Maxwell speed distribution

yall

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!

mfb

The usual derivation assumes that the particles are in contact with some external reservoir and that the total energy can vary a bit. In the limit of infinite particles, I would expect that an exact energy gives the correct result, too.