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If is it a closed system and a random process, it is unlikely to have the same entropy, it will likely have more entropy, and it will never ever have less.

No. Entropy is a state function. Random, non-random, deterministic, chaotic, quantum, spooky, etc., all processes lead to the same entropy for the same state. This is what Sewell seems to be missing in his appendix.

If is it a closed system and a random process, it is unlikely to have the same entropy, it will likely have more entropy, and it will never ever have less.

No. Entropy is a state function. Random, non-random, deterministic, chaotic, quantum, spooky, etc., all processes lead to the same entropy for the same state. This is what Sewell seems to be missing in his appendix.

I stated the second law of thermodynamics and you said "No"?

$\frac{dS}{dt} \ge 0$

Of course entropy is a state function. It is a measure of the thermodynamic state at a particular point in time. If time x occurs before time y, the state at time x will always be have less or equal entropy to the state of entropy at time y, for a closed system.

"all processes lead to the same entropy for the same state"

The state changes with time. That is like saying the entropy at time x is the same as the entropy time x. Maybe all of your systems are assumed to already be at equlibrium, but even then the statement is a strech. Maybe you have invented a way to stop time (travel the speed of light), but I doubt it.