All this is very well for the computer, but what about the real world? Well, the experimental data shows that real gases do have speed distributions very closely given by Maxwell's result; moreover, there is plenty of evidence to show that real gases, initially prepared in a variety of nonequilibrium states, do approach eventually the Maxwell distribution as they become "thermalized." So the question is, how do real gas molecules "lose their memory" of their initial positions and velocities? Put slightly differently, don't two real molecules, when they collide, "compute" their resulting velocities to "infinite" precision?
The answer is, in fact, no! Such "infinite precision" is physically impossible because of the (quantum-mechanical) uncertainty principle, which states that x and p (the position and momentum) for a molecule cannot be defined simultaneously to a better acuracy than that given by
, where
Planck’s constant h is of the order of 
J.s. Now, for an air molecule at
room temperature, the momentum may have a value of the order of 
kg.m/s, and the molecule's size
may be of the order of 
m. This means
that one cannot simultaneously specify the momentum to better than,
say, one percent, and the (center of mass) position to better than a
fraction of the molecule's size, without violating the uncertainty
principle! Put another way, the simultaneous 15 significant figures
for x and v used by the computer do not even
exist for a real, physical (quantum-mechanical) molecule. One
should keep only about two decimal places--and the real system would
then be even less microscopically reversible that the one in
the computer simulation.(This idea may seem startling, but, in fact, it is just another way to state the quantum-mechanical "solution" to a paradox of classical statistical mechanics: How can the entropy of a closed system increase? The answer is that it can if it is defined as a "coarse grained" quantity: that is, the classical phase space of position and momentum is divided into cells and we declare that all points inside a given cell represent essentially the same microscopic state. Classically, however, this is arbitrary, because there is no natural scale for such a phase-space cutoff; but quantum mechanics provides a natural scale, in the form of Planck's constant. All we have to do is use cells of area h in every two dimensional x-p slice.)
In spite of the above, it may be too much to claim that quantum mechanics solves the classical reversibility paradox, with which so many physicists (most of all, Boltzmann) struggled for so long. Rigorously, one can only say that the uncertainty principle shows that the classical reversible-dynamics description is really not accurate already after the first collision; but one cannot just conclude that the "correct" way to run the simulation would be simply to treat the molecules classically but keeping only two decimal figures in every collision. Rather, the truly correct thing to do would be to use quantum dynamics throughout, but this is extremely difficult--unworkable, really, if one wants to use more than a couple of wavepackets. Fortunately, the validity of the Maxwell equilibrium distribution can also be established (in a very different way) for a quantum-mechanical system of molecules. Unfortunately, one can also ask difficult questions regarding entropy growth in quantum statistical mechanics. But this is, really, quite another story.