Unconvenient consequences of the logic behind the second law of thermodynamics

if entropy is decreasing maybe your memory is just working "backwards"

I think the key to the puzzle is likely to be here: there's likely to be some principled reason why agents embedded in physics will perceive the low-entropy time direction as "the past", such that it's not meaningful to ask which way is "really" "backwards".

You should think of entropy as a subjective quantity. Even by thinking in terms of different fluctuations, you're imposing a frame/abstraction that isn't part of the actual structure of reality. Entropy is a property of abstractions, and quantifies the amount of information lost by abstracting. The universe, by itself, does not have any entropy.

Space itself is symmetrical, so it is equally possible that the world is behaving in the way we think it is, with objects falling down and that the world is behaving in a super wild and improbable way: the earth is above us and things actually fall up.

The situation is symmetrical. We define past and future in terms of entropy. Your weird and improbable universe is an upside down map of the same territory.

Also, note that large fluctuations are much much less likely than small ones. The "we are in a random fluctuation" theory predicts that we are almost certainly in the smallest fluctuation we could possibly fit in. (Think spontaneously assembled nanocomputer just big enough to run a single mind that counts as an observer in whatever anthropic theory we use. ) Even conditioned on our insanely unlikely experience so far, this hypothesis stubbornly predicts that the world we see isn't real. (I am not sure whether it predicts that our experiences were noise, or we are on a slightly larger nanocomputer that is running an approximate physics simulation.)

Clearly some sort of anthropic magic or something happens here. We are not in a random fluctuation. Maybe the universe won't last a hyperexponential amount of time. Maybe we are anthropically more likely to find ourselves in places with low komolgorov complexity descriptions. ("All possible bitstrings, in order" is not a good law of physics, just because it contains us somewhere).

Wikipedia has a good summary of what Entropy as used in the 2nd law actually is: https://en.m.wikipedia.org/wiki/Entropy

The key points are:

1. We define the direction of time by the increase in Entropy.
2. "If we wait long enough" rhetorically hides the scope involved. To have merely 10 Helium molecules congregate in the center 1% of a sealed sphere, assuming it takes them 1/10 of a second to traverse that volume, will require 10^99 seconds. Which is 10^82 times the current age of the universe. Or that it will happen approximately 10 times before the heat death of the universe.

If you believe that our existence is a result of a mere fluctuation of low entropy, then you should not believe that the stars are real. Why? Because the frequency with which a particular fluctuation appears decreases exponentially with the size of the fluctuation. (Basically because entropy is S=log W, where W is the number of micro states.) A fluctuation that creates both the Earth and the stars is far less likely than a fluctuation that just creates the Earth, and some photons heading towards the Earth that just happen to look like they came from stars. Even though it is unbelievably unlikely that the photons would happen to arrange themselves into that exact configuration, it's even more unbelievably unlikely that of all the fluctuations, we'd happen to live in one so large that it included stars. Of course, this view also implies a prediction about starlight: Since the starlight is just thermal noise, the most likely scenario is that it will stop looking like it came from stars, and start looking just like (very cold) blackbody radiation. So the most likely prediction, under the entropy fluctuation view, is that this very instant the sun and all the stars will go out.

That's a fairly absurd prediction, of course. The general consensus among people who've thought about this stuff is that our low entropy world cannot have been the sole result of a fluctuation, precisely because it would lead to absurd predictions. Perhaps a fluctuation triggered some kind of runaway process that produced a lot of low-entropy bits, or perhaps it was something else entirely, but at some point in the process, there needs to be a way of producing low entropy bits that doesn't become half as likely to happen for each additional bit produced.

Interestingly, as long as the new low entropy bits don't overwrite existing high entropy bits, it's not against the laws of thermodynamics for new low entropy bits to be produced. It's kind of a reversible computing version of malloc: you're allowed to call malloc and free as much as you like in reversible computing, as long as the bits in question are in a known state. So for example, you can call a version of malloc that always produces bits initialized to 0. And you can likewise free a chunk of bits, so long as you set them all to 0 first. If the laws of physics allow for an analogous physical process, then that could explain why we seem to live in a low entropy region of spacetime. Something must have "called malloc" early on, and produced the treasure trove of low entropy bits that we see around us. (It seems like in our universe, information content is limited by surface area. So a process that increased the information storage capacity of our universe would have to increase its surface area somehow. This points suggestively at the expansion of space and the cosmological constant, but we mostly still have no idea what's going on. And the surface area relation may break down once things get as large as the whole universe anyway.)

First, a little technical precisation: Poincaré's recurrence theorem applies to classical systems whose phase space is finite. So, if you believe that the universe is finite, then you have recurrence; otherwise no recurrence.

I think that your conclusion is correct under the hypothesis that the universe exists from an infinite time, and that our current situation of low entropy is the result of a random fluctuation. 

The symmetry is broken by the initial condition. If at t=0 the entropy is very low, then it is almost sure that it will rise. The expert consensus is that there has been a special event (the big bang), that you can modelize as an initial condition of extremely high entropy.

It seems unlikely that the big bang was the result of a fluctuation from a previously unordered universe:  you can estimate the probability of a random fluctuation resulting in the Earth's existence.  If I recall correctly Penrose did it explicitly, but you do not need the math to understand that (as already pointed out) "a solar system appears from thermal equilibrium" is extremely more likely that "an entire universe appears from thermal equilibrium". Therefore, under you hypothesis, we should have expected with probability of almost 1, to be in the only solar system in existence in the observable universe.

I wish to stress that these are pleasant philosophical speculations, but I do not wish to assign an high confidence to anything said here: even if they work well for our purposes, I feel a bit nervous in extrapolating mathematical models it to the entire universe.

If the origin of our current universe-state is just entropy randomly getting insanely-unlikely low and then bouncing back, that would make Big Bang just the random moment where the entropy bounced back.

Like, if you could travel back in time in some magical time machine, you would see the stars being unborn, universe getting smaller, and then at a completely random moment, the time machine would stop and say "this is as far in the past as I can go; from here it is a future in both directions: your time or the antichronos". And that moment itself would be nothing special: universe much simpler than we are used to, but not literally zero-size. Just how much back the entropy went, before it bounced. Far enough that we had enough time to evolve. On the other hand, every second of the past is extra improbability, so even "five minutes after the universe was zero-size" would be astronomically more likely than "exactly when the universe was zero-size", and it wouldn't make a difference for our ability to evolve.

From this perspective, the theories of Big Bang are just an attempt to reverse-engineer how far back could the entropy go in the hypothetical extreme case... but in our specific past, it most likely didn't go back that far. Like, on the picture with the red curve, Big Bang is a description of the lowest point on the scale, but the curve is more likely to have bounced a few pixels above it.

(Epistemic status: I don't understand physics, I am just trying to sound smart and possibly interesting.)

I don't think your graph is accurate. In our universe, entropy can reverse, but it's unlikely, and an entropy reversal large enough to take us from heat-death to the current state of the universe is unimaginably unlikely. Given an infinite amount of time, that will eventually happen, but your graph should be entropy spikes surrounded by eternities at minimum entropy. Given that more-accurate graph, the probability that we're in a decreasing-entropy part of the graph is ~100%.

This looks like:

1. Consider an artificial system (that doesn't obey the 2nd law of thermodynamics, but that isn't explicitly stated)
2. Such a system will fluctuate in phase space
3. One aspect of a system is the amount of Entropy in that system
4. Therefore, Entropy must decrease to arbitrarily low levels eventually.
5. Hey, what if the real world also didn't obey the 2nd law?

Without some bridge to address why the 2nd law (which 100% of tests so far have confirmed) might not apply, I don't see any value in the original post?

This argument proves that

• Along a given time-path, the average change in entropy is zero
• Over the whole space of configurations of the universe, the average difference in entropy between a given state and the next state (according to the laws of physics) is zero. (Really this should be formulated in terms of derivatives, not differences, but you get the point).

This is definitely true, and this is an inescapable feature of any (compact) dynamical system. However, somewhat paradoxically, it's consistent with the statement that, conditional on any given (nonmaximal) level of entropy, the vast majority of states have increasing entropy.

In your time-diagrams, this might look something like this:

I.e when you occasionally swing down into a somewhat low-entropy state, it's much more likely that you'll go back to high-entropy than that you'll go further down. So once you observe that you're not in the maxentropy state, it's more likely that you'll increase than that you'll decrease.

(It's impossible for half of the mid-entropy states to continue to low-entropy states, because there are much more than twice as many mid-entropy states as low-entropy states, and the dynamics are measure-preserving).