Ok last time on the how often should two nuclear submarines smack into each other question. This particular sum managed to get me and Luke into the Wall Street Journal print edition here and their blog here. Which was nice.
So the sum is a mean free path calculation. Where the particles are submarines and the box they are in is the submarine patrol zone or the whole Atlantic depending.
The size of a submarines patrol area is generally set by the requirement to have a surface support vessel within 24 hours. If one of these could travel at 20 knots an hour it could cover an circle with radius of 900 kilometers. The maximum operating depth is not much more than 600m and the submarine has to operate about 100m below the seasonal thermocline. A stealth mode speed is about two meters per second. The size of a nuclear submarine is about 150m length by 12m diameter plus a conning tower of about 8m high by 12 meters long. We modeled this cylinder as a sphere, which on these scales should not be too bad.
A support ship can support up to 14 subs. So we took it that there were 14 in its particular area in the small sum. If there were 14 gas particles wandering around a box the size of the Atlantic (big) or a ships support area (small). from a depth of 100m to 600m. At a speed of 2 meters a second. How often would you expect one to hit another is what we worked out.
Nuclear submarines have to stay 100 meters below the seasonal thermocline. We took the thermocline to be zero based on this
The model "mean free path" used is derived here
Now these are big ifs, thinking of things as being random gas particles is a big IF. People thought derivatives acted like random particles (the Black–Scholes model) and they just lost a lot of money. So I think this is more "fun application" of physics rather than "rigorous analysis". Thanks to Luke and Carl who crunched the numbers and got the data (i.e who did all the work).