This is actually a very hard question, at least for me. First, I guess you must mean that what's divided up in different ways is the kinetic energy. The net momentum is a vector, not a number, and it changes on every elastic collison with a wall, so it can't be what you mean.
Assuming that there are many elastic collisions, after a while there will be a whole range of possible distributions of energies. Regardless of the starting values, on the average the distribution of values will follow a Maxwell-Boltzmann form, so in the long run the collision rates will be the same.
In the short run, it sounds like a harder problem. It would help to define a little better what's meant by the rate. Is it supposed to be the typical probability per time in some very short time of having a single collision? If so, I guess I could figure that out and answr your question, but then the rte will change after the first collison as the energy distribution changes.
p.s. I tried doing a calculation for a very simple case: 3 equal-mass particles, either with all the energy in one particle or with the energy evenly distributed. At least first time through the calculation, I got that the even distribution case wins, with 2/31/2
times as high a net initial collision rate.
(published on 02/19/2008)