all over the place. This demonstrates the concept well, but I realized that this experiment can be used to model other things as well, after seeing a Java applet from my alma mater that demonstrated this concept at http://ccl.northwestern.edu/netlogo/models/Mousetraps .

For example, Peak Oil. If you clicked on the mousetrap URL and ran the Java app, notice the graph on the bottom, measuring the number of balls in the air. Now compare this with the curve that is used to demonstrate the concept of Peak Oil. Don't they look similar? It's because similar concepts are involved.

Oil was discovered in 1859. At first, only a handful of Pennsylvanians used the oil. But these people talked to others about this wondrous substance, and other people started to dig for oil. These in turn did wondrous things with the oil, such as invent and operate automobiles and aircraft, and this led others to want the oil. This kept spreading all over the world, resulting in a growth in the use of oil that resembles the up side of the Peak Oil bell curve. However, as people began spreading around drilling for the oil, less and less oil was available to find and drill for. This puts a damping effect on the growth of oil production, until eventually the damping will exceed the growth and turn it into an exponential descent or decay. The same thing happens with mousetraps or uranium, as the chain reaction runs out of mousetraps to spring or uranium atoms to fiss.

In both cases, the determining differential equation is the same: y' = Ay(C - y) . C is the total capacity of the system, and A is the rate at which neutrons or ping pong balls are generated per neutron or ball strike. Solving the equation and graphing the solution results in a bell-like curve called the logistic curve.

The equation can be used to describe other behavior as well. Take people living in a city around 1900. They live in decent neighborhoods. But some get dissatisfied and move out farther from the city so they can have more room. These people are like ping pong balls that get ponged and hop out to land on another mousetrap. This causes more people to move out. This accelerates as the once livable neighborhood deteriorates in the inner city. Eventually the suburbs get crowded and people move out even farther from the city. This causes the inner suburbs to deteriorate as well. Look at the Java applet again. Doesn't that red blob remind you of a typical city with development going on? It's fueled by oil, of course.

The equation can be used to describe bacteria in a Petri dish, viruses in the body of a large animal, the growth of the Internet, the growth of automobiles, and in fact, the growth and decline of just about any product or fad. It's an equation we need to live with. We live in a world that moves on in logistic curves, and Peak Oil happens to be one of these curves.