Hooray, mutual imparting of understanding. That’s the best kind of thread.
Define “evolve backwards”.
As for the “initial conditions of the Universe”, consider this: Any given physical process can only “remember” stated of lower entropy. In the entirety of the Universe, here, there, elsewhere, past, present, and future, there must be some point at which the entropy is at a global minimum. All points in the Universe will then regard that point as the origin of the Universe, since no point can “remember” any point beyond that, since that would require some point remembering a point of higher entropy.
Or, to put it another way: The fact that entropy is greater in the future and less in the past is not an observation about the Universe, but the definition of “future” and “past”.
I’m not sure if you’re actually asking me to explain what I meant by this in the context where I used it, or if you’re just making the point here, as in the rest of your post, that the notion of “backwards” is defined as the direction of lower entropy.
If you actually are asking me to explain what I meant by the question/statement I made: let us assume that the laws governing particle behavior are deterministic; this isn’t essential to what I was saying, but makes it easier to say. Furthermore, suppose they are T-symmetric. Then, just as much as we can calculate the state of a situation at later times from its state at earlier times, so can we calculate its state at earlier times from its state at later times. My rhetorical question for Pasta then was this: Why will a random configuration almost certainly be such that, were we to calculate corresponding future configurations from it, they would be of higher entropy, but were we to calculate corresponding past configurations from it, they would not be of higher entropy, despite the laws governing these calculations being time-symmetrical (and entropy being invariant under time-reversal)?
Yeah, that’s sort of what I was getting at in Post #15.
But let’s see if I can expand my understanding of entropy a bit further…
Supposing I did happen to witness the anomaly of all the air in a box spontaneously rushing to one side. I wouldn’t observe all the details of all the microscopic configurations involved, but I would see the macroscopic picture. Then, afterwards, the box will be in a state of lower entropy than it started in; however, I will be able to remember that higher entropy state. Granted, my own brain will have undergone an increase in entropy, but given that all I committed to my brain was the macroscopic outline of what went down, couldn’t we arrange for this to occur in such a way as that the total entropy of the situation decreases but I continue to remember it?
I.e.,
State A) Air distributed all over box, me with pleasant nothings in my head, high entropy
State B) Air squished all to one side of the box, me with memories of State A in my head, low entropy
So then we would have an example of the universe “remembering” (and, not just that, but my own brain remembering) in one state information about a higher entropy state. Is there anything wrong with this, and, if not, what implications does it have for the notion of the arrow of time being simply defined as the arrow of entropy increase?
The simplest answer is, you didn’t happen to witness that. Such events are so mindbogglingly improbably that we can basically just say that they don’t happen, at all.
Well, ok… But that hardly alleviates my definitional concerns. Supposing I witnessed a more moderate, more probable spontaneous decrease in a system’s entropy; then, as in my above post, what would be the implications for the notion that nothing can “remember” higher entropy states and that the arrow of time is simply defined so as that the future is the direction of entropic increase?
I’ll give this a slight bump, I guess. What appears to conflict to me now are the following two relatively oft-encountered statements:
A) The Second Law of Thermodynamics is a probabilistic law; it’s highly unlikely for the entropy of an isolated system to decrease over time, but there is always a tiny possibility of this occurring.
B) The arrow of time is defined so as that the direction of increase in time is the direction of increase in entropy
If B) is true, then it seems A) can’t be quite right; the Second Law of Thermodynamics would not be a probabilistic statement, but rather a tautology. If B) were true, then there would simply, by definition, never ever be any situations in which entropy (of the universe as a whole, I guess?) did not strictly increase with time. But the common understanding of what the Second Law is doesn’t seem to be quite like this.
Furthermore, as explained above, it doesn’t seem quite right to me to say that low-entropy states of the universe are incapable of “remembering” any information at all about high-entropy states of the universe.
So, I have my doubts that a naive reading of B) is tenable; the proposition needs some elaboration to work, it seem to me. But others might know better and might be able to explain away the apparent conflict.
What people keep forgetting is that symmetries apply to the laws of physics, not to the solutions. The law of gravity is spherically symmetric, but orbits are ellipses.
Sure. The symmetry of the physical law, in that case, doesn’t demand that every solution is itself spherically symmetric, just that the space of solutions is spherically symmetric (i.e., rotating or flipping a situation which satisfies the law of gravity gives another situation which satisfies the law of gravity). I’m not sure where in this thread anyone has made a mistake regarding this, though.
Right in the OP. The law of gravity is symmetric under time reversal. The space of solutions is symmetric under time reversal. The solutions themselves are not, but that’s not a problem.
My OP was rather messed up, as I later in the thread realized, because I had mis-analyzed the law of gravity (as “Objects get closer over time”). But I guess I’m still not seeing in the OP where I thought solutions were themselves supposed to be symmetric simply because the underlying laws were symmetric; it’s true I thought solutions to the law of gravity had to have objects getting closer over time and that this was problematic, but the reason I found this problematic was precisely because it would imply that the space of solutions itself was not T-symmetric (because it contained solutions where objects got closer over time but didn’t contain solutions where objects got further apart over time), conflicting with my understanding that the simple laws of physics were supposed to be T-symmetric. My confusion arose precisely because I did mis-state the essence of the law of gravity as something which was not actually T-symmetric.
At any rate, although the OP was full of quite embarrassing confusion on the nature of the law of gravity, and thus had mistakes in it (though not, I think, mistakes of the sort you are describing), that’s all been cleared up now. In the discussion of entropy, the Second Law, and the arrow of time, have there been any mistakes of the sort you are describing?
No, but entropy (and thermodynamics in general) is a different sort of thing altogether.
The laws of thermodynamics are derivable from statistical mechanics, just as Kepler’s laws are derivable from Newtonian mechanics and gravitation. What you’re looking for in “entropy” is related to something called the “phase space volume”. The phase space of a classical mechanical system is a symplectic manifold, so it comes with a natural volume form, and thus a measure, which tells us “how likely” a system is to be found in one region of phase space or another.
Alright, I’ll look into that. But when it comes to deriving the Second Law from statistical mechanics, out of curiosity, what assumption, exactly, does one need to make to get around Loschmidt’s paradox, that non-time-symmetric conclusions should not be inferable from time-symmetric underlying laws?
You’ve got it sort of backwards. You need to not assume that you have perfect information. You don’t know everything about the system, and you don’t know precisely what happens instant-to-instant.
I’m afraid I don’t understand… could you clarify what you mean by this? I wasn’t saying anything about me, personally, having any information, or knowing anything about any system, etc. I was just noting that the Second Law of Thermodynamics, in a straightforward form, is non-T-symmetric (it states that certain histories of the universe are more probable than their time-flipped duals). To derive it by statistical mechanics, one has to make some non-T-symmetric assumptions (for example, assume something about initial states without making the corresponding assumption for final states). So I was curious what sort of non-T-symmetric assumptions, exactly, one would make in order to derive the Second Law with statistical mechanics.
And you’ll find out when you look into it, as you said you would.
Not even a sneak peek, a hint at what the answer will be? Alright, fair enough. Any particular sources you recommend I read?
Also, would it be possible to get some clarification on the definitional status of the arrow of time, as I was asking before? The question, from posts 24, 26, and 27, of whether the Second Law is a probabilistic one or a definitional tautology?
This is sort of an ancient bump, but I find I still have the same questions I had in post #27 and #33, and I’d still like to know the answer, so… I’ll bring them back to attention and give it another shot.
[And, again, for anyone reading this anew, I apologize and am most greatly embarrassed by my mistaken assertion at the top that the behavior of gravity is not T-symmetric. I’ve seen the error in that now; no need to point it out again.]
I’d say that the naïve reading of (B) isn’t quite right, and also that we don’t know how to get around that paradox. Like Fr. Flanagan always said, it’s a mystery.
Don’t worry too much about that, at least it gave me a small perk of confidence to be able to answer one of your questions for a change, so maybe you can take comfort in knowing that at least some good came from your (perfectly understandable) confusion… 
Regarding your question about the second law and the arrow of time, however, I’m afraid an answer is not that easily obtainable, or at least not that I know.
In your two statements, however, I believe A) is the more fundamental one – you have to keep in mind that, however often encountered, the thermodynamic arrow of time isn’t the only one known to physics: another would be the cosmological arrow of time, defined by the expansion of the universe, and yet another is given by the T-asymmetry of the weak interaction (in kaon decay), or even the time arrow of radiation, which merely refers to the fact that radiation is emitted outwards (from a source) as time progresses into the future, rather than the past.
Thus, it is not just (the increase of) entropy that defines the arrow of time, and your statement B) isn’t necessary to define its direction.
The question, now, essentially is which of those arrows of time can be considered fundamental – i.e. what’s really responsible for the easily observed macroscopic time asymmetry of the universe.
The thermodynamic arrow, I think we can rule out, seeing as how a spontaneous violation of the second law would lead to a regionally negative arrow of time – not that there’s anything wrong with that, in principle, but let’s try and find something a bit more neat first.
The cosmological arrow, seeing as how it takes into account the universe pretty much as a whole, might be an obvious candidate – and indeed, there are models in which a contracting universe, i.e. one with a negative cosmological time arrow, also leads to a negative thermodynamic time arrow (this is, however, not the majority view). Still, the question then is – how does this, i.e. the universe’s expansion, affect us at all? How does the cosmological arrow of time translate into our own personal sense of past and future?
Then there’s the electromagnetic arrow, which is perhaps a bit difficult to grasp – the emission of a photon isn’t different from its absorption, and thus, a person watching this on film couldn’t tell a difference between forward and backward, and Maxwell’s equations are obviously time symmetric. Still, there exists a difference, not in the laws, but in the observed state – electromagnetic radiation is emitted in the form of ‘retarded’ waves, i.e. we choose the retarded Green’s function as a solution to the field equations rather than the advanced one in order to protect causality. Atoms can emit retarded photons and lose energy, they cannot emit advanced photons and gain some. However, this is an empirical fact, not something fundamentally required.
Thus we’re left with the T-symmetry violation in kaon decay: the kaon can decay into two pions, which is a violation of P-symmetry; since the system is its own charge conjugate, this is also a violation of CP-symmetry, and thus, T-symmetry must be violated to conserve CPT-symmetry. Now, this is about as fundamental as it gets – it’s a manifest T-asymmetry in a fundamental interaction, the weak force. But again, there’s the question of how this relates to our everyday experience – most people give a rat’s arse about kaons and can still tell you the time pretty well.
The answer to that, as far as I know, is to study the interdependency of the time arrows – for instance, the kaon time arrow might give rise to the cosmological one (somewhat counterintuitively) because of the overall matter-dominance of the universe: glossing over the details, a C-conjugation replaces a particle with its antiparticle, and thus, reactions having a higher cross-section than their CP-counterpart (equivalent to a T-asymmetry) favour either matter or antimatter over the other (matter, in our universe). The introduction of a cosmological arrow of time, then, might serve to conserve the macroscopic CPT-invariance in a matter-dominated universe.
From there on, things are a bit subject to debate, as far as I can tell – the expansion of the universe is responsible for a condition of thermodynamic non-equilibrium, which leads to the thermodynamic arrow of time (since evolution to a higher state of entropy is only problematic in an equilibrium case, via the Fluctuation Theorem); or, the Big Bang might serve as a ‘reflective barrier’, preventing the development of advanced electromagnetic waves, setting the electromagnetic arrow of time. (The interdependency of the various arrows of time is better explained in this paper by John G. Cramer – which I’ve drawn from quite a bit in the preceding discussion --, which focuses on the question of whether the electromagnetic or the thermodynamic arrow of time is more fundamental.)
Perhaps a good read on the subject would be The Physical Basis of the Direction of Time, by H.D. Zeh.
I apologize if I got something of the above wrong, my understanding of the problem is far from complete, and I’ve been writing this post off and on for most of the day during the outage, so I might have gotten confused a time or two.