Maxwell speed distribution
I want to know if the Maxwell speed distribution is the following.
An ideal gas system of n particles, say constrained to the unit box, has the phase space ([0,1]^3 x R^3)^n. That is, [0,1]^3 for the position of a particle, R^3 for the velocity, and all to the n since there are n particles. Now in this space we can take the surface of constant energy say E=n/2, so that the average energy of a single particle is 1. This surface has finite surface area, so we can put a uniform probability distribution on it, and ask what the distribution of the first particle's velocity is.
Is said distribution the Maxwell speed distribution, in the limit as n->infinity?
In other words, is the Maxwell speed distribution just the distribution for the velocity of a particle found in a system chosen uniformly over all systems of the same energy E?
Thanks in advance!