Grim Data for Grim Modeling: Oppenheimer and The Halifax Explosion

Mushroom cloud over Halifax. Click to enlarge. In my previoius post I described some of my thoughts about the teaching of differential equations that model death in some way - via overpopulation, disease, and war - what I call Grim Modeling.

Perhaps the grimmest modeling of them all, however, is modeling associated with the design and delivery of atomic weapons - your basic WMD.

An essential step in any mathematical modeling is the correlation of the model output with the world, i.e. you need to have some data that supports your model. This is especially true if the model is to be used as a starting point for an extrapolated prediction. By this I mean that t --> infinity, or a parameter is taking on a value much larger (or smaller) than was the case for the situation that produced the data being used to check the model.

Could there have been any mathematical exercise more fraught with danger than the modeling done by J.R. Oppenheimer and his team of scientists on the Manhattan project? In a very short time (the summer of '42 ), a group of physicsts, chemists, and engineers managed to develop the theory of nuclear reactions in an atomic bomb, the engineering required to design and construct a deliverable weapon, and model the effects that the atomic blast would have on a city and its population.

Model the effects of an atomic blast? What type of model was this, and, more important, what data could possibly be used to validate the model as one that could be trusted? This was not an idle concern - some on the Manhattan Project believed that an atomic bomb blast would start a chain reaction that would spread through the earth's atmosphere - in effect blowing up the earth.

Oppenheimer's search for blast data took him to a most unusual location: Halifax, Nova Scotia. In a terrifying incident that is still very little known (at least outside of Canada), a French munitions ship exploded in Halifax harbour towards the end of World War 1. The ship was loaded with

Click to read more ...

Grim Modeling: Overpopulation, Disease, and War

Battle of TrafalgarI often wonder how appropriate it is in an Intro. Diff Eq. course to discuss the darker side of modeling - the business of modeling death and destruction via war, terrorist activity, or plain, old-fashioned man-made disaster. (I have no qualms about this in our Chaos and Fractals course - after all, one of the goals of the course is to consider the ramifications of the pervasiveness of mathematical modeling in all aspects of society, and especially in areas where religion, philosophy, and politics intersect.)

But back to Differential Equations - basically a mathematics course where students get their first real concentrated exposure to applied mathematics and modeling via basic calculus.

It seems OK to discuss population models such as the Logistic equation, where growth rates go negative for population values exceeding the carrying capacity of the ecological niche. The logistic equations is, after all, a sterile shorthand for what is an accepted "law of the jungle," at least for animals: if there ain't enough food, you die.

And there doesn't seem to be any queasiness among students when a Lotka-Volterra model is used for a Predator-Prey system. This again may be explained because it is a fact of life on the other side of the human/animal divide - eat and be eaten, in an endless cycle of life and death. Or maybe it's because the use of cute function names - R(t) and F(t) for rabbits and foxes - provides a welcome relief from reality due to the abstractness of the notation: we don't feel the rabbits' pain, or wince at the growth of the Rabbit-Fox interaction term when dR/dt<0.

Epedemiology is a fertile source of models for Diff. Eq., but even here what is being modeled may be especially stark. (See Deterministic Modeling Of Infectious Diseases:Theory And Methods, a thorough review of disease modeling by Helen Trottier and P. Philippe of the Univ. of Montreal). In this case, an ambiguous use of English can go a long way towards not having to think about the grim reality of what the models often predict.<

Click to read more ...

Modeling the Arms Race

Nuclear Proliferation game box-top. Click to enlarge.Following up on Tom and Meredyth's presentation and post: the Saperstein article referenced dealt with modeling interactions among nuclear states. A recent article by William C. Potter, director of the Center for Nonproliferation Studies at the Monterey Institute of International Studies, entitled The Second Last Chance: American Power and Nuclear Nonproliferation points out the need for non-proliferation modeles and theorists to consider the effects of regimes and terrorist agents on our notions of non-proliferation.

Potter's article is an extensive review of the book Nuclear Terrorism: The Ultimate Preventable Catastrophe by Graham Allison (Harvard Kennedy School of Government). Because of its call for the U.S. to change its current theories and practices of approaching non-proliferation, the connection with Saperstein's model is an important one. I don't know whether Saperstein, or any think-tanker worrying about such modifications has the ear of the government. At the very least, the chaotic behavior of nuclear-arms possessing states as predicted by the Saperstein model will certainly be more prevalent as the highly non-linear interactions of terrorist agents and rogue states are factored into the model.

(Note: the image at the top of the post is from the card game Nuclear Proliferation by the FlyingBuffalo company. From the blurb on the box: "It's a sarcastic, humorous look at the futility of Atomic Warfare in the post-cold war 1990s.")

Chaos: Good For More Than Absolutely Nothing

Originally Posted by Meridyth Mascio and Tom Plick

Chaos, War, and Disarmament

Debate has raged over whether wars are due mainly to long-term causes (oppression, economic depression, nationalistic zealotry) or whether they only find their genesis in freak events (the murder of a ruler, the sinking of a fleet). Many believe the former; among these people was Lewis Richardson, who, in the 1930s, developed a simple model of an arms race between two countries. He related the size of a nation's stockpile to its willingness to go to war, for two reasons: Firstly, arms could be easily qualified, whereas the "feelings" of a nation were not easily transformed into numbers. Secondly, the larger a nation's stockpile, the greater its chance of winning a war, and thus, the greater its chance of starting a war. (When each nation's stockpile is sufficiently large, the system becomes crisis-unstable: each nation wants to attack the other(s) first, lest the other(s) make the first blow.)

Richardson's model was the first attempt at modeling an arms race mathematically. Many analysts point out the inapplicability of Richardson's model to modern conflicts - the model mishandles many situations, because of its simplicity and the assumptions made to secure that simplicity.

Others after him, including Alvin Saperstein, took Richardson's idea and elaborated on it, in an attempt to model reality more accurately. Saperstein modeled an arms race between three nations under two systems - in one, the nations were allowed to ally as they saw fit, so that a superpower might find the other two nations allied against it. In the other system, the nations were not permitted to ally, and thus were at odds with both of the other nations. Saperstein's model was complex, and non-linear; recall that non-linearity breeds chaos.

Saperstein showed the system permitting alliances to be stable - in time, he says, each country's stockpile will dwindle to zero. Saperstein uses this observation in support of encouraging cooperation between countries instead of competition.

More interesting than the alliance system is the system that disallows alliances.
In the independent-nations scenario, several outcomes are possible, depending on the initial stockpiles of the nations and the "fear and loathing" values between the nations, plus many other parameters defined by Saperstein.

1. Strong stability: Each nation's stockpile decreases to zero (that is, in the limit, not necessarily in any finite amount of time).
   
2. Weak stability: Each nation's stockpile decreases to some non-zero amount.
   
3. Weak chaos: The system has a strange attractor, but its basin of attraction is small, and so the system is still largely crisis-stable.
   
4. Strong chaos: The system has a strange attractor, whose basin encompasses the entire system.
   

According to Saperstein, the strange attractor signals crisis instability, since once on a strange attractor, what happens in the long-term is anyone's guess. The way to maintain peace, he says, is to avoid the appearance of the strange attractor. To him, knowing chaos is useful for avoiding it in world affairs.

You can see our Powerpoint presentation again here.