Ensembles, Ergodicity, and Entropy

Wherein our author continues his attempts to understand stat mech, as inspired by the critiques of Arun and Lumo.

What is the relationship between the ensembles of statistical mechanics and the physical systems that they represent? In the case of a classical system, its coordinates qi and momenta pi (i=1…N) are supposed to have definite values at each point of time, so that the state of a system at an instant should correspond to a single point in the 2 N dimensional Hamiltonian phase space (actually, symplectic manifold). In statistical mechanics, however, we represent such a system by the whole ensemble of points in the manifold that is compatible with the macroscopic description of the system. Moreover, essential thermodynamic properties of the system such as temperature, entropy, and free energy are defined (statistically) only in terms of the ensemble, not in terms of an individual phase space point (or microstate).

So how is the unicity of the classical state to be reconciled with the fact that real physical systems in equilibrium have temperature, entropy, etc.? There are two traditional explanations in statistical physics. Ergodicity is the notion that for typical macroscopic systems, the individual p’s and q’s are evolving very rapidly compared to the time for, say, a temperature measurement. Consequently, the phase space point representing the system is moving rapidly in its phase space and exploring a lot of it, so any temperature (say) measurement is averaging over a large number of different phase space configurations. It turns out that such time averages are usually hard to compute, so the other approach, Gibbs idea of an ensemble of systems, has proven more fruitful. In that approach, the macroscopic system is represented by the complete collection or ensemble of microsystems (a volume of the phase space, in effect) compatible with the macroscopic parameters.

This does still pose the problem of unicity vs. the ensemble, however. How can a macroscopic classical system, with its presumably unique location in phase space, be compatible with averaging over a volume. One attempted solution is the ergodic hypothesis, which posits that the time average and the ensemble average are equal. This is hard or impossible to prove. The second approach, which I like better, is based on the idea that if a system is in equilibrium (the only conditions under which temperature and entropy are well defined), some states are overwhelmingly more likely than others.

Suppose for example that we have a known amount of an ideal gas confined to known volume at a known temperature, and we wish to measure the pressure. In our ensemble of microstates that is compatible with our known parameters, there are a small number for which no gas particles will impinge upon our pressure sensor during the measurement period, and consequently that the pressure sensor will read zero. In this view, the predictions of stat mech are purely probabilistic – the most probable reading for the pressure is p = nRT/V, but there is a small probability of zero.

So, given a microstate (say ten atoms in a box, all of which happen to be in the left half of the box at the moment), does it make sense to talk about its entropy? Only, I think, if you happen to slip in a partition at that moment, which confines them to that side. It’s like Feynman’s joke: “Driving to work today, I saw a license plate with the number XKL-395. What do you suppose the probability of that is?” It’s the same, of course, as AAA-000, or any other plate, even if our pattern recognition facilities find it less special.

So where does that leave me? All the way back to Arun, at least, and part of the way to Lubos, though with Lumo I often have trouble deciding exactly what he does mean. Is a fluctuation that puts all the oxygen atoms into one half of the room enough to make a drastic reduction in entropy? If we treat the room contents as a perfect gas, the answer is no. If we treat it as a real gas with realistic diffusion times, the answer could be different.

There are lots of books on stat mech, but most avoid much talk of the conceptual foundations, which might be one reason a lot of physicists are a bit vague themselves. One that is devoted the the subject is the almost century old Paul and Tatania Ehrenfest book.