Quantum Question
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 gmalivuk
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Re: Quantum Question
Treating entropy as a measure of disorder is a pretty typical convention in physics. A maximal entropy system is "uniform" only at a macroscopic (anthropic) level. If every particle has a random position and velocity, that's not very ordered at all on a microscopic level.
 doogly
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Re: Quantum Question
tomandlu wrote:ucim wrote:Naively, it seems to me that time's arrow is a result of the third law and the uncertainty principle, and as such, is a macroscopic thing. Consider an explosion; a classical case of increasing entropy. During the explosion (like at most any other time), the future is likely to be more disordered.
Jose
Is "more disordered" the right description? Disorder could be regarded as the domain of minimal entropy. Aren't maximal entitled systems incredibly uniform? They're only really disordered from an anthromorphic pov.
I like your "why we can't unexplode" hypothesis, but it's slightly irrelevant to my burbling, since I'm pondering whether the uncertainty might disappear if quantum particles are time agnostic.
Quantum mechanics *is* time reversal invariant (except for some weak interactions, which are mostly irrelevent for the macroscopic limit.)
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Re: Quantum Question
So.... could you unexplode a bomb by reversing time? Or would the particles likely pick other paths upon interaction?doogly wrote:Quantum mechanics *is* time reversal invariant (except for some weak interactions, which are mostly irrelevent for the macroscopic limit.)
Jose
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 doogly
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Re: Quantum Question
The paths of bomb shrapnel are not sensitive to quantum mechanics. Much more just classical mechanics. All you'd have to do is reverse time, yeah.
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Re: Quantum Question
Maybe not the big pieces, but what about the paths and interactions of the individual molecules responsible for the chemical reaction that made the explosion in the first place? And the molecular bonds that held the (now) pieces together in the first place?
Jose
Jose
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 doogly
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Re: Quantum Question
Yup. That's why chemistry is boring. Not quantum enough.
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Re: Quantum Question
Ok, could you unexplode an atomic bomb?
Jose
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 Eebster the Great
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Re: Quantum Question
ucim wrote:Ok, could you unexplode an atomic bomb?
Jose
Time symmetry is pretty much conserved outside of kaon physics. I don't think CPviolation plays much of a role in nuclear bombs.
Re: Quantum Question
Either I don't understand, or we're talking about different things.
Consider two subatomic particles that are on a collision course. Their interaction, should they get close enough, is governed by QM. When they "bounce" off of each other, there will be a certain probability that they will bounce at a certain angle, but it isn't a certainty that they will. In this particular case, they did. Now, reverse time. These two particles are coming at each other from the opposite direction, and
1: They will "bounce" exactly back at the same angle, retracing their path during the interaction, or
2: There will be a certain probability, but not a certainty, that they will bounce back at that angle, thus not retracing their history.
Choose one.
I don't think this is a case where time symmetry is involved, as (AIUI) time symmetry says the laws are invariant under time reversal, not that the states are invariant. It seems to me that choice 2 would be consistent with QM, and be the source of time's arrow.
Where have I gone wrong? (Other than the conceit that we can talk about particles in the first place)
Jose
Consider two subatomic particles that are on a collision course. Their interaction, should they get close enough, is governed by QM. When they "bounce" off of each other, there will be a certain probability that they will bounce at a certain angle, but it isn't a certainty that they will. In this particular case, they did. Now, reverse time. These two particles are coming at each other from the opposite direction, and
1: They will "bounce" exactly back at the same angle, retracing their path during the interaction, or
2: There will be a certain probability, but not a certainty, that they will bounce back at that angle, thus not retracing their history.
Choose one.
I don't think this is a case where time symmetry is involved, as (AIUI) time symmetry says the laws are invariant under time reversal, not that the states are invariant. It seems to me that choice 2 would be consistent with QM, and be the source of time's arrow.
Where have I gone wrong? (Other than the conceit that we can talk about particles in the first place)
Jose
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 Eebster the Great
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Re: Quantum Question
ucim wrote:(Other than the conceit that we can talk about particles in the first place)
In order to talk about possible histories of particles in isolation, I think you need local realism, which isn't going to happen. MWI is the closest, but it definitely doesn't have true particles the way you mean, and except for CPviolating terms in the weak interaction (and, hypothetically, in the strong interaction, though this has not been observed), it is time symmetric.
The CPviolating terms turn out to be pretty irrelevant in practice, except I guess for neutral kaon decay. They are responsible for the "weak arrow of time". The way we actually experience time (storing memories, etc.) is due to macroscopic phenomena which are governed by thermodynamics. Entropy is higher in the future and lower in the past, by definition. This is the "thermodynamic arrow of time." The reason we can make this distinction in the first place is the existence of a relatively low entropy point in time (roughly corresponding to inflation).
A universe at thermodynamic equilibrium would have no meaningful past or future direction of time, not even in the weak sense, as CPT symmetry is still conserved. In other words, switching every particle with its mirrorimage antiparticle would be equivalent to switching the direction of time.
Re: Quantum Question
I think the thing Jose is trying to ask about is exactly that entropic arrow of time. Entropy always goes up, so if you have a simulation of the universe that correctly models that property, and then you reverse the momenta of everything in that simulation, entropy is still going to go up in the future of the simulation, rather than going down as the simulation returns precisely to the same state it was previously. Right?
I think the specific question is whether quantum mechanical randomness is at fault for that necessary progression of entropy; whether if, in a strictly deterministic universe, exactly reversing the momenta of everything would result in a future state identical to a previous state (lower entropy and all), and the reason we would not expect that in our universe is because it's not strictly deterministic like that.
I think the specific question is whether quantum mechanical randomness is at fault for that necessary progression of entropy; whether if, in a strictly deterministic universe, exactly reversing the momenta of everything would result in a future state identical to a previous state (lower entropy and all), and the reason we would not expect that in our universe is because it's not strictly deterministic like that.
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 doogly
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Re: Quantum Question
Pfhorrest wrote: Entropy always goes up, so if you have a simulation of the universe that correctly models that property, and then you reverse the momenta of everything in that simulation, entropy is still going to go up in the future of the simulation, rather than going down as the simulation returns precisely to the same state it was previously. Right?
Entropy doesn't always go up, it tends to. If you cherry pick a state, like one with a bunch of shrapnels all with momenta pointing back in towards a bomb, you can get entropy to go down. Cherry picking initial conditions makes entropic arguments very deflated.
As for the prior claim, it does precisely break down at the point where your quantum state can be described as a particle with a classical trajectory. Uncertainty will bork up the replayability of that trajectory. But if you are drinking deep and doing a quantum state that is evolving with e^iHt, you can do t. It's cool. Cauchy surfaces and all that.
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Re: Quantum Question
doogly wrote:Entropy doesn't always go up, it tends to. If you cherry pick a state, like one with a bunch of shrapnels all with momenta pointing back in towards a bomb, you can get entropy to go down. Cherry picking initial conditions makes entropic arguments very deflated.
So if you cherry pick a state exactly like whatever state the universe happens to be in right now (where e.g. bunches of shrapnels are flying away from bombs and similar entropyincreasing processes are all in motion), except with all with momenta reversed, do you have a state of the universe that, if allowed to evolve normally, would evolve into a lowerentropy state? Would it then keep evolving to a lowerentropy state the longer it's allowed to continue evolving, because e.g. the moment after your cherrypicked state with shrapnels pointing back in toward a bomb still has those shrapnels pointing back in and so should continue to be a similarly special state, as should the next moment, and the next moment, and so on indefinitely? Or does something interfere with that and set entropy back toward increasing, disrupting the apparent reversal of time you set in motion when you flipped all the momenta?
The answer to that last question seems to be obviously yes, but what exactly is that something?
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 doogly
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Re: Quantum Question
If you did the time reversal to the entire universe, you'd replay back to the big bang, so pretty low entropy. If it were just your little bomb, eventually things would come in and interfere and your system would have its specialness aggregated away.
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Re: Quantum Question
So the only thing that stops a system that's locally evolving toward a lowerentropy state from continuing to do so is the fact that the rest of the universe, which is meanwhile in the process of evolving to a higherentropy state, interferes? And the rest of the universe just happens to be on a trajectory through the configuration space that's evolving toward a higherentropy state, but not every possible universe will always tend to always eventually evolve toward higher entropy? In fact, it seems from what you're saying that for every possible state of the universe that is on its way toward higher entropy, there is a corresponding possible state of the universe (namely, the one with all its momenta reversed) that is on its way toward lower entropy, and we just happen to be in one of those states of the universe that's on its way toward higher entropy.
You seem more knowledgeable than me about these things in general, but that all seems counter to everything I've learned before about the stochastic origins of the second law, namely that the universe always tends toward higher entropy because there are simply more possible higherentropy states than lowerenergy ones, so a random walk through the configuration space will tend to wander into higherentropy states just because of the odds.
You seem more knowledgeable than me about these things in general, but that all seems counter to everything I've learned before about the stochastic origins of the second law, namely that the universe always tends toward higher entropy because there are simply more possible higherentropy states than lowerenergy ones, so a random walk through the configuration space will tend to wander into higherentropy states just because of the odds.
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 Eebster the Great
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Re: Quantum Question
By definition, high entropy states are far more numerous than low entropy states. It is exceedingly unlikely for entropy to drop at a large scale. As I said, the reason we have this asymmetry is that the Big Bang was relatively low entropy (in a sense; things get kind of complicated when you bring in gravity and inflation).
Re: Quantum Question
That doesn't seem to address my question to doogly at all. What you're saying sounds like my prior understanding of the situation, but what he's saying sounds like it contradicts that: like it implies that there are just as many possible states of the universe that, left to their own devices, would tend toward lower entropy, as there are possible states of the universe that tend toward higher entropy. I thought that all possible states of the universe always tend toward higher entropy over time, but he seems to be saying one just like ours but with all the momenta reversed would just evolve back to the big bang again.
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Re: Quantum Question
Pfhorrest wrote:it implies that there are just as many possible states of the universe that, left to their own devices, would tend toward lower entropy, as there are possible states of the universe that tend toward higher entropy.
If you pick a random configuration with a given low entropy, then you'll find that both its evolution and its time reversed evolution will (probably) yield higher entropy results. The bijection you're trying to establish only works in our special little part of the universe, where the most common evolution is a gradient of slowly increasing entropy, i.e. it only works on a small subset of all possible states.
Re: Quantum Question
This implies that, given a certain state of the universe, there is a single definite resulting future which can be known in advance, because the "final" state of the timereversed universe is known in "advance". There would be no arrow of time. (Entropy is a statistical property of a bulk system, so doesn't really count.)doogly wrote:If you did the time reversal to the entire universe, you'd replay back to the big bang, so pretty low entropy.
But QM states (roughly) that, given a certain state of the universe, the future states are not predictable at all, except in a probabilistic sense at every interaction. This would imply that, given the timereversed state of the universe as a starting point, the big bang is not at all guaranteed, and is in fact unlikely. This creates an arrow of time.
It seems to me that you have to pick between these two. Are you sure the first one is the correct one?
Jose
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Re: Quantum Question
If you understand the state of the universe as the whole wave function, then its temporal evolution is deterministic (see: Schrödinger equation) and it is time reversible.
Randomness only appears when you try to force said wave function into a classical notion of particles with fixed positions and velocities.
Randomness only appears when you try to force said wave function into a classical notion of particles with fixed positions and velocities.
 Soupspoon
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Re: Quantum Question
I see it as just as the forwards prediction is subject to quantum indeterminacy that makes the exact state of the future fuzzy, in a reversed situation the backwards 'destrapolation' is also subject to quantum indeterminacy that makes the exact state of the past fuzzy.
(Needs to be resolved to chaotic divergence, perhaps, but it's perfectly possible to have been dealt the cards you have by having them be landed in front of you because of people throwing them at your table from several directions, rather than originating from the one deck controlled by the dealer (if you weren't there to witness it) so while you might assume a high probability of rewound time placing those cards neatly back in the hand of the dealer, it might not have been in so many different ways. Though it might be wise just to restrict your imaginative scepticism as to whether the dealer might have bottomdealt some of the cards after taking a sneaky peek, naturally, as a close alternative of the orthodox estimate of the starting setup.)
(Needs to be resolved to chaotic divergence, perhaps, but it's perfectly possible to have been dealt the cards you have by having them be landed in front of you because of people throwing them at your table from several directions, rather than originating from the one deck controlled by the dealer (if you weren't there to witness it) so while you might assume a high probability of rewound time placing those cards neatly back in the hand of the dealer, it might not have been in so many different ways. Though it might be wise just to restrict your imaginative scepticism as to whether the dealer might have bottomdealt some of the cards after taking a sneaky peek, naturally, as a close alternative of the orthodox estimate of the starting setup.)
Re: Quantum Question
Tub wrote:If you pick a random configuration with a given low entropy, then you'll find that both its evolution and its time reversed evolution will (probably) yield higher entropy results.
That is as I expect, but then, I also expect it of any given state, not only lowentropy ones. What doogly's saying seems to contradict that, and also to imply that if you pick a random highentropy configuration (or at the least, a random selection from the set of highentropy configurations that resulted from letting lowentropy configurations evolve for a long time) and timereverse it, you would expect it to (not even probably, but definitely) evolve back to a lowentropy configuration. That is surprising to me and I don't understand how it can be so in light of everything I think I understand about entropy.
One possible point of confusion that arose in my mind while writing the above: is the evolution of a momentareversed version of a given configuration the same thing as the timereversed evolution of the original configuration? I'm thinking yes, but I've been explicitly talking about the former, while you've been explicitly talking about the latter, and while it seems implied that they're the same thing I'd like to be explicit that we really are talking about the same thing.
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 Zamfir
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Re: Quantum Question
Let's take the standard toy of thermodynamics, a closed box with gas molecules. Let's say that the molecules move perfectly deterministic, and let's say there are two kinds of molecules, split 5050.
Box 1 has a truly random chosen state. You expect, almostcertain, that the 2 kinds of molecules are well mixed. You can adopt all kinds of statistical measures on exactly how well mixed they are, and the box will score high and stay high over time. The measure will bob and down like noise, but that's not really meaningful.
Box 2 initially has the 2 kinds of molecules divided in the left and right halves of the box. Then you let the system evolve, and the molecules mix. Fairly soon, it will hit the same values on the mixing measures as the other box, and it will stay there just as box 1. The boxes are now effectively the same.
Now we exactly reverse the movement in both boxes, and we wait until each box is split in separated halves. Both boxes will get there, eventually. Box 2 goes fast, exactly as long as you have it running. Box 1 will take longer, by a time that is utterly beyond comprehension.
You see what I mean? A wellmixed box is almost the same as a truly random box, but not quite. And exact time reversal is basically the one (rather hypothetical) method to tell them apart.
Now, the 'heat death' of the universe is still box 2. In 10^100 years or whatever, the universe might look like it's just a random point in state space, but actually it will still be relatively close to the big bang compared to a truly random state. In a truly random state, entropy is just as likely to go up as go down, no matter whether you go forward or backward in time.
Box 1 has a truly random chosen state. You expect, almostcertain, that the 2 kinds of molecules are well mixed. You can adopt all kinds of statistical measures on exactly how well mixed they are, and the box will score high and stay high over time. The measure will bob and down like noise, but that's not really meaningful.
Box 2 initially has the 2 kinds of molecules divided in the left and right halves of the box. Then you let the system evolve, and the molecules mix. Fairly soon, it will hit the same values on the mixing measures as the other box, and it will stay there just as box 1. The boxes are now effectively the same.
Now we exactly reverse the movement in both boxes, and we wait until each box is split in separated halves. Both boxes will get there, eventually. Box 2 goes fast, exactly as long as you have it running. Box 1 will take longer, by a time that is utterly beyond comprehension.
You see what I mean? A wellmixed box is almost the same as a truly random box, but not quite. And exact time reversal is basically the one (rather hypothetical) method to tell them apart.
Now, the 'heat death' of the universe is still box 2. In 10^100 years or whatever, the universe might look like it's just a random point in state space, but actually it will still be relatively close to the big bang compared to a truly random state. In a truly random state, entropy is just as likely to go up as go down, no matter whether you go forward or backward in time.
Re: Quantum Question
Yeah, but this kind of dodges the question, which was illuminated better by Pfhorrest (reversal of momentum). I am considering a system where there are quantum particle interactions. If momentum is reversed, we either know what the next state must be (time reversal), or we don't (quantum randomness). Is there a difference?
Jose
Yes  agreed. But this works classically too. Classically, there is only one "choice": forward or backwards. With QM, a new choice is made at every QM interaction. My question is whether or not these choices are constrained to be the reverse of what they originally were, in a {time  momentum} reversed situation.Zamfir wrote:A wellmixed box is almost the same as a truly random box, but not quite.
Jose
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Re: Quantum Question
Assuming determinism and no cpviolations, time reversal gets you back to exactly the previous state, not to a different random lowentropy state. In a hypothetical classical world, time reversal means turning those particles around by negating all velocities.
In a quantum world, time reversal is a bit different. You don't have particles, you have a wave function. And the parts of the wave function that you'd identify as particles do not have a clearly defined position and/or momentum, so you cannot just turn them around the way you'd do in a classical world.
If you're asking what'd happen in a quantum world if you did something classical like reversing a clearly defined momentum, then I'm afraid you're not going to get a good answer, because the question is mixing two incompatible models. It's a bit like asking how to score a royal flush in a game of chess.
In a quantum world, time reversal is a bit different. You don't have particles, you have a wave function. And the parts of the wave function that you'd identify as particles do not have a clearly defined position and/or momentum, so you cannot just turn them around the way you'd do in a classical world.
If you're asking what'd happen in a quantum world if you did something classical like reversing a clearly defined momentum, then I'm afraid you're not going to get a good answer, because the question is mixing two incompatible models. It's a bit like asking how to score a royal flush in a game of chess.

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Re: Quantum Question
As an alternate interpretation of the discussion;
We are given an isolated system that has recently gone from low to high entropy.
What transformations need to be applied in order to have the system evolve back to its initial state over positive time?
We are given an isolated system that has recently gone from low to high entropy.
What transformations need to be applied in order to have the system evolve back to its initial state over positive time?
 Zamfir
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Re: Quantum Question
Yes  agreed. But this works classically too.
Sure, I was only talking about a classical system here. I get the impression that some of phorests questions were not just about quantum effects. In particular, this part:
That is as I expect, but then, I also expect it of any given state, not only lowentropy ones. What doogly's saying seems to contradict that, and also to imply that if you pick arandom highentropy configuration (or at the least, a random selection from the set of, highentropy configurations that resulted from letting lowentropy configurations evolve for a long time) and timereverse it, you would expect it to (not even probably, but definitely) evolve back to a lowentropy configuration.
I tried to illustrate the difference between "a random highentropy configuration" and "a random selection from the set of highentropy configurations that resulted from letting lowentropy configurations evolve for a long time". For a typical system from stats mechanics, there's a reasonable mixing time T that makes these two things effectively the same by any realworld test.
But that's no longer true if you consider the thought experiment of exactly reversing momentums. That (hypothetical) exactness would make the whole concept of entropy shaky. What would be the point of recognizing ' irreversible' processes, if there was a reverse button?
Quantum effects might be the final killer of that reversibility, but just classical uncertainty is already enough.
 Soupspoon
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Re: Quantum Question
You can go to a 'messy' state quite easily from many possible 'defined' ones. Though not any exact messy one, because that's just a state defined as messy in exactly that way. You have to accept that it's a large group of finite(ish) nontrivial possible outcomes that (for example) define themselves as two wellmixed fluids.
Reversed, to seek the origin, the precise number of states that might relate to the above example's nominal starting position (say, all fluid molecules on the left are type A, all on the right are type B) may still have a large group of possible states, that is the distribution within the assigned half alone and it's a smaller large group of states. By a magnitude possibly equivalent to your metric fucktonne or so.
The resolution of the probability distribution over (forward) time getting from oneexactunmixed position to oneexactmixed position is equal to that of (reversed) time getting from the latter to the former. But anyoneofallpossibleunmixed moving towards anyoneofallpossiblemixed is a lowmany to highmany thing that covers much of the area under the probability curve (wide and mediancentred to cover most of where the outcomes clump), whereas the reverse needs to land in a much smaller area (far narrower, and somewhere in a tailingoff outlying region with negligable height to boot).
Whatever happens will happen, but the more particular the outcome is, the less you should expect it. If your unmixed state is "any state where an identifiable half is Aonly, the remainder Bonly" and your 'mixed' state is a pinpoint regular lattice¹ of alternating A and B, then neither is likely to progress to the other (in favour of the group of amorphously irregular states) but the effort needed by your enslaved Maxwellian demon to engineer that desired change is more with the (allegedly) 'mixed' destination.
Or something like that. Maybe.
¹  Which, if you think about it, that's just a specific gerrymandered version of the "defined half A, orher half B".
Reversed, to seek the origin, the precise number of states that might relate to the above example's nominal starting position (say, all fluid molecules on the left are type A, all on the right are type B) may still have a large group of possible states, that is the distribution within the assigned half alone and it's a smaller large group of states. By a magnitude possibly equivalent to your metric fucktonne or so.
The resolution of the probability distribution over (forward) time getting from oneexactunmixed position to oneexactmixed position is equal to that of (reversed) time getting from the latter to the former. But anyoneofallpossibleunmixed moving towards anyoneofallpossiblemixed is a lowmany to highmany thing that covers much of the area under the probability curve (wide and mediancentred to cover most of where the outcomes clump), whereas the reverse needs to land in a much smaller area (far narrower, and somewhere in a tailingoff outlying region with negligable height to boot).
Whatever happens will happen, but the more particular the outcome is, the less you should expect it. If your unmixed state is "any state where an identifiable half is Aonly, the remainder Bonly" and your 'mixed' state is a pinpoint regular lattice¹ of alternating A and B, then neither is likely to progress to the other (in favour of the group of amorphously irregular states) but the effort needed by your enslaved Maxwellian demon to engineer that desired change is more with the (allegedly) 'mixed' destination.
Or something like that. Maybe.
¹  Which, if you think about it, that's just a specific gerrymandered version of the "defined half A, orher half B".
 doogly
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Re: Quantum Question
What's uncertain in quantum mechanics is the outcome of a classical measurement, not the evolution of a state.
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Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
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 Soupspoon
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Re: Quantum Question
If you can't tell for sure what the current state is, or what state this becomes, you are effectively riding a probability curve.
(I'm sure there's something that says that you can't, despite the uncertainty of the initial and concluding states, take the large gamut of individual possibilities of the two endpoints and then work out the far smaller gamut of the combined possibilityfunction. You must be left with a less constrained spaghettimess of possible start/intermediate/end states so that you can't derive a disallowablyaccurate classical measurement through induction from two or more allowedaccuracy ones.)
(I'm sure there's something that says that you can't, despite the uncertainty of the initial and concluding states, take the large gamut of individual possibilities of the two endpoints and then work out the far smaller gamut of the combined possibilityfunction. You must be left with a less constrained spaghettimess of possible start/intermediate/end states so that you can't derive a disallowablyaccurate classical measurement through induction from two or more allowedaccuracy ones.)
 Eebster the Great
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Re: Quantum Question
We can imagine devising a precise set of coordinates and momenta of gas molecules in a box such that to any casual observer, it appears random, but in matter of fact, they are all coordinated just so they will all happen to end up near the same corner of the box, with the rest of it a perfect vacuum. There is no contradiction in this idea, it's just really unlikely.
Similarly, if we put a bunch of gas molecules in one corner of a box and wait a bit, we will reach the timereversal of such a state. This state is the "wellmixed" state mentioned above. It appears random, but if we allow time to run backward for a bit, we will see that it is a very special state; one which evolved from all the gas molecules being in the same corner just minutes ago. The vast majority of states are not like this.
Because, in practical terms, one seemingly random configuration of gas molecules cannot be distinguished from another, we say these are "microstates" of the same "macrostate." By definition, a macrostate has high entropy if it could be due to any of a large number of microstates. This is what we mean by saying that high entropy states are more likely. And this is how classical thermodynamics is consistent with timereversal symmetry.
In quantum mechanics, in order to talk about multiple possible histories of particles under different conditions, we need something called counterfactual definiteness, and to get this, we need to lose locality. But it turns out that we can preserve determinism here if we really want to by allowing nonlocal interactions between particles in something like (a relativistic extension of) Bohmian mechanics. So here there is again no contradiction; all particles behave deterministically (albeit nonlocally), and if we reverse time, they will just go backwards (including the unobservable pilot waves). If we want something a bit less speculative, we can use the Multiple Worlds Interpretation, which simply takes the Schrodinger equation at face value. This interpretation is still deterministic and so time reversal is still easy to understand, though the existence of multiple universes makes it confusing. If we want something a bit closer to our intuitions, we can talk about wavefunction collapse, but then you lose determinism, and time reversal in the "objective" sense of running the universe backwards becomes meaningless.
Similarly, if we put a bunch of gas molecules in one corner of a box and wait a bit, we will reach the timereversal of such a state. This state is the "wellmixed" state mentioned above. It appears random, but if we allow time to run backward for a bit, we will see that it is a very special state; one which evolved from all the gas molecules being in the same corner just minutes ago. The vast majority of states are not like this.
Because, in practical terms, one seemingly random configuration of gas molecules cannot be distinguished from another, we say these are "microstates" of the same "macrostate." By definition, a macrostate has high entropy if it could be due to any of a large number of microstates. This is what we mean by saying that high entropy states are more likely. And this is how classical thermodynamics is consistent with timereversal symmetry.
In quantum mechanics, in order to talk about multiple possible histories of particles under different conditions, we need something called counterfactual definiteness, and to get this, we need to lose locality. But it turns out that we can preserve determinism here if we really want to by allowing nonlocal interactions between particles in something like (a relativistic extension of) Bohmian mechanics. So here there is again no contradiction; all particles behave deterministically (albeit nonlocally), and if we reverse time, they will just go backwards (including the unobservable pilot waves). If we want something a bit less speculative, we can use the Multiple Worlds Interpretation, which simply takes the Schrodinger equation at face value. This interpretation is still deterministic and so time reversal is still easy to understand, though the existence of multiple universes makes it confusing. If we want something a bit closer to our intuitions, we can talk about wavefunction collapse, but then you lose determinism, and time reversal in the "objective" sense of running the universe backwards becomes meaningless.
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