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I've read a bit about the 2nd law of thermodynamics.

As I understand it, no process can happen if it's not increasing the overall entropy of a system, therefore the universe is always going towards more entropy.

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Right (statistically speaking, of course, which I'm sure you understand).

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And as I understand that, in a system with total entropy, there's no time.

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Right -- starting with a state of thermodynamic equilibrium (maximum entropy), you won't be able to define a thermodynamic arrow of time. There will be occational fluctuations from equilibrium, but these take place equally often in either direction of time (evolving "backwards" or "forwards").

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Now my question is, is this change towards entropy slowing, accelerating, or neither? Put another way, does the 2nd law imply that the universe will at a certain point reach total entropy, or could it be the case that even though it has more entropy as time progresses, it can never reach a state of total entropy since this change is being slowed by orders of magnitude?

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It's very difficult to be precise here since the observable universe isn't a closed system -- things can, for example, fall beyond the cosmological horizon and then never interact with us again. However, off the top of my head I'd tend to think "locally" that the increase of entropy is slowing -- stars are entropy producing factories (eventually producing lots of thermal radiation, heavy elements, and black holes), and the first generation of stars were extremely short-lived compared to the ones currently burning.

Ultimately in standard cosmology, though, we will reach a state of "asymptotic de Sitter space," which will be in a state of thermodynamic equilibrium at some incredibly minute temperature.
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A second question - if we can choose what process to let happen (say use an engine vs using horsepower, etc), does that mean we get to choose to some extent the speed at which the entropy of a system progresses to?

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Yep. Who knows, maybe if hyper-intelligence eventually saturates the cosmos, we'll be able to postpone the heat death of our corner of the universe for a very long time.

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And a third, related one: If a system has an X "level" of entropy, does that mean that in average, all processes will yield Y progress in entropy? In other words, given a set amount of entropy, is the average speed at which processes produce entropy fixed, or can it vary?

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It can vary.