This is the fundamental assumption of statistical mechanics:
In an isolated system in thermal equilibrium $^1$, all accessible microstates are equally probable.
But why does it mention the condition thermal equilibrium?
As Daniel V. Schroeder mentions:
Any large system in the equilibrium will be found in the macrostate with the greatest entropy.
Equilibrium is the state where there can only be that macrostate which has the greatest multiplicity that is the greatest entropy.
Also, check this pic taken from here; it clearly mentions thermal equilibrium at $q_A =4$; there is only one macrostate corresponding to $q_A= 4$ in thermal equilibrium and not more than one.
But the ergodic hypothesis says otherwise; it says in thermal equilibrium, all microstates are equally probable. Now, among those accessible microstates, there are also many which correspond to $q_A\ne 4$; these microstates define such macrostates which have lesser entropy than the macrostate to that of $q_A=4$ in the example. So, how can it be possible for those microstates of lesser-entropic macrostates $q_A\ne 4$ to exist at thermal equilibrium? For, there can only be one macrostate i.e. $q_A= 4$ at thermal equilibrium.
The famous line goes as:
Entropy is maximum at equilibrium.
You can't expect microstates of lesser-entropic macrostates in thermal equilibrium, isn't it? Equilibrium exists only at the macrostate having the greatest number of microstates; ie. $q_A= 4\;.$
So, how can I say, all microstates are equally likely in thermal equilibrium if microstates other than that of the macrostate having greatest entropy $q_A= 4$ can't exist in thermal equilibrium?
Best Answer
A macrostate is a set of microstates. Some microstates are thermal, others are not.
Without the assumption of being in thermal equilibrium you can't assume anything since any possible microstate is possible. And lots of possibilities macrostates could be picked.
Usually you want to group your macrostates according to a state variables such as pressure, volume, total energy or something like that.
And when you break your 30 microstates into three groups: A, B, and C you can ask yourself if each group is classified according to a state variable such as pressure, volume, total energy or something like that.
And even if it is, then all you might know is the state variable, and even that maybe not precisely. For instance the volume isn't precisely known since the exact locations of all the many parts is not known.
Now even when the microstate is a particular microstate, and that microstate is assigned specifically to a particular macrostate, that doesn't tell you how that macrostate assigns probabilities to the microstates in it
And you can assume that each microstate in the macrostate is equally likely. But that is just an assumption. If energy is conserved, then the dynamics will always keep the total system at a configuration with that fixed initial energy, so it doesn't change from any state to just any state.
It means so much more. Firstly, it requires that a macrostate is a probability distribution on microstates. Secondly, the macrostate is specified by some (macro) state variables. Thirdly, the particular distribution specified by the hypothesis requires that the space of all microstates be partitioned (partitioned by different values of the state variables) and each partition has an equal probability assigned to every microstate in that part of the partition. I always imagine different floors of a mansion, where your variables constrain you to a different floor and each room in a floor is equally likely.
Now, thermal equilibrium doesn't mean having the greatest multiplicity. You could have $N$ particles of some gas at a certain pressure $P_0$ and volume $V$ and that could be one macrostate, and there might be macrostates with a larger multiplicity with the same $N$ and $V$ and larger $P_+\gt P_0$ but there isn't enough energy for those $N$ particles to have that pressure, there just isn't enough kinetic energy to spread around to get them to the $P_+\gt P_0$ macrostate variable.
If you want to go to the mansion example. Imagine that you have two mansions and one person can go up a floor if (and only if) the other person goes down a floor. When they are both by some stairs then one can go up and the other can go down. But if that places one of them into a floor with trillions more rooms available than the other one had, then they are way way more likely to stay in the configuration with the one stuck in the floor with way more rooms.
So energy can be exchanged between the two people, but the additional energy spends most of it's time with one of them having more energy if they can exchange it. Eventually they could get to a level where one gains as many rooms as the other one loses. And that joint collection roughly is the macrostate of the combined system.
And when that happens we say they are thermal equilibrium. And the thing they have in common, temperature, is how many additional rooms they gain per bit of energy. Maybe one has stairs that are longer, so going up/down one flight for it is going up/down 5 flights for the other. But maybe the rate at which the floor have more rooms changes with floors at a different rate. There could still be a $\textrm dS/\textrm dE$ in common.
The macrostate could, in principle change from thermal equilibrium and more to have to two subsystems be at different temperature instead of the same temperature. But that would require bouncing around until you are by some stairs going down instead of towards any of the many more options on staying on the same floor, and then continuing to do the improbable floor after floor until the temperature of the two subsystems are very out of line.
And even if it happened, it could just go back to equilibrium. The idea is that for a large enough system, the time to wait for such a thing to happen is just really really long.