I disagree that this is the only way cause and effect have been defined. Cause and effect were words being used by people long before the concept of entropy was invented. No dictionary definition of cause or effect will talk about entropy. It is physicists trying to understand why we perceive time moving only in one direction that co-opted the words the mean what you are talking about. Just like physicists use the word work in a very specific way that relates to the non scientific use of the word. They co-opted the words for decent reasons but to blithely state that not beveling in entropy is the same as not beveling in cause and effect is papering over a lot of assumptions about what those words mean.
No, the old definitions were in terms of information. And the original definitions of entropy didn’t reference information, just heat and temperature. But we’ve since realized that entropy is the same thing as information, and so in retrospect, we can say that the old definitions are about entropy.
I should probably bring up the Fluctuation Theorem here. It says that it is possible for entropy to decrease, but this is much, much more improbable than entropy increasing. However, it still doesn’t fully resolve Loschmidt’s Paradox, since the fluctuation theorem is still time-reversible (and, indeed, requires the fact that the laws of physics are time-reversible to be proven.) The resolution appears to be that the universe started out with very low entropy, so this sets the arrow of time in the high-entropy direction.
No the old definitions did not talk about information. Seriously look up the words.
Well, not really. What do you mean when you say that the Universe “started out” that way? You’re attaching a label to one of the bounds of the Universe: Why that particular label attached to that particular bound?
Does a system in total equilibrium and therefore at a maximum entropy experience time?
But isn’t that the crux of the problem? Statistical physics alone would imply that the entropy increases in both temporal directions; that’s fixed by making the entropy low in the past. But what’s ‘the past’? Thermodynamics doesn’t say, at least not that I could see. And, of course, even if this line of reasoning works, we’ve merely replaced the question ‘why does the arrow of time point that way?’ with ‘why was entropy low in the past?’. It’s not immediately clear to me that much has been gained.
I think it’s also a bit too glib to say that all definitions of cause and effect are framed in terms of information. Sure, if you say that what we can remember is the past, then this implies that the past is what we have information about, in some sense. But causality is usually assumed to be a primitive notion—remember Hume: we can’t say that just because A always preceded B, A causes B. To which one reply is that A has causal powers, dispositions, that, if actualized (i.e. under the right circumstances), make B happen; causality is counterfactual, i.e. if A had not happened, B would not have happened. I don’t see any relation to thermodynamics here.
And then there’s of course phenomena which are time-asymmetric, such as certain weak decays. Granted, those don’t have anything to do with our everyday perception of time, but I don’t see any way to reduce the difference they make between past and future to thermodynamics, either.
There are three known arrows of time in physics: There’s the cosmological arrow of time, which defines the past as being the time when the Universe is smaller, and the future as the time when the Universe is bigger. So far as anyone can tell, this arrow of time has no real significance whatsoever, and if the Universe were at some point to stop expanding and start contracting, nothing else would change.
Then there’s the weak-force arrow of time. This one is extremely subtle, and has never even been directly observed. We assume that it exists, because we’ve observed (very small and subtle) violations of CP symmetry, and we’re confident that CPT symmetry is exact, so there must be a T asymmetry that corresponds to the CP asymmetry. This must have at least some effects, but it’d be very surprising for them to be significant in almost any context.
And finally, there’s the thermodynamic arrow of time. This one is everywhere. All of the other “arrows of time” we think we know of are really just examples of the thermodynamic arrow of time. So, for instance, when we say “We have the Second Law because our Universe happened to start in a low-entropy state”, well, that’s wrong. Our Universe doesn’t happen to have that trait; rather, the lowest-entropy time is the one that we call the “start”, because that’s what “start” means.
Does that arrow reverse if we impose a left-right (mirror) reversal on the universe? If so, it seems unlikely to be related to the arrow of causality.
I am also interested in the answer to this question, especially if “time” is replaced with “time’s arrow.”
It reverses if we impose a left-right reversal and a matter-antimatter reversal, at the same time. Doing either one of those reversals without the other makes a large enough difference to the weak force that the subtle asymmetries from doing both are lost in the shuffle: We don’t know where the T asymmetry actually comes from.
It seems you could never reach total equilibrium, just approach it asymptotically. The direction you are coming from on your asymptotic approach is the past, and the future is closer to the asymptote.
If a system were at equilibrium, then no, it would not have a thermodynamic arrow of time, but that’s not relevant, as such a system couldn’t exist. Even if it could come into being, quantum fluctuations would quickly create differing levels of entropy (miniscule ones, to be sure), that would prevent it from staying in complete equilibrium.
Does time flow more slowly in systems near equilibrium?
If a system is very near equilibrium and random fluctuations push it away from equilibrium, does time go backward in that moment?
But the question is still valid. When a system is near equilibrium (i.e., the universe approaches its “heat death”) does the arrow of causality dwindle to almost nothing? (The human arrows of memory and volition would be completely missing since advanced life requires the consumption large amounts of negative entropy.)
ETA: Ninjaed by Blue Blistering Barnacle.
I guess we could flip it around.
Is there any evidence that time moves faster in a system far from equilibrium?
We can create (sub)systems far from equilibrium and compare them to (sub)systems close to equilibrium.
I don’t know that “time moving faster” actually means anything in this context.
Time is a function of spacing out spacelike events so that they don’t happen all at once. Time will always take the same “amount of time”. It takes 1 plank time for casualty to propagate 1 plank length.
We are used to time being a concept that means something like minutes or seconds, when those terms are just the emergent effects of the underlying principles of the increasing entropy and information in a system.
It being a situation where there is much less free energy to be utilized, any events would progress much slower, but the rate of time passing, that of how long it takes for a disturbance to propagate 1 plank length, will always be the same, it cannot change, pretty much by definition.
We are already in the asymptotic side of entropy as it is. At the time of the big bang, entropy was much lower than it is today. How much lower would be hard to express without quite a number of exponents. The difference between the entropy in the first 10^-37th seconds compared to now is much greater than the difference in entropy between now and what we would consider the heat death of the universe. Would it make any sense to ask if time moved faster closer to the big bang?
If you have a random fluctuation that lowers the entropy, that would not reverse time. If it kept lowering, then that would be similar to that of time flowing, but it would not be the same thing, especially as it would not be something that was sustainable. On the other hand, a sudden fluctuation lowering the entropy could mean that work can be performed off of that new imbalance. A big enough fluctuation, and you get yourself a new universe.
The strength of time’s arrow is not the same as “speed” of “time,” as k9bfriender acknowledges: “If you have a random fluctuation that lowers the entropy, that would not reverse time.”
In the sense we’re speaking of, time does “move slower” in the absence of free energy, cf. a bear hibernating. Or
I think so! Sure, the speed of light was the same then but from what I hear, the arrow of causality was operating very violently!
I don’t understand the last few posts other than k9bfriender’s. Time always moves the same rate for everyone/everything inside the same reference frame. There is no reason to think that time moved slower or faster close to the big bang. Physicists have calculated how long it took for each aspect of the development of the universe; they use the same time scale throughout. Time never goes backward under any known circumstance. Nor do equilibria states affect time.
Re the first quoted sentence: I explicitly acknowledged in my post that the speed of light is constant, and was informally using “speed of time” to mean “strength of time’s arrow.”
Re the second quoted sentence, and assuming “time” means “time’s arrow”: Cite? 
I guess I’m trying to come to grips with how a statistical process describing the group behavior of vast collections of particles sets the “arrow of time”, without affecting its “intensity”.
And now I’m told the arrow won’t reverse when the collection of particles happen to randomly behave in a “backward” fashion.
The whole thing sounds very “tail wagging dog” to me.
What would it mean for “time to move more slowly”? It goes at only half a second per second? In a Universe close to equilibrium, physical processes will occur only very slowly, but time itself is unaffected.