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msumm

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A micro-canonical ensemble is one in which the probability of finding the micro-state in any part of S is equally likely (the density function is constant over S).

In Pathria's Statistical Mechanics Section 2.3 it is explained that the ensemble average of any physical quantity is the same as the time average of that quantity (for a microcanonical ensemble). I understand/agree with that statement, but I don't understand the route he took to explain it. Particularly, when he says "the time average of any physical quantity must be the same for evey member of the ensemble." Why should that be true for _any_ physical quantity? I agree that it should be true for the physical quantities used to descibe the macrostate in the first place (E in this case), but not necessarily true for others. Using Pathria's example consider the hypershell of phase space consisting of all points with energy in the range [E,E+dE]. Since we haven't specified the temperature, aren't there some points in there which will traverse a trajectory with constant T and mu, while others will have a trajectory with (different) constants T' mu' such that they both have roughly the same energy? If so, the time averages of those quantities are not the same.

After typing this I'm starting to suspect that the problem is that the macrostate is supposed to specify ALL measurable macro properties, in which case I agree with the quote above and my example is bad. Is that right? If so, then it seems strange that Pathria consistently picks out macrostates by giving an energy range only, since that wouldn't really specify a macrostate, right?