Ensemble Avg, Statistical Mechanics

[msumm]

Given a macro-state M of a system, let S denote the potion of the phase space that has the macro-state M.

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?

 

[0xDEADBEEF]

Ok I didn't quite follow you but the answer might be this: The statements are about almost all trajectories in the mathematical sense. As a corollary to a theorem by Poincaré the usual trajectory comes arbitrary close to every point in energetically allowed phase space if we wait long enough. So whatever your second trajectory is, the first one will get as close to it as it can at some point in time, and whatever physical property is connected to it will similarly get as close to it as possible, if we assume that there is a continuous map from the phase space to that property.

 

[msumm]

But a difference of energies between members of the ensemble is allowed as well. Consider the example Pathria is using when explaining this stuff -- an ensemble of states with energy in the range [E - delta, E + delta] (i.e. the energy is not locked at a particular value). Any element of the ensemble with energy E-delta will stay in that hypersurface while another element with energy E+delta will stay in its hypersurface. So both will maintain constant energies of different values. So how can he say the time average of any variable (like E) is the same for all elements of the ensemble?

 

[msumm]

I wonder what the assumptions are in that theorem/corollary by Poincare? I ask because if you think of a 2D pendulum as an example, the bob of the pendulum can swing about any axis in a plane (say the "x-y plane"). If I start such a pendulum swinging about the x-axis only it will continue to swing around the x-axis only, just like a 1D pendulum. Thus, it will never reach most points in phase space with same erergy (for example the points reached by a pendulum swinging about the y-axis with same energy).

In terms of my original question, what I'm saying is that the time average x-velocity of different memebers of an ensemble will be different.