What situations in classical physics are non-deterministic? (2018)
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Also, major shout out to the “Big Picture” book referenced in the question. It is one of my favorite books bar none.
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.
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.
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".
Read. Then post.
>I repeat, this is a math result, not an argument about physical systems.
Did you even care to read the title of the post?
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.
Sorry, I didn't take my crazy pills today.
To me it is very clear that the question involves physics from the start.
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.
>What situations in classical physics are non-deterministic?
Is "statistical mechanics" contained within "classical physics"?
Yes or no? No need for a nonsensical philosophical essay.
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.
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.
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.
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?
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.
Not a different question, not an essay, not hand-waving, just focus on that very concrete question.
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.
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.
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.
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.
For example: https://physics.stackexchange.com/questions/39632/nortons-do...
see https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is... which was linked to from that new question:
> 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.
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
It has no reason to roll unless the placement was uneven, and if it was uneven, it would not break determinism.
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.
TL;DR: Magic breaks Newton's laws
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.
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.
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 ?
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).
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.
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 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
One does not need to go to atomic theory for this assumption to be wrong in reality.
In the model, you have perfect spheres...
How?
I've only seen "magical wand" arguments.
Norton's Dome is nonsense to me.
Whatever takes the ball out from its state of equilibrium/rest.
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.
Norton's Dome is an example where multiple solutions exist that have the exact same initial conditions but still develop differently.
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?
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?
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.
[0] https://mitpress.mit.edu/9780262553452/structure-and-interpr...
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 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.
https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_t...
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.
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.
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.
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.
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.
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).
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.
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.
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!
Could you expand on how this is possible?
It's a testament to the power of exponential growth.
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 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
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.
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
See this gif for example https://gereshes.com/2019/02/18/chaos-and-the-double-pendulu...
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.
The examples in this article all seem to involve infinite forces and speeds... I don't think they're as interesting as they are made out to be.
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.