Quote from: iamnotback on March 03, 2017, 01:49:15 AM

Quote from: dinofelis on March 02, 2017, 03:11:41 PM

That really doesn't matter. My cup of hot water is also entangled with the rest of the universe, but that doesn't increase its entropy with respect to me.

Quantum mechanics models the microstates superimposed, because otherwise measuring the position at any point sample in time is aliasing error (a random result). Analogously we must model the entropy as superimposed into the future in our macro perspective.

You are losing me here. I can read that 10 times, and still not know what it might mean. I try to wrap my mind around what you mean with "microstates superinposed". If the system is isolated and in a pure state, it is of course in one single state (which may be a superposition of "base states" in which I like to express this state, but it is still just one single quantum state). If the system is entangled (which it most certainly is) with the rest of the universe, it is entangled, which means exactly what I wrote earlier: several individual quantum states of the cup are, well, entangled, with states of the environment. And that is exactly the entropy we are talking about: the number of those states that is involved. If there were only state, the entropy would be 0, and if there are many, the logarithm of 2 of that number of states is the entropy of my cup (I'm assuming uniform distribution, which is not necessarily true, but which would diminish the entropy if not uniform).

There's no such thing as "cumulative entropy over time". I think that is the error you are making, by thinking that the entropy is about the *entire history - and future* of a system. No, entropy is the ignorance of the *current state*, not about its entire past and future. That doesn't make sense, because if that were the case, entropy wouldn't be a time-dependent concept, and the second law could not even be formulated !

It is as if you were saying that the "velocity of an object is the whole of movement that an object did and will do in the future" or something. But if that is the case, you couldn't talk about the acceleration (the *change* in velocity) and not write down Newton's second law ! Velocity is instantaneous, now, and doesn't care about the state of motion yesterday or tomorrow. If it did, velocity would be an a-temporal notion, and you couldn't ask how velocity changes over time.

In the same way, entropy is an "instantaneous" notion, because the second law of thermodynamics tells us how entropy needs to CHANGE over time. If all our past and future ignorance were already included, the entropy tomorrow would be the same one as today, as all ignorance would already been included, as well today as tomorrow.

I think you are confusing the notion of entropy itself, with the notion of dynamics. If I know the dynamics of an isolated system perfectly well (which is very rare), then there's a kind of Liouville's theorem that tells me that my knowledge of the microstate (even if imperfect) is conserved. When I know something about the microstate today, that knowledge is still pertinent tomorrow, and my entropy about that micro state didn't grow.
If however, my knowledge of the dynamics of the system is not perfectly accurate, OR the system interacts with the environment (of which, by definition, I don't consider the dynamics), then my knowledge of the microstate today will probably get lost gradually over time: my entropy of the system grows, until it reaches its full value of my total ignorance of it, given by one or another canonical statistical ensemble, constrained only by a few macroscopic parameters.

This "loss of knowledge over time" is intimately related to the second law. It is always related to "the environment" of which the unknown microstate is much more entropy-rich than my system, and is an infinite source (for all practical purposes) of ignorance (= entropy).

This is also why the notion of "entropy of the universe" is problematic. The universe cannot be entirely observed from the outside, and cannot connect to "its environment" ; as such, this notion breaks down.