Pushed Around by the Stars | Hacker News Reader

Pushed Around by the Stars

eighteenthelephant.com

34 pointsthaumasiotes31 comments

Yeah I'm not sure about this math. Midway through the article the author correctly notes that their original argument was wrong. The relevant factor here is not the gravitational force, which scales as 1/d^2, but the tidal force, which scales as 1/d^3. But the author claims that after they make this correction, it only changes the result by about 30% or so. But given that they would have to be off by a factor of 1/d (which, in their example is ~4 x 10^-18). Now there's admittedly a non-linear factor here, but even still I have a hard time believing that a change in one of the inputs by 18 orders of magnitude will only result in a 30% change in the end result.

Another hint that something is wrong with the math is that the author does not have a term for the width of the balloon, which is a relevant quantity when calculating the tidal force.

Even though the math here is dodgy and the example is somewhat artificial, the same idea is super important when accounting for the effect of Galactic tides on the kinematics of stars and clusters in the Galaxy. Tides play a role in disrupting open clusters and dispersing young stars into the Galaxy.

The point he made is that because the discrepancy in angle grows exponentially, the change from 1/d^2 to 1/d^3 has only a small effect on the time until the change builds up to O(1).

I don't think there's the exponential dependence they claim. I think a more correct derivation would look like this:

Let's suppose that we place a 1 g mass at 4 ly. How does this affect the internal dynamics of a helium balloon? The relevant change is the tidal acceleration induced by the 1 g mass:

a_tide ~ GMl/r^3

where l is the mean free path in the balloon (l ~ 1 / n sigma), where n is the number density and sigma is the cross section of a helium atom. At room temperature, we have l ~ 4 x 10^-5 m.

Now, we want to calculate the change in angle induced by the tidal force across a single mean free path. This is (ignoring factors of order unity)

delta phi ~ delta x / l ~ al / v^2

Noting that the mean velocity is v ~ sqrt(kT / m_He), and substituting in the tidal acceleration, we have

delta phi ~ GMml^2 / kTr^3

Given these numbers this gives us

delta phi ~ 10^-78

At this point it should be pretty clear that this is not going to be a big effect. But let's see how long we would need to follow these interactions until the delta phi had built up to be large enough that a collision that would have happened doesn't. This is:

phi_crit ~ r_He / l ~ 10^-6

Now, I think a critical error that the author made is in assuming that the delta phi's build up exponentially. There is a non-linear dependence here, but I don't think it's as fast as described. Consider, for example, an atom bouncing head-on between two stationary atoms. Classically, the atom would continue indefinitely. How long would it be before an atom perturbed by delta phi would miss one of the stationary atoms entirely? There will be a linear increase in delta phi over time, along with a non-linear increase of order ~y / r_He where y is the vertical displacement. But this non-linearity is essentially a cosine, which is very close to zero initially. So the non-linear term will be essentially irrelevant until the linear term has grown to be large enough (i.e., of order unity) that the non-linear cosine term can take over. So it will require of order 10^78 collisions, which given the velocity of the atom of ~1000 m/s, means that it will take ~10^68 years for the atom to miss.

Now, due to other non-linearities in the system it would probably in practice be much less than this. But it would certainly be nowhere close to the few microseconds claimed by the author.

I think the dependence is clearly exponential. The discrepancy in the scattering angle should grow by a factor of O(mean free path/radius of scatterer) per scattering event.

I think that for extremely small perturbations the discrepancy will be linear until it reaches a critical threshold, at which point the trajectories diverge exponentially. Basically, you need to get to the point where the path of one atom diverges by ~atomic radius. But until that happens, there is no non-linear mechanism to increase the divergence.

For example, if you just consider a billiard ball bouncing between two walls, the divergence in trajectories will only ever grow linearly. But for very small perturbations, the case of collisions between atoms is effectively the same as collisions between walls --- the curvature of the scattering atom can be neglected. After all, the curvature here is a second order term, and when the perturbation is of the order of 10^-68, the non-linear factor will be (10^-68)^2 ~ 10^-136.

> I think that for extremely small perturbations the discrepancy will be linear until it reaches a critical threshold

An error of epsilon on the angle of a path toward the scatter will cause an error of ~(distance to scatter/size of scatter) epsilon in the angle from the center of the scatterer to the "impact point". So the errors should grow exponentially in the number of scatterings.

The (effective) curvature of the scatterer CANNOT be neglected -- it is crucial. Modeling this with large flat reflectors is not appropriate.

I spent a bit more time thinking about this, and yes you are correct. Each scattering event increases the discrepancy by ~mean free path / atomic radius, which is about five orders of magnitude. So even starting from a 10^-78 discrepancy, it only takes ~10 collisions to get an order unity discrepancy. I was thinking that the discrepancy due to collisions was negligible compared to the accumulated tidal acceleration, but this is clearly wrong.

About three orders of magnitude, not five, but yeah.

I'm glad this makes sense; thanks for writing!

pfdietz is correct. The key thing is that divergences always multiply, leading to an exponential dependence on number of scattering events. There isn't a transition between a linear and an exponential dependence. I think where you're getting confused is that an exponential can be initially approximated by a linear function (i.e. exp(x) ~= 1 + x), but that doesn't mean that the process is linear.

The dominance of the exponential growth is also why the tidal correction, as massive as it seems, gets swamped and ends up not mattering much.

-- Raghu Parthasarathy

> Everything affects everything else, and the correct questions to ask are how much?

I wish this kind of statement would be applied to biology, indeed there is system biology, but it does not seems to be used widely.

I read avidly everything which is published in neurodegenerative diseases. Most of academic publications claim that one or another of ten of thousands molecules have a positive influence on these diseases.

Clearly the biotech or pharmaceutical industries does not trust most of these publications as little of these suggestions are translated in clinical trials.

I do not want to say that the biotech/pharma industries are more rational either as it happens that:

- They seems to apply one molecule to multiple unrelated diseases until they find a suitable target (which more than often they do not find)

- They apply repetitively the same molecule to the same disease even when clinical trials show it is not effective (for example memantine and ALS).

> Everything affects everything else, and the correct questions to ask are how much?

I don't think this is true. It's true that interstellar mass movements affect atomic motions on earth (the theory is well understood!), but it's not true that the movements affect, say, whether a particular computer program correctly meets a given spec: they're simply at different levels of explanation. One can't even imagine a theory that connects them. Maybe the interstellar motion could cause a bug in a particular computer, but we can consider a computer program's properties independent of a particular execution on a particular hardware. Furthermore, we can even test this using an ensemble of physical computers in the presence of error-producing radiation or whatever.

This video by Veritasium on the unexpected effects of the cosmic rays is pretty interesting: https://odysee.com/@veritasium:f/how-distant-galaxies-mess-w....

yeah, pretty funny how a cosmic ray can flip an election.

and frustrating that ECC ram is not standard

>One can't even imagine a theory that connects them.

You are saying that atomic motions on earth cannot affect computer programs?

Or are you saying that some atomic motions on earth cannot affect computer programs?

If the latter, then how, theoretically, would you divide them into those that can and those that can't?

The distinction I'm making is not between different types of atomic motions, but between properties of an algorithm (eg whether it meets a spec) and the physical instantiation of the algorithm running on some computer: ie software vs hardware. Hardware is subject to physics (cosmic rays etc), but software is not. There are lots of examples like this where the subject of interest occurs at different levels of emergence/analysis.

I wouldn't be going down this tangent if the author didn't make such strong, overbroad statements that amount to "everything affects everything," and instead said something like "small, distant phenomena can have surprisingly non-negligible effects" or something, but I get the impression the author would argue against this change on fundamental (vs pedantic) grounds.

For what it's worth, I like the story referenced in the blog's name https://eighteenthelephant.com/2013/08/02/167/

Your comment is a good one, and I don't have time for a long reply, but I'll point out that the motivation for my being a bit "overblown" is the unfortunate and very common tendency of many people to have a nonsensical "null hypothesis" that there's precisely zero effect of something on something else, as if it's at all sensible that there are zero connections. I work, for example, on microbiome research, and there's an unending stream of papers testing the "hypothesis" of "the gut microbiome affects X" rather than "what's the magnitude of the effect of the gut microbiome on X". I'm writing quickly, but hopefully this makes sense.

Glad you like the blog name, by the way!

Thanks for the reply. I would be interested in your longer reply, if you get the chance!

I get where you're coming from now that you mention the microbiome. That similarly reminds me of headlines like "antidepressants alter the brain," specifically variations that imply something groundbreaking; on some level it IS impressive that physical effects can be observed, but on another, it's almost trivially true that it affects the brain, because where else do thoughts originate? In fact, speaking of depression, there's probably studies exactly like what you described where depression is X.

I think this could be usefully tied back to the software/hardware distinction I raised. Some might say that, in a lot of cases, depression is due to faults in "software" -- negative thinking, assuming the worst, etc, -- or at least is better treated as though it were. This is opposed to faults in "hardware" -- insufficient production of serotonin, or whatever (and I'm sure there cases where this is the best explanation/treatment).

So maybe a certain microbiome state is like a computer running on the Mars rover where its hardware is subjected to fault-inducing radiation? The software/hardware distinction probably isn't as clean in the brain compared to our silicon-based computers! We know so little in this area that it probably does make sense to proceed as though everything affects everything, as you say -- but I also think it's important to allow that it may not.

>strong, overbroad statements that amount to "everything affects everything,"

Perhaps the author did not mean to include abstractions as "things" in this context?

I think you're discussing metaphysics and I don't see the connection to the article.

I guess gravity doesn't affect faith, hope, or charity, but is that news, or actionable?

First, I'd say it'd be an impoverished scientific worldview that excludes algorithms just because it's not a "thing" per se!

But I could have just as well chosen a "thing"-ier example. For example, do quarks/the strong nuclear force affect the melting point of metals? There's probably a large "parameter space" where it literally has zero effect (in some part of parameter space atoms cease to exist, so it is a leaky abstraction!). But you can have melting phenomena in substrates that don't even have quarks. In fact, to bring it back to software, you can get melting phenomena in a pure algorithm like the Ising model.

(I won't comment on what the author meant to include -- he responded to me earlier, and maybe he'll respond to this too.)

You sound like you are champing at the bit to get into the argument about whether there is a difference between rain in "reality" and rain in a simulation.

All I have to say about the question of whether there is a difference between physics and metaphysics is to quote "Reality is that which, when you stop believing in it, doesn’t go away".

Which is really the same thing as Samuel Johnson kicking a stone: "I refute it thus."

Whether or not you like the distinction, it remains.

And while only the author can say what they intended, I didn't see any hint in the article of this topic.

Conversely you now seem focused on getting into an argument about simulations -- I gave an example of what I meant that doesn't involve simulations and is purely physical but you didn't comment on that.

It's true I've focused on something that wasn't the meat of the original post -- but it's also true that I challenged a direct quote that is arguably the most profound claim in the post, and that's what I'm most interested in ¯\_(ツ)_/¯

>Conversely you now seem focused on getting into an argument about simulations

I gave a definition of reality, and definitions can't really be debated; thus, I don't see them as encouraging argument.

>I gave an example of what I meant that doesn't involve simulations and is purely physical

I'm uncertain what that was. Do you mean your "thing-ier example"? That compares simulated melting with actual melting?

Yes to your first question: the example was that quarks (sub-nuclear particles) can be justifiably ignored completely in understanding/predicting some physical properties of matter, like the melting point of metals (even though metals are composed of quarks). Or more generally, quantum chromodynamics mostly has no effect on solid state physics -- and these are both sub-branches of physics! When you consider all of science/reality there are many such levels where phenomena appear at only some levels.

I don't really understand where you're going with the metaphysics/reality discussion. It seems you consider software part of metaphysics? The laptop I'm typing this on is real in Johnson's sense, as are the patterns of photons projected from the screen, and these can't be understood much of the time without having some understanding of the software it's running.

Just to establish some common ground: I agree that metaphysics is different from physics. I also agree that a thing is different from a simulation of the same thing. I get that software is more abstract than the particle interactions described in the article, and that that is perhaps a distracting difference in the example I first used, as you pointed out. If there's an interesting discussion/disagreement here it's somewhere in the neighborhood of the question: "is math part of reality?" That's something of an open philosophical question https://www.journals.uchicago.edu/doi/abs/10.1086/289138. I think that's even more tangential than my original comment though.

You seem to be arguing that "everything affects everything else" is false because, moving to a concrete example, if Kristin Kreuk is beautiful, then she's beautiful in an objective sense that can't be affected by the motion of space dust.

But this doesn't matter, because every time her beauty is assessed, the assessment itself is affected by space dust. Her objective, Platonic beauty is only relevant in that it affects how often she is assessed to be beautiful. But this is precisely the statistical description that the OP argues for. In the world of things that happen, everything affects everything else.

It seems in this example you do accept that there's an objective reality (beauty) that isn't affected by space dust? If so, I don't understand why you say that doesn't matter, and instead attach significance to the camera noise or whatever measurement artifacts that space dust causes. A central goal of science -- and making sense of the world more generally -- is to distinguish between such measurement artifacts and more interesting theories that describe what's really out in the world.

I would reserve the usage "X affects Y" to where there's an interesting theory involving both, and I had the impression the OP did as well.

> It seems in this example you do accept that there's an objective reality (beauty)

Think about why I might have picked beauty for the example.

I don't know, I did think it was an odd example, but you also referred to beauty in an objective sense twice. Are you saying... everything is subjective?

Objective vs. subjective is a slippery concept.

Everybody can have different opinions on a given quality of a set of things. Is this art attractive. Is this product worth more than it costs.

But those "subjective" opinions can all be functions of objective, measurable properties.

And if you poll or interview everybody, you can construct an average, a statistical distribution, a model, etc. which is objective in the sense that it describes the opinions, which objectively exist and are about objective reality.

Yes, I agree that even "subjective" topics like beauty are nonetheless based on object qualities and aspects can be studied in an objective way through surveys etc.

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