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Considering a closed system with an ideal gas(in a low entropy state) inside, then are following statements correct?

The gas is in a certain state we can assign an entropy value to.

Let X be the set of all states which are of a lower entropy value compared to the current state.

Y the set of states of higher entropy value etc.

We can assign a probability value to each of the states within X and Y, giving us the chance of the gas reaching any of those states from the current state.

Question: Are there states with exactly zero probability, which cannot be reached at all from the current state(without having to go through other states)? In general, are all states equally probable or do the probabilities to reach a given state depend on the current state of the gas?

The probability of reaching a state of higher entropy contained in Y are higher than reaching one in X.

However, it is not impossible to reach a state within X, therefore (rarely) decrease entropy. Hence, it is more likely for entropy to increase, rather than decrease but not impossible for it to decrease.

At some point, as entropy keeps increasing more often than not, we reach the equilibrium state of the gas.

Question: Is the equilibrium state of the gas really the maximum entropy state? If yes, then do all other states which are contained in the set of X have exactly zero probability of occurring?

Or is the equilibrium state a state where the probabilities of reaching one of the states contained within X is equal to reaching one of the states contained in Y?

Therefore, there are states of higher entropy still, which are not equilibrium states and where reaching one of those states from the equilibrium state is just as likely/unlikely as reaching a state contained within X of lower entropy.

If this was the case, then even the statement of "entropy is more likely to increase than decrease" would not be true any more once the equilibrium state is reached.

Question: If above is not the case, then is there a quantum mechanical proof for it?

Question: Can the gas being at maximum entropy fall back to its initial state of minimum entropy just out of sheer coincidence? Extremely low chance, but possible?

Before you close my thread, this is what Susskind explains 24:44 within this video

also in this video, he states that entropy ALMOST always increases but not always at 16:44 into the video

The gas is in a certain state we can assign an entropy value to.

Let X be the set of all states which are of a lower entropy value compared to the current state.

Y the set of states of higher entropy value etc.

We can assign a probability value to each of the states within X and Y, giving us the chance of the gas reaching any of those states from the current state.

Question: Are there states with exactly zero probability, which cannot be reached at all from the current state(without having to go through other states)? In general, are all states equally probable or do the probabilities to reach a given state depend on the current state of the gas?

The probability of reaching a state of higher entropy contained in Y are higher than reaching one in X.

However, it is not impossible to reach a state within X, therefore (rarely) decrease entropy. Hence, it is more likely for entropy to increase, rather than decrease but not impossible for it to decrease.

At some point, as entropy keeps increasing more often than not, we reach the equilibrium state of the gas.

Question: Is the equilibrium state of the gas really the maximum entropy state? If yes, then do all other states which are contained in the set of X have exactly zero probability of occurring?

Or is the equilibrium state a state where the probabilities of reaching one of the states contained within X is equal to reaching one of the states contained in Y?

Therefore, there are states of higher entropy still, which are not equilibrium states and where reaching one of those states from the equilibrium state is just as likely/unlikely as reaching a state contained within X of lower entropy.

If this was the case, then even the statement of "entropy is more likely to increase than decrease" would not be true any more once the equilibrium state is reached.

Question: If above is not the case, then is there a quantum mechanical proof for it?

Question: Can the gas being at maximum entropy fall back to its initial state of minimum entropy just out of sheer coincidence? Extremely low chance, but possible?

Before you close my thread, this is what Susskind explains 24:44 within this video

also in this video, he states that entropy ALMOST always increases but not always at 16:44 into the video

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