Thermodynamics – Why Does Entropy Tend to Increase if All Microstates Are Equally Probable? How to Understand Entropy Increase

Question

One of the assumptions of the thermo textbook I'm reading is that the probability of any given microstate of a system is equally likely to occur. This does not mean that the macrostates of the system are all equally probable, since there will usually be several microstates corresponding to a particular macrostate which is expressed as the multiplicity of the macrostate.

1. Under the assumption that every microstate is equally probable, the formula for entropy is $S=k_B\ln(\Omega)$. Hence, $\Omega=e^{S/k_B}$.
2. The second law states that every thermodynamic system tends toward higher entropy. Since the exponential function is increasing, higher entropy implies higher mutliplicity.
3. As the multiplicity of a system increases, the number of macrostates with higher multiplicity decreases.

But doesn't this imply that there should be a steadily decreasing probability of entropy increasing as there are fewer and fewer states with higher entropy? And once the system achieves maximum equilibrium, isn't the system almost certainly going to revert to a state with lower entropy? There must be something I don't understand.

Answer 1

2) The second law states that every thermodynamic system tends toward higher entropy. Since the exponential function is increasing, higher entropy implies higher mutliplicity.

3) As the multiplicity of a system increases, the number of macrostates with higher multiplicity decreases.

I don't think I fully understand your complete argument, but I think your flaw might be in one of these two points.

It is true that entropy and multiplicity are related, but it seems to be like you are considering the change in one of these values as a cause and the change in another one of these values as an effect, when in fact they are exactly the same thing.

For the purposes of this question, you can essentially reword the second law as "the most likely thing will definitely happen". So the system will just go to whatever macrostate has the most microstates because that is what is most likely. It doesn't really matter how many macrostates exist for some multiplicity.

Of course you can get into finer details. If there is a single macrostate with the largest multiplicity, and if there are fluctuations that causes our system to be in a different macrostate, then the entropy would technically decrease. But then the system will just move back to the higher entropy state right after. It will not keep decreasing in entropy. And even then these fluctuations are probably not large enough to be concerned with. Imagine the highest entropy state as a stable equilibrium, where any small perturbations won't kick us out of equilibrium, and on average we can choose to ignore the perturbations.

Like I said, I don't know if I fully understand your question, so I'm sorry if this answer seems like I'm just throwing information and thoughts out to see what sticks.