Per Mark's request, I'll provide an answer.
First, I think it's not possible to obtain such a formula. One can of course naively try to extrapolate various classical formulae but all of these attempts are bound to fail. Here's why: even when constructing relativistic quantum mechanics one finds that the theory is not consistent. For example, in non-relativistic quantum mechanics one has a position operator that can be used to get information on precise position of the particle (up to the uncertainty given by Heisenberg's uncertainty principle). But when one includes relativity into the picture the theory stops being consistent. This reflects the fact that to localize the particle with great precision one has to make experiments with increasingly high energy and at certain point the energy is enough for the appearance of new particles. In fact, creation and annihilation of particles is inevitable in relativistic regime (and in some systems it's not even clear what particles should be and one has to talk about fields instead). The situation is even more pronounced in statistical physics where there are huge number of particles present.
More importantly, there's no need to get such a formula. Consider systems for which it would be useful. Such systems would have to be extremely non-classical (like neutron stars, black holes interior, quark-gluon plasma, etc.) and the concept of velocity would have no meaning as there's no way to observe individual particles of these systems; which is in contrast to the classical case where you can test Maxwell distribution by letting the particles out of the box one by one and seeing how fast they are (the actual experiment is lot more sophisticated of course, but that's unimportant here).