Some questions about todays lecture

(1) "This times averages = microstate averages part... what are we referring to by <A>, and is the delta just 1 for the correct N,V,E and 0 for the incorrect N,V,E?"

Yes, exactly. That delta just means that we only sum up A for those configurations that have the right N, V, E.

A is any property we want to calculate that comes from microscopic configurations. It could be something like "the average number of hydrogen bonds a water molecule makes with its neighboring molecules" or "the average rate at which molecules bounce off of the walls of the container (which could be used to calculate the force, and hence, the pressure)". It is just a general property that, given a list of velocities and positions of all the atoms, we could have a recipe to calculate it.

So all we are saying is that one way to measure A would be to, say, check the instantaneous molecular configuration every minute, calculate A for that configuration, and then tabulate it. We would do this for a long time, so we get a long list of A values. Then we take the average of them. This is the time average.

The alternative way we can do this, at equilibrium, is find out what the N, V, and E values are for that same system. Then we just take *all* of the microscopic configurations that have those values, in a very systematic way, and we average A over those. This average will be equal to the time average because the rule of equal a priori probabilities says that we are equally likely to pick any one of these microstates at any moment in time.


(2) "Can you say S = kB ln Omega(N,V,E) = S(N) + S(V) + S(E)?"

You can only say this in the case that we can write Omega(N,V,E) = Omega(N)Omega(V)Omega(E). In general this is not the case. The reason is, the number of ways of arranging N molecules in a volume V with energy E is usually not separable. For example, the system today in class, where we had a bunch of toy molecules that could each either be in an energy=0 or energy=1 state. If there are N molecules, and we want to find out how many ways there are to have a total energy of E, then the number of ways is N!/E!(N-E)!. There is no way to separate this into two functions f(E)f(N).