Confused about micro, macro, ensemble, accessible, possible ect. states

Question

I'm hopelessly confused about accessability of microstates and macrostate. Let's imagine ideal gas box, A, with parameters:

• $T_\mathrm{A}$ (temperature)
• $P_\mathrm{A}$(pressure),
• $V_\mathrm{A}$(volume),
• $N_\mathrm{A}$(atoms)

So the enerrgy $E_\mathrm{A}$ of the system A is defined by the stating those parameters.

Let's add another system B with parameters: $T_\mathrm{B},\ P_\mathrm{B},\ V_\mathrm{B},\ N_\mathrm{B}$. They are for the time being do not interact. They are all in equilibrium.

System A has only one macrostate $(T_\mathrm{A},P_\mathrm{A},V_\mathrm{A},N_\mathrm{A})$, since all params are fixed. Yes, there are many accessible microstates for it to realize the macrostate. The same could be said about system B.

Now the confusion begins. Let's join the systems together! Immediately

1. Energy of the combined system C is equal to $E_\mathrm{C}=E_\mathrm{A}+E_\mathrm{B}$.
2. System C is not in equilibrium $T_\mathrm{A}\neq T_\mathrm{B},\ P_\mathrm{A}\neq P_\mathrm{B}$ etc. but... System C has many equally probable - accessible microstates, but system goes (starts to search) to equilibrium and attains it at some more accessible and more probable macrostate which clearly would not be the same as in the beginning.

I mean... why system C will search for equilibrium if all the states are equally probable and we are at ONE right now (non-equilibrium state)?

Why we have two systems with ONE macrostate but combined sistem (it looks like) will go through many macrostets before it attains equilibrium?

What is the difference between:

1. ensemble and accessible microstates?
2. ensemble and one particular macrostate?.
3. How do you call all microstates from the bulk of which you cut out small portion which you call "accessible microstates"?
4. Does the phrase "accessible microstates" imply only one possible macrostate always?
5. When does "accessible microstates" imply that there are many macrostates possible?

P.S. I know that the question sounds and feels confused. I can't formulate it clearer... I'm lost.

Answer 1

In reference to your questions;

for '1.', for the microcanonical case, they are the same, and the term accessible is added for the situations where a constraint may be placed.

for '2.' the ensemble corresponds to the macroscopic macrostate defined by the phase variables utilized.

for '3.' for a macrostate there can be a subset of microstates possible given by eg $\Omega(E)$ which gives you the number and that number has to include only the accessible.

for '4.' in the microcanonical example it is typical to refer to the accessible microstates in reference to a particular macrostate.

I think that you are reading too much into 5. The combined system, has a different energy, and therefore a different macrostate with a set of equally probably microstates. These microstates which correspond to the new macrostate will be reached as it approaches equilibrium. (it is stating in other words that the new system has a different macrostate $E_c$ than the previous system and that it will deviate from either systemA or systemB during the 'search')