Microcanonical Distribution, Equally Weighted Microstates, Entropy and Virtual Work

The microcanonical distribution for a classical gas involves a fixed number of particles N and a fixed total energy E. The energy may be distributed in different ways among the N particles such that its total is E and each distribution or microstate is said to have the same weight or probability. We argue that the idea of equal weight follows directly from Newtonian mechanics. For a dense gas, the collisions in the gas are Newtonian and if there were a greater weight to have e1,e1,e1 (for three particles) than 3e1,0,0, there would need to be a reason for this coming from Newtonian physics. In other words, this is not a statistical issue. Equal distribution of energy among particles has important consequences because one may immediately write P(ei)P(ej) = P(ei+ej), if one relaxes the total energy constraint (canonical distribution). This leads directly to P(ei) = C exp(-ei/T). The underlying idea, however, is present in the microcanonical distribution. The idea that energy distributions carry the same weight also leads to the notion of the maximization of the number of distributions or the maximization of entropy for the gas. Thus we argue that maximization of entropy subject to a constraint on total average energy is not an independent statistical idea, at least for the usual classical gas, but follows from Newtonian mechanics. In (1), it is argued that the equal weights for microstates in a microcanonical gas follow from the notion of virtual work being zero. We argue that the equal weights follow from Netwonian mechanics which does not show any preference for e1,e1,e1 versus 3e1,0,0 for three particles