With the Tour de France now running (Monday is a rest day), watching the race brought back some mathematical thoughts that I have had over the years. I did a Google search on both regular Google & Google Scholar, but wasn't able to find anything about the following problem, although I did find some interesting discussions about the game theory of 2 cyclists sprinting against one another and/or slipstreaming behind one another. This is a classical case in cycle track racing with 2 people. I'm curious about the possible modeling the entire peloton of 100-200 riders as a kind of gas and/or liquid, moving through another (much lighter) gas, in such a way that some elements of the behavior of the entire ensemble of cyclists is captured. Anyone who has watched bike racing for more than a few moments quickly notices that most of the cyclists coalesce into blobs called "pelotons". This is because it is much more efficient (requiring perhaps only 66% as much wattage) to travel in groups, sharing the load of breaking the wind, than it is to travel individually. However, blobs occasionally break off ("break away") because they want to go faster, with larger groups being able to go faster more efficiently than smaller groups. I was thinking that there ought to be some way to model this behavior in an idealized way. One idealization might have the cyclists simply _falling_ straight down through the air, but being able to move sideways to form slipstreaming groups (parachuting cyclists??). In such a model, a more energetic cyclist simply becomes a heavier cyclist. However, he can still team up with another cyclist & fall even faster. Thus, although an individual cyclist might reach a terminal velocity V_T, two such cyclists teamed together might reach a faster terminal velocity V_T'>V_T, and an even larger ensemble of cyclists might reach a terminal velocity V_T">V_T'. There needs to be a kind of "attractive force" between cyclists which tends to clump them together, but this force is quite short-range because the ability of one cyclist to shade another from the wind does not extend that far behind him/her. So if two cyclists are pulled apart far enough (some distance D_0), then the force falls completely to zero. One could possibly model this force as a Gaussian if this made the problem simpler, because the tails of the Gaussian are so small. One could also conceive that if a sufficiently massive (i.e., powerful) cyclist were to be grouped with a much less massive (i.e., weak) cyclist, the weak cyclist might still not be powerful enough to keep up with the massive cyclists even with the advantage of being shaded from the wind. We also have a problem with the modelling of how the wind-shielding is shared among the cyclists. One could conceive that the peloton is some kind of gas, where the molecules/cyclists are bouncing around inside the peloton, but lose energy when they reach the "surface" of the blob & become exposed to the wind. They then fall slower than their associates until the "attractive force" (absence of retarding force) allows them to catch up again. This should at least ensure that the peloton assumes the correct shape, which is a very long, thin (very small multiple of D_0) line. The attractive force needs to be large enough that the peloton is, in some sense, "held back" by the less massive/less capable cyclists, and moves at a slower speeds as a result. The problem is that any larger ensemble should almost _always_ fall faster than a smaller ensemble, which would eliminate the possibility of breakaways entirely. Perhaps there are several different species of cyclist molecules, with different masses & attractive forces. (Assume for the moment that the cyclists don't form into color-coded teams!!) I.e., some cyclist molecules are "better"/"fitter", so they have slightly heavier masses (they have larger terminal velocities even when on their own) and slightly smaller attractive forces (they are more willing to break away). Then if statistically a group of these more massive, less attractive cyclists molecules found themselves together at the front of the peloton, then they could indeed "break away", and possibly fall faster than the larger peloton. --- I was hoping for an extremely simply model with only perhaps 1-3 parameters which could explain some of the simpler aspects of peloton behavior. As you can see, I haven't completely worked out the model, but I thought that since bike racing is over 100 years old, some mathematician/physicist must have looked at this type of model before. Perhaps someone on this list can provide a pointer?