This came up as a tangent from this question, which is itself a tangent from a discussion on The Hidden Complexity of Wishes.

Suppose we have a perfect cubical box of length 1 meter containing exactly 1 mol of argon gas at room temperature.

At t=0, the gas is initialized with random positions and velocities drawn from the Maxwell-Boltzmann distribution.

Right after t=0 we perturb one of the particles by 1 angstrom in a random direction to get the state .

All collisions are perfectly elastic, so there is no viscosity [edit, this is wrong; even ideal gases have viscosity] and energy is conserved.

For each possible perturbation, we run physics forward for 20 seconds and measure whether there are more gas molecules in the left side or right side of the box at t=20 seconds (the number on each side will be extremely close to equal, but differ slightly). Do more than 51% of the possible perturbations result in the same answer? That is, if is the predicate “more gas molecules on the left at t=20”, is ?

This is equivalent to asking if an omniscient forecaster who knows the position and velocity of all atoms at t=0 except for 1 angstrom of uncertainty in 1 atom can know with >51% confidence which side has more gas molecules at t=20.

I think the answer is no, because multiple billiard balls is a textbook example of a chaotic system that maximizes entropy quickly, and there’s no reason information should be preserved for 20 seconds. This is enough time for each atom to collide with others millions of times, and even sound waves will travel thousands of meters and have lots of time to dissipate.

@habryka thinks the answer is yes and the forecaster could get more than 99.999% accuracy, because with such a large number of molecules, there should be *some* structure that remains predictable.

Who is right?

In the 2D case, there’s no escaping exponential decay of the autocorrelation function for any observable satisfying certain regularity properties. (I’m not sure if this is known to be true in higher dimensions. If it’s not, then there could conceivably be traps with sub-exponential escape times or even attractors, but I’d be surprised if that’s relevant here—I think it’s just hard to prove.) Sticking to 2D, the question is just how the time constant in that exponent for the observable in question compares to 20 seconds.

The presence of persistent collective behavior is a decent intuition but I’m not sure it saves you. I’d start by noting that for any analysis of large-scale structure—like a spectral analysis where you’re looking at superpositions of harmonic sound waves—the perturbation to a single particle’s initial position is a perturbation to the initial condition for every component in the spectral basis, all of which perturbations will then see exponential growth.

In this case you can effectively decompose the system into “Lyapunov modes” each with their own exponent for the growth rate of perturbations, and, in fact, because the system is close to linear in the harmonic basis, the modes with the smallest exponents will look like the low-wave-vector harmonic modes. One of these, conveniently, looks like a “left-right density” mode. So the lifetime (or Q factor) of that first harmonic is somewhat relevant, but the actual left-right density difference still involves the sum of many harmonics (for example, with more nodes in the up-down dimension) that have larger exponents. These individually contribute less (given equipartition of initial energy, these modes spend relatively more of their energy in up-down motion and so affect left-right density less), but collectively it should be enough to scramble the left-right density observable in 20 seconds even with a long-lived first harmonic.

On the other hand, 1 mol in 1 m^3 is not very dense, which should tend to make modes longer-lived in general. So I’m not totally confident on this one without doing any calculations. Edit: Wait, no, I think it’s the other way around. Speed of sound and viscosity are roughly constant with gas density and attenuation otherwise scales inversely with density. But I think it’s still plausible that you have a 300 Hz mode with Q in the thousands.

I think you’re probably right. It does seem plausible that there is some subtle structure which is preserved after 20 seconds, such that the resulting distribution over states is feasibly distinguishable from a random configuration, but I don’t think we have any reason to think that this structure would be strongly correlated with which side of the box contains the majority of particles.

The variance in density will by-default be very low, so the effect size of such structure really doesn’t have to be very high. Also, if you can identify multiple such structures which are uncorrelated, you can quickly bootstrap to relatively high confidence.

I don’t think “strong correlation” is required. I think you just need a few independent pieces of evidence. Determining such independence is usually really hard to establish, but we are dealing with logical omniscience here.

For example, any set of remotely coherent waves that form in the box with non-negligible magnitude would probably be enough to make a confident prediction. I do think that specific thing is kind of unlikely in a totally randomly initialized box of gas, but I am not confident, and there are many other wave-like patterns that you would find.

Predicting the ratio at t=20s is hopeless. The only sort of thing you can predict is the variance in the ratio over time, like the ratio as a function of time is μ(t)=0.5+ϵ , where ϵ∼N(0,σ2) . Here the large number of atoms lets you predict σ2 , but the exact number after 20 seconds is chaotic. To get an exact answer for how much initial perturbation still leads to a predictable state, you’d need to compute the lyapunov exponents of an interacting classical gas system, and I haven’t been able to find a paper that does this within 2 min of searching. (Note that if the atoms are non-interacting the problem stops being chaotic, of course, since they’re just bouncing around on the walls of the box)

https://www.sciencedirect.com/science/article/abs/pii/S1674200121001279

They find Lyapunov exponent of about 1 or 2 (where time is basically in units of time it takes for a particle at average velocity to cover the length of the box).

For room temp gas, this timescale is about

^{1}⁄_{400}seconds. So the divergence after 20 seconds should increase by a factor of over e^8000 (until it hits the cieling of maximum possible divergence).Since an Angstrom is only 10^-10 m, if you start with an Angstrom offset, the divergence reaches maximum by about a tenth of a second.

Do you know how to interpret “maximum divergence” in this context? Also, IIRC aren’t there higher-order exponents that might decay slower? (I just read about this this morning, so I am quite unfamiliar with the literature here)

Hm, this is a good question.

In writing my original reply, I figured “maximum divergence” was a meter. You start with two trajectories an angstrom apart, and they slowly diverge, but they can’t diverge more than 1 meter.

I think this is true if you’re just looking at the atom that’s shifted, but not true if you look at all the other atoms as well. Then maybe we actually have a 10^24-dimensional state space, and we’ve perturbed the state space by 1 angstrom in 1 dimension, and “maximum divergence” is actually more like the size of state space (√12+12...1024 times=1012 meters).

In which case it actually takes

twotenths of a second for exponential chaos to go from 10^-10 to 10^12.Nah, I don’t think that’s super relevant here. All the degrees of freedom of the gas are coupled to each other, so the biggest source of chaos can scramble everything just fine.

Hmm, I don’t super buy this. For example, this model predicts no standing wave would survive for multiple seconds, but this is trivial to disprove by experiment. So clearly there are degrees of freedom that remain coupled. No waves of substantial magnitude are present in the initialization here, but your argument clearly implies a decay rate for any kind of wave that is too substantial.

Yeah, good point (the examples, not necessarily any jargon-ful explanation of them). Sound waves, or even better, slow-moving vortices, or also better and different, diffusion of a cloud of one gas through a room filled with a different gas, show that you don’t get total mixing of a room on one-second timescale.

I think most likely, I’ve mangled something in the process of extrapolating a paper on a tiny toy model of a few hundred gas atoms to the meter scale.

The goal is not to predict the ratio, but to just predict which side will have more atoms (no matter how small the margin). It seems very likely to me that any such calculation would be extremely prohibitively expensive and would approximately require logical omniscience.

To clarify this, we are assuming that without random perturbation, you would get 100% accuracy in predicting which side of the system has more atoms at t=20s. The question is how much of that 100% accuracy you can recover with a very very small unknown perturbation.

Is this supposed to involve quantum physics, or just some purely classical toy model?

In a quantum physics model, the probability of observing more atoms on one side than the other will be indistinguishable from 50% (assuming that your box is divided

exactlyin half and all other things are symmetric etc). The initial perturbation will make no difference to this.Quantum physics. I don’t see why it would be indistinguishable from 50%.

Agree that there will be some decoherence. My guess is decoherence would mostly leave particle position at this scale intact, and if it becomes a huge factor, I would want the question to be settled on the basis being able to predict which side has higher irreducible uncertainty (i.e. which side had higher amplitude, if I am using that concept correctly).

Citing https://arxiv.org/abs/cond-mat/9403051: “Furthermore if a quantum system does possess this property (whatever it may be), then we might hope that the inherent uncertainties in quantum mechanics lead to a thermal distribution for the momentum of a single atom, even if we always start with exactly the same initial state, and make the measurement at exactly the same time.”

Then the author proceed to demonstrate that it is indeed the case. I guess it partially answers the question: quantum state thermalises and you’ll get classical thermal distribution of measurement results of at least some measurements even when measuring the system in the same quantum state.

The less initial uncertainty in energy the faster the system thermalises. That is to slow quantum thermalisation down you need to initialize the system with atoms in highly localized positions, but then you can’t know their exact velocities and can’t predict classical evolution.

Decoherence (or any other interpretation of QM) will definitely lead to a pretty uniform distribution over this sort of time scale. Just as in the classical case, the underlying dynamics is

extremelyunstable within the bounds of conservation laws, with the additional problem that the final state for any given perturbation is a distribution instead of a single measurement.If there is any actual asymmetry in the setup (e.g. one side of the box was 0.001 K warmer than the other, or the volumes of each side were 10^-9 m^3 different), you will probably get a very lopsided distribution for an observation of which side has more molecules regardless of initial perturbation.

If the setup is

actuallyperfectly symmetric though (which seems fitting with the other idealizations in the scenario), the resulting distribution of outcomes will be 50:50, essentially independent of the initial state within the parameters given.That is the question is not about the real argon gas, but about a billiard ball model? It should be stated in the question.

Tangential.

Is part of the motivation behind this question to think about the level of control that a super-intelligence could have on a complex system if it was only able to only influence a small part of that system?

There are a lot of assumptions in “omniscient forecaster knows the position and velocity of all molecules at t=0” that make the answer “probably possible to calculate, probably not in real-time on current hardware”.

Edit (motivated by downvote, though I’d have preferred a textual disagreement): I actually fight the premise. “omnicient forecaster” is so far from current tech that it’s impossible to guess what it could calulate. Say it only has 32 bits of precision in 3 dimensions of position and velocity, so 24 bytes for each of 6x10^23 particles. 1.4x10^25 bytes.

Call it 2 yottabytes. There’s no way we can predict what such a being might or might not be able to calculate, to what precision.