Thanks for the reference to the video. I watched it a few weeks ago and was befuddled by it. How can the ball just randomly start rolling in a random direction? It seemed to me that an obvious explanation would be that there is air flow in the environment and with the ball balanced in an unstable position that some air movement would easily nudge the ball off balance. I understand the diff eq of motion with the singularity but it seems to me that a ball balanced at the apex of any radially symmetric convex surface would eventually commence rolling, due to fluctuations in the air flow.
Norton’s Dome plays fast and loose with the math. It could be a halfway decent way of modeling a ball that randomly started moving, but that’s not actually how anything works.
Norton’s dome is a valid paradox, in the sense that the math really does admit two valid equations of motion. The link you provided doesn’t dispute that fact (other than commenters pointing out that you need a proportionality constant to make the units work out).
My favorite intuitive explanation for the presence of the paradox is well summarized by the Wikipedia article on the dome: “To see that all these equations of motion are physically possible solutions, it's helpful to use the time reversibility of Newtonian mechanics. It is possible to roll a ball up the dome in such a way that it reaches the apex in finite time and with zero energy, and stops there. By time-reversal, it is a valid solution for the ball to rest at the top for a while and then roll down in any one direction.”
Calling it a valid paradox is questionable, there’s a little mathematical sleight of hand required for the particle to actually stop in finite time. It doesn’t work for say particle sliding up a sphere.
To work the curvature of the dome is infinite is at the apex, which then breaks many things. There’s a lot of disagreements around this paradox and much older related examples because Newtonian physics is somewhat ill defined: https://philsci-archive.pitt.edu/8833/1/dome_v3.pdf
Since many thought experiments allow for objects to be composed of infinite one-dimensional points that somehow form higher-dimensional bodies, an infinite curvature could be interpreted as the apex being a single point supporting the point at the bottom of the ball. Both points should be perfectly aligned and in perfect equilibrium.
It has no reason to roll unless the placement was uneven, and if it was uneven, it would not break determinism.
I don't know if you can actually stitch the equations together though because they have different initial values, albeit in higher order derivatives than Newtonian mechanics cares about.
> If we start at an arc length of 1/144 for example, it will run up the dome and arrive at the apex in 1 second. As we’ve seen, it has zero velocity and zero acceleration at this point, but moves off after anyway because it still has a positive value for snap.
Norton's dome is not a valid paradox, it just exploits in an overly complicated way the fact that almost no handbook of physics bothers to present a complete set of axioms for the classical mechanics (and even less for relativistic or quantum mechanics). So the "paradox" is based on the sloppy teaching of physics from a mathematical point of view.
The form of the Norton's dome does not matter. The so-called paradox is just a random example of the fact that there exist multiple functions of a variable that have in the origin the same values for the function and for the first 2 derivatives, e.g. various pairs of polynomials of the 4th order.
Therefore if you accept any function of time as describing a possible motion, you can always find motions that at some moment in time have the same position, velocity and acceleration.
This is not an example of indeterminacy in classic mechanics, because one of the axioms of the classic mechanics is that all the forces that exist in nature are such that the state of a mechanical system is completely determined by the positions, velocities and accelerations of its components (in other words, a mechanical system must be described by a system of differential equations of the second degree that has a unique solution).
There is no difficulty of imagining other kinds of forces, for which this assumption is not true, but a theory where such forces exist is no longer the Newtonian mechanics, in the same way as any geometry where Euclid's axiom of parallels is not true is no longer an Euclidean geometry.
If Newtonian mechanics were a correct model for the World, a ball would remain forever on the top of the dome, without ever falling. In reality, even assuming the validity of Newtonian mechanics, the main reason why any attempt to test this experimentally would fail is the thermal motion, due to which a ball can never be at rest, so it would always start immediately to fall in a random direction.
The violation of the axioms is why the so-called different solutions are not solutions within Newtonian mechanics.
On the other hand the argument that the initial state could be obtained by launching the ball towards the top, and then time reversal would demonstrate a valid solution, it is also wrong, because the so-called solution cannot be obtained by time reversal.
If the ball is launched with only enough energy to reach the top, so it will come to rest, then it requires an infinite time to reach the top. Reversing the time means that the ball will remain on the top for an infinite time, without falling, as expected.
The real paradox is why and how a stationary object in a perfect world that obeys Newton's laws suddenly started moving. There's only mass and gravity, not even thermal or atomic effects allowed. The best justification the author gives is that it happened and this thought experiment doesn't explain why, how, or even when.
If you watched the video, you would have the answer. Well, an answer. Which is that there does not need to be a cause! Even in an idealized dome with no air, friction, or external forces.
As it does for you for different reasons, this also matches my lay intuition of physics: sometimes things just spontaneously occur, and a system in dynamic equilibrium simply will not hold still forever.
The point is that this is not discussing a physically realizable situation, but an idealized one. The engineering and manufacturing precision that would be required to actually achieve this setup are infinite and unattainable. In the idealized setup there is no air, no surface imperfections, no deviations from central positioning, etc.. And yet despite this idealized perfection, a case can still be made that under this construction the ball might spontaneously move in an undetermined direction. The discussion of whether that case “holds water” and what that means if so is an abstract philosophy discussion rather than one with any obvious practical implications.
> How can the ball just randomly start rolling in a random direction?
Because that's legal according to the laws of motion. The intuitive answer is that it's the time reversed situation to a ball being carefully rolled UP the dome so that it stops and comes to rest on the apex. The shape function of the dome was carefully constructed so that this process takes finite time. So if it's legal in one direction it must be legal in the other.
Obviously this is a statement about math and not physics (since the underlying physical theory here is, after all, wrong!) What we thought were a bunch of well-constructed rules for classical dynamics turn out to have some holes.
>The intuitive answer is that it's the time reversed situation to a ball being carefully rolled UP the dome so that it stops and comes to rest on the apex.
That's nonsense. The arrow of entropy always goes forward. Sure, the ball comes to the top of the dome to rest but it also carries direction, momentum and a lot of other properties that you have to put in as well in your hypothetical entropy-arrow-now-goes-back scenario.
This is high-school grade physics, come on. It's surprising some people still take John Norton seriously, not because of the dome, but because of his many other "controversial" takes on physics that fail miserably on their foundations.
Seriously, you don't seem to know much about things you speak confidently about.
Norton's dome is a surprising mathematical situation in very conventional classical mechanics. It doesn't matter what else Norton has done, this observation is trivial to verify for every undergrad maths/physics student.
This has absolutely nothing to do with entropy or the arrow of time.
The mathematical situation is of no practical relevance because it's "density zero": Generic deviations will destroy this peculiar behaviour.
>you don't seem to know much about things you speak confidently about
Good one, chap! How about you argue with substance instead ...
Explain, what makes the ball suddenly start rolling down the dome? Do not hand-wave, just give a direct answer to this question, based on your purported understanding of the problem.
> Good one, chap! How about you argue with substance instead ...
Considering they were replying to a post that was, effectively, arguing "nuh uh!", their response seems reasonable.
> Explain, what makes the ball suddenly start rolling down the dome?
That's _literally_ the entire point. Nothing does. There is nothing that causes the ball to start rolling. But the Newtonian laws of physics indicate it will.
> But the Newtonian laws of physics indicate it will.
Pedantically: they indicate it can. The situation where the ball spontaneously starts rolling[1] at any specific moment in time, without any application of force or interaction with any other part of the system, are perfectly legal and well-defined by the laws of motion. They just can't be predicted determinically.
[1] FWIW it's not even a ball in this case, as the rotational mechanics of a sphere with non-zero moment of inertia would destroy the very carefully constructed function required for the potential energy field.
Nothing. The Newton equations predict that, given the Norton potential, there are two possible trajectories that solve them.
The next state is not uniquely determined by the prior state, so asking what makes the ball roll shows that you don't understand the claim (non-determinism ) at hand.
I'm trying to wrap my head around your logic. So, I'll go step by step to make sure I get it.
If you were able to perfectly balance the ball on a perfectly constructed dome, blah blah, would the ball stay static indefinitely or would it start rolling down some arbitrary path?
The point is that the equations don't tell you what would happen. Both options would be valid according to the equations.
This is completely contrary to our intuition about Newtonian mechanics. The question "given this situation, what would happen?" typically has a unique answer is typical. If it does, we have determinism. The observation of Norton's dome is that mathematically this question does not have a unique answer in all situations.
Are you asking about the real world? Then the answer is that you can not perfectly craft the dome and what happens depends on the imperfections.
Are you asking about a fictional universe governed by the Newton equations and nothing else? Then I can not answer your question because the question builds on a faulty assumption: That this universe is deterministic and that what is determines what will be. Mathematics shows that to not be the case.
The only possible answer to your question in the second case is: It can not be known or predicted what the ball will do.
It is a concrete question! Here's a potential energy function describing a "hill". Here's an object on the hill at this location. How will it move? The question is well formed and complete. And it has more than one answer!
The arrow of what now?[1] This is classical dynamics we're doing.
I repeat, this is a math result, not an argument about physical systems.
[1] Edit as this was clearly missed: THIS IS SARCASM. Thermodynamics and statistical mechanics are excellent theories and worth studying as they tell us deep and profound things about the natural world. This particular novelty is a result from classical dynamics where they don't apply. The "arrow of time" in Newtonian mechanics is absolutely reversible, and there is no Newtonian idea of "entropy".
did you? the title and content of this post is about math results. you should really consider the possibility that you're very wrong here.
the discussion is about hypothetical results from classical mechanics, which, along with the rest of physics, is a mathematical model that may be incongruous with observations.
I suggest you read the HN guidelines, you are quite abrasive and aggressive in your posts.
Regarding your post about entropy. The reason it does not apply is because entropy is a concept from statistical mechanics which is about the statistics of ensembles of many (even non-classical) particles. It's a concept invented after Newton dynamics, but does not apply to describing the equations of motion of a single particle (try to define the entropy of the single particle system). Time reversal is a core tenent of Newton dynamics.
Maybe you should read my most, I didn't say that statistical mechanics is nonclassical. I said statistical mechanics does not apply to the discussion of a single particle rolling up or down a slope. Tell me which of the states has more entropy the one with the particle at the top od the Norton dome or the one at the bottom?
And yet, the video in question seems to make it _very_ clear that this has been debated over and over, across various papers and people, and _nobody_ has been able to provide proof as to why it's wrong.
It's an entirely nonsense argument. Akin to arguing that algebra is nondeterministic with "zero divided by zero is a random number, because any number times zero is zero".
In the case of classical physics, we come to a singularity in which there are several solutions for how the system resolves. This doesn't make classical physics nondeterministic, this simply means if you come to such a solution, then classical physics have no answer for what happens next.
Intuitively, it seems to me that those examples of classical "non-determinism" are radically different from the quantum ones, in the sense that quantum physics theorize non-determinism, while those situations are merely "left out" by classical theory. (I'm not a physicist, if any one reads this, I'd like to know what they think :)
By "left out", I mean that there are multiple solutions to the equations of motion which are compatible with the initial values of the situation.
I guess this could also explain why there is such an association in this thread between non-determinism and non-predictability ?
> By "left out", I mean that there are multiple solutions to the equations of motion which are compatible with the initial values of the situation.
It's worth noting the distinction between a model and the thing the model describes. It's not "cheating" to note that while a model could admit multiple solutions only one could be valid in the original system.
In a very specific sense, eliminating the other solutions is still part of solving the model, just with discrete logic rather than e.g. calculus.
1) When we say the ball is at rest, and let’s grant it can be, doesn’t that mean velocity, acceleration, jerk, etc are all 0? And thus it will never move? There’s a single solution governing the ball if we say it’s truly at rest.
2) We can’t determine when a space invader will suddenly appear to us, but that isn’t some fundamental indeterminism, that’s just limits to the speed of light.
Quantum mechanics (potentially) has a radically different indeterminism than these in some of the interpretations (Copenhagen, GRW), where even some FTL and infinitely precise oracle couldn’t predict. Its fundamental randomness (in some interpretations).
I'm not sure if this will make you feel any better, but there's an interesting mathematical corner case at work with Norton's Dome that's responsible for the breakdown in the intuition you're expressing in (1).
You could formalize this intuition as the statement that, if I'm trying to describe a function f(t) and I know (a) the value of f and its derivative at t=0 along with (b) a second-order differential equation that f has to satisfy, this should be enough to nail down the entire function.
A big theorem, which many people just call something like "existence and uniqueness of solutions of ordinary differential equations", says that in most ordinary situations this is indeed true, and basically for the reason you probably intuitively think: you can imagine using the differential equation to make tiny "updates" to the value of f to move a little bit forward in time, and take the limit as the size of your time increment goes to zero. (You can read more about it in this somewhat technical Wikipedia article: https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_t....)
But there is a condition on the theorem which limits its scope: the right side of the differential equation has to be something called "Lipschitz continuous". The vast majority of differential equations that appear in Newtonian physics satisfy this condition, but the equation you get in the Norton's Dome example doesn't, and this is what's responsible for the lack of uniqueness in the solution. It turns out that there are many different trajectories for the particle that satisfy both the initial condition and the differential equation.
What relevance does this have to the actual universe? Personally, I think very little; it's a fact about a model of physics, not a fact about the actual universe. There are all sorts of reasons why you can't literally build Norton's Dome: matter is not actually continuous because it's made of atoms, and classical physics isn't an exact model of the universe anyway. But it's interesting to see that a feature of Newtonian physics that we usually take for granted isn't actually always true.
Wow thank you. You’re right there’s likely some hidden assumptions that I had taken for granted that a unique solution is relying on when solving DE’s. I will have to read up to make this more clear mathematically, but at least mathematically I think you’ve answered my concern about 1). Now whether things break down when we model a potentially discrete world with continuous math, that’s for another day like you said. And what it means for something “at rest” to start moving if all its position derivatives are 0. But those might be more philosophical.
You're welcome! There's actually one more point that I thought of after sending that reply, which since you mentioned it again I should maybe flag.
Totally apart from physics, it may seem intuitively plausible that if you have a function f and (a) all f's derivatives exist everywhere, and (b) f(0), f'(0), f''(0), etc. are all zero, then f must be the zero function. This is actually also not true! For a counterexample, you can look at this article on bump functions: https://en.wikipedia.org/wiki/Bump_function.
In classical mechanics it’s quite common to say something is “at rest” and then instantaneously acted upon by a force, so just because something is at rest doesn’t mean it will never move. It usually just means dr/dt = 0 in a particular coordinate system.
In fact[1] “at rest” doesn’t even mean the thing isn’t moving, accelerating or even that the rate of acceleration isn’t increasing. It just means than an inertial frame exists from which you can conveniently assume that these things aren’t happening, and Newton’s first law tells us that indeed such a frame always exists.
I’m not sure I buy the example of the Norton’s Dome though. I get the technical argument from continuity but it seems weak to me. Lots of differential equations (eg the wave equation) have a trivial solution where u(x,t)=0 for all x and t - that doesn’t actually translate to us not knowing whether or not something will move in the real world. As you have said it will fundamentally always move or always not move given a particular set of initial conditions - we may just be lacking enough information to say which of those cases may hold, but that doesn’t make it non-determanistic - just it appears so to us.
[1] I got this from Kleppner and Kolenkow, which is an amazing book if you’re interested in Classical Mechanics
Norton's Dome isn't spherical, so it's a bit more sophisticated than you imply. In classical physics, you could roll a ball up the dome so that it comes to a rest at the apex in finite time. Thus, due to time symmetry, it's also theoretically possible for a ball at rest on the apex to suddenly roll down the hill in a non-deterministic way. This is certainly unrealistic, as you say, but that's what classical physics predicts.
Rather than accepting non-determinism, this could also entail that the existence of non-deterministic classical solutions implies that such configurations don't actually exist, eg. you can't construct Norton's Dome. It turns out that that's correct, and the same with the space invader example, eg. no such thing as unbounded acceleration.
What is "magical wand" here? This is simply Newton's equation and a specific dome shape. Trivially no perfect bodies can be built experimentally, and it's specific properties are not robust to perturbation, but there is absolutely nothing magical or strange going on here.
Do you understand the difference between Norton's dome and other domes with balls at rest on the top?
The very point of non-determinism is that the next state doesn't follow uniquely from the current state, causality is broken. Thus there is no "cause" for one or the other trajectory. This is a feature of the Newtonian equations, whether you like it or not.
The question is about classical (newtonian) physics, which is a mathematical model, rather than the real world. The question is asking when is there non-determinism in the newton mathematical model.
You've got it backwards. Newtonian theory does not rule out perfect spheres (or any other mathematically perfect shape), hence the possibility of non-determinism.
It also better rule out general fractals as well, cause some shit will happen between these as well. Not fully sure, but it feels like Newtonian doesn’t really work with anything non-finitely jagged, cause geometry has extremums or simply insane behavior at everything non-linear non-finite.
Chaotic dynamics was taught before Norton’s Dome. Three body problem, articulated pendulum, etc. Are those out of vogue as examples of non-determinism?
Chaotic dynamics aren't necessarily non-deterministic. Chaos is about small changes to initial conditions causing large changes in the future, but the exact same initial conditions can still deterministically lead to the exact same future state.
Norton's Dome is an example where multiple solutions exist that have the exact same initial conditions but still develop differently.
Of course it does, you could build one yourself. It's just a dome with a specific shape.
Of course the ball will choose a way down based on tiny physical forces which we can't eliminate in the real world. Fundamentally, Norton's Dome is non-deterministic in classical physics, but reality is quantum.
So, for example, for large enough r, the gravitational force \sqrt(r) will exceed the free fall accelleration g?
More importantly, does this additional branch of solutions that satisfies the initial conditions, survive under the small deformations of this dome shape? The perfect Dome shape certainly does not exist.
Usually, when we say something doesn't exist in nature, we mean it's fundamentally incompatible with our 3 spatial dimensions as they exist, passes through itself, or is infinite along some dimension, requires infinitely thin surfaces, etc.
But by your definition, even something as simple as a cylinder or sphere doesn't exist in nature, because basic mathematical relationships like radius to circumference won't survive "small deformations".
I don't know what point you're trying to make. Norton's Dome "exists in nature" as much as a sphere or a cube does. If it doesn't exist in nature, then no geometric form does.
>Usually, when we say something doesn't exist in nature, we mean it's fundamentally incompatible with our 3 spatial dimensions as they exist, passes through itself, or is infinite along some dimension, requires infinitely thin surfaces, etc.
Well, that's what we're saying. The distinction lies in ideal/perfect, vs imperfect. For example, does a perfect cube shape exist? If you closely examine any cube in existence, it has small deformities if you look close enough, as the very edges and corners which make up a cube are made of atoms, which are non-cubical in shape (not to even mention quanta). A perfect cube relies on an cubical shape at infinite scale, but as you mentioned above:
>when we say something doesn't exist in nature, we mean it's fundamentally incompatible
>or is infinite along some dimension
Norton's dome requires an infinitesimal point of sorts for the math to work out. Does that exist in reality? Idk, but it certainly seems dubious.
> Well, that's what we're saying. The distinction lies in ideal/perfect, vs imperfect.
No, that's not. "Exist in nature" doesn't mean "perfect". Totally different concepts.
> Norton's dome requires an infinitesimal point of sorts for the math to work out. Does that exist in reality? Idk, but it certainly seems dubious.
A cone requires a point at the pointy end. Does that exist in reality? I've certainly seen a lot of objects we call "cones". And they were pointy.
The point is, if you say Norton's dome doesn't exist then you mean cubes don't exist. And we all agree cubes do exist. A perfect Norton's dome doesn't exist, just like a perfect cube doesn't exist. But a regular Norton's dome certainly does exist. Just like a regular cube. Again, it's not an exotic shape. But it doesn't need to be perfect to exist -- otherwise nothing would exist at all!
In the sense that one cannot create an ideal dome and make an experiment, whether a point particle placed exactly at the top later randomly starts to fall.
One has to study if this class of solutions survives deformations of the ideal dome, to make such an experiment (neglecting quantum effects).
But there's no point in performing such an experiment.
As I already explained, this is only non-deterministic in classical mechanics. And the world is quantum. It is an entirely theoretical distinction to begin with. It does not require experimentation.
Nevertheless, you can construct such a dome the same as you can construct a sphere. It's just a regular old geometric shape.
For any dome-like shape, you can start a marble at the bottom and roll it up with some initial speed. If you roll it with insufficient initial speed it'll turn around and come back down. If you roll it too hard, it'll overshoot the peak. By continuity, there must exist some initial condition where it stops at the top.
Now, here's the thing that makes Norton's dome special: For a typical dome shape it'll take an infinite amount of time before that marble stops at the top. If you plot the position as a function of time it'll have some type of sigmoid-like shape. However, for the special case of Norton's dome, you can make it settle at the top in a finite amount of time where it'll sit for the rest of eternity. In other words, if you plot the position as a function of time, there will be some critical time after which its position is constant.
Now, the clever thing to do now is to realize that Newton's laws are time reversal symmetric which means that any motion forward in time could equally well happen backwards in time.
So, you're allowed to take any position plot and flip it horizontally; this is also going to be a valid trajectory.
For any typical dome shape this is not a problem. For a typical dome shape you have a sigmoid-like solution which, when flipped, is still sigmoid shaped. In particular this means that there is no finite time at which you can place the marble at the top of the dome and have it roll off. At any finite time, the marble will be slightly off the top and have a small nonzero speed.
Norton's dome is different. If you flip its trajectory horizontally you'll see that there are many moments in time where you can start the marble at the top to have it abruptly start rolling off the top at some later time. This is the paradox. You can choose to have it sit at the top for one second and then start rolling or sit at the top for one minute and then start rolling.
Unlike other domes, Norton's dome seems to violate our intuition for how initial conditions work. In all cases the marble starts at the top with zero initial speed and yet falls off the top att different moments.
The answer given in Stack Exchange and the Wikipedia article linked to in one of its comments provide explanations, though whether you find them convincing is, of course, up to you!
The Wikipedia article says there are solutions to the classical equations of motion: one in which the ball remains stationary forever, and then all those where, after an arbitrary period during which the ball is stationary, it rolls off the dome in an arbitrary direction. What makes this indeterminate is that the analysis of a single initial state yields multiple possible outcomes.
The article goes on to say "Notice in the second case that the particle appears to begin moving without cause and without any radial force being exerted on it by any other entity, apparently contrary to both physical intuition and normal intuitive concepts of cause and effect, yet the motion is still entirely consistent with the mathematics of Newton's laws of motion so cannot be ruled out as non-physical."
This raises the question of what we mean by 'physical', and whether theories of physics define the physical or describe something existing independently. I will leave that to the more philosophically minded; for myself, I will just note that as there are cases (and physically realizable ones at that) where classical physics gives answers that are not merely indeterminate but outright wrong (the ultraviolet catastrophe being a canonical example), I don't think anything of consequence hangs on this particular case.
I looked over the explanation on the Wiki and, thinking about it, I'd say that the surface around the tip of Norton's Dome is not a continuous function, which alone is enough for me to disqualify it.
Intuitively, that's a sufficient explanation to me, or at least a sufficient start of one. IANAPhysicist, so I'll ask here: are there any examples of surfaces or phenomena in classical physics that are defined by a discontinuous function, and are something you'd actually expect to see existing in the real world? Things seemingly discontinuous until you zoom in close enough don't qualify.
Pontus gave a good explanation of the physics. Mathematically the statement is simply that there are different trajectories with the same initial conditions that solve the same differential equation. This is easy to calculate and has nothing to do with "people wanting it". It's simply a mathematical fact. What do you not find convincing here?
This is the first time I'm encountering Norton's dome, and I'm not particularly academic, just really like learning about math -- so I'm hoping a friendly HNer can help me out here.
Is norton's dome essentially describing a saddle point? Is the only reason it's nondeterministic because at that point things go to infinity? If we're in the world of mechanics, wouldn't it be up to the machine to determine what to do at that point? Implementation defined, one might say?
Not really. It's a dome/hill shape, not a saddle, and nothing in particular goes to infinity. The only thing special about the shape is that it's constructed such that there exists a way to kick a ball straight up the slope and (if you're impossibly precise), it'll come to rest at the top in finite time.
This in itself is fine, but starts feeling real weird once you are familiar with the time-reversability of physical systems. If you time-reverse this system, you end up with a ball that sits at the top for an arbitrary period of time, and then suddenly just rolls down for no deterministic reason.
If you considered a bunch of different runs of this, some where the ball starts at the top, stationary, and some where it's kicked up to stop at the top (from various locations, at various times), they all start at the exact same conditions in the time-reversed system. So why do they do different, unpredictable things?
I'm a bit out of my depth, but I think I recall Gerald Sussman talking about dealing with a similar problem using an operator that indicated that evaluated to exactly one out of a finite set of elements but can be evaluated to any in the set.
If I'm recalling, it was ok because each of the elements of the set were differentiable functions, but the operator produces a set of possible results, but doesn't specify which. Sort of like a monad, it seemed.
I want to say he was referring to a quadrature, but I honestly can't recall and wont be able to spare the time to hunt down the talk until later.
Coincidentally, I just got a copy of structure and interpretation of classical mechanics just the other day, so hopefully I'll get some more appreciation for the problem.
isn't the point of the chaos that as time progresses the predictability of outcomes severely deteriorates?
or maybe a better frame:
any simulation the seed would still result in the same answer, since the computation is deterministic, but the system being simulated is not likely to behave the same at that time under the same initial conditions.
> isn't the point of the chaos that as time progresses the predictability of outcomes severely deteriorates?
Yes. And that's what the GP said.
Predictability deteriorates because small errors in the initial conditions grow in proportion to the total value up to the point they can more than explain the entire value.
> but the system being simulated is not likely to behave the same at that time under the same initial conditions.
No, that part is wrong. Chaos is not about non-determinism.
Yes, predictability deteriorates because the sensitivity to initial conditions increases over time. We can't measure or create initial conditions to infinite precision so at simulation will be accurate only to our measurement precision.
Those examples are deterministic. The issue with them is that we can not precisely measure their starting positions and the locations of the objects diverge quickly with minuscule different starting positions. If we could measure the starting position of either of those problems with 100% precision we can predict what they would do.
Extreme sensitivity to initial conditions occurs at the level of interactions of molecules in a gas. Errors in position or velocity accumulate exponentially over very short time scales. It's so sensitive that moving a single atom on the other side of the universe would, by the small change in its gravity, cause O(1) changes in the position of molecules in less than a millisecond.
> It's so sensitive that moving a single atom on the other side of the universe would, by the small change in its gravity, cause O(1) changes in the position of molecules in less than a millisecond.
Small errors in position/velocity are amplified exponentially at each collision. Air molecules collide on average about once every 200 picoseconds. So, an error of one part in 10^1000 will build up to an O(1) difference in about O(log 10^1000) collisions, or maybe ~1 microsecond (the "less than a millisecond" claim was being very conservative.)
It's a testament to the power of exponential growth.
What I meant wasn't that moving the atom causes a FTL propagation of an effect, but that in two situations that differ just by the position of said atom, the changes become visible that quickly (after any imagined classical propagation delay).
Isn't the 3 Body Problem non-deterministic in classical physics? or am I misunderstanding it?
example of a possible misunderstanding might be (don't know if the follow statement is true), its only non deterministic due to our inability to calculate the initial conditions exactly, but if we could calculate the initial conditions exactly, it would no longer be non-deterministic. It's only from modern physics (i.e. not classical), that we understand that its impossible to measure the initial conditions exactly, classical physics might have expected that its simply due to lack of ability, vs impossibility.
I think this is a trick question. I don't think it's even been settled that quantum mechanics are truly, for sure probabilistic. The universe's true underlying nature (deterministic or not) is still unknown. Additionally, classical mechanics inherently assumes states can be predicted if initial conditions are known.
I think more practical questions would be along "what classical situations are non-deterministic from a human perspective or in-practice?", which would lead to questions about what is calculable or not with the tools and knowledge we have now
> I don't think it's even been settled that quantum mechanics are truly, for sure probabilistic.
Since you can never prove or disprove the existence of "God" or some other hidden global variable deterministically moving the universe, yes, nothing can ever be settled. Scientists don't find that line of reasoning particularly interesting or compelling to dwell on.
You're right, I didn't mean to add so much emphasis on 'knowing'. I meant settled in the sense of 'settled science', or being as reasonably sure as we can--barring any flying spaghetti monsters. Far from being uninteresting to scientists, I'd argue that this question of whether quantum mechanics is truly probabilistic or hides deeper deterministic mechanisms is one of the most profound topics in physics
Isn’t heat transfer modeled using stochastic processes? Why is it considered deterministic?
BTW I have absolutely no idea of physics, I just know about this because of finance where stochastic processes are used for pricing and heat transfer is used as an example
"Heat", in classical thermodynamics, is a derived abstraction. It's defined as the sum over a bunch of classical energies distributed among particles that behave deterministically. So this is just the measurement trick: we can't measure it therefore it's behavior is "random" from our perspective. But that's a practical limit and nothing to do with "determinism" in a mathematical sense.
When we consider things at the macro scale, we may find properties that don't appear at the micro scale, this is called emergence, and that's what thermodynamics is about.
Particle motion may be deterministic, and importantly, time-reversible, when we have too many particle to individually consider, the rules change, and that's when you are talking about entropy and temperature. To consider an extreme example, our brain is made of subatomic particles, and yet, psychology is not at all like particle physics. The same can be said of finance, where global economics have laws that don't apply to individual transactions and vice-versa.
It is like shuffling a deck of card is considered random, though it is not at all the case, in fact, a skilled magician can control the shuffle and pretty much order the deck in any way he likes. But for the purpose of playing cards, it is considered random, and theory is built on that.
Thermal phenomena like heat transfer can arise in systems that are deterministic at the microscopic level. Indeed, at the time of Boltzmann (pre quantum) one of the major questions is how to go from deterministic but complex & unpredictable classical particle dynamics to continuum models like the heat equation. Kinetic theory is one piece of that bridge between scales.
In more recent times these questions are still studied, e.g., within mathematical physics / ergodic theory circles. Look up "Lorentz gas", "Fourier law", etc. Usually to get anything interesting one needs to hook these systems up to "reservoirs", which are usually stochastic. In principle one could replace the reservoirs by another large, chaotic classical system but that makes the mathematical questions too hard, and having some randomness in a small corner of the system and studying how its influence spreads is still very challenging but more tractable.
If you attach a pendulum to a pendulum to a pendulum. You will get chaotic behavior. There's videos out there. Because my lamen thinking is that if each pendulum is just spinning in one direction (dimension). which is predictable. Then why does adding more dimensions causes the chaos. Don't get it.
Chaos is still deterministic in classical physics. If you start out with the same initial conditions, the system always evolves in the same perfectly deterministic (but chaotic) way.
They are connected and as such influence each other. This creates non-trivial evolution which contains what may be called “continuous tipping points” that heavily depend on the past and also create a whole worlds of futures in the tipping vicinity. Ofc that is just a discrete analogy, because it’s all continuous and it works for any situation. You may see it as tipping points at infinitely many discrete calculations.
Also, major shout out to the “Big Picture” book referenced in the question. It is one of my favorite books bar none.