9 ms·

What situations in classical physics are non-deterministic? (2018)

57 points by squircle 2y ago

ttyprintk 2y ago
Chaotic dynamics was taught before Norton’s Dome. Three body problem, articulated pendulum, etc. Are those out of vogue as examples of non-determinism?
- yorwba 2y ago
  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.
  - postalrat 2y ago
    I haven't seen a convincing explaination why Norton's dome is non-deterministic other than people wanting it.
    - chrsig 2y ago
      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?
      - kadoban 2y ago
        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?
        chrsig 2y ago
        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. [0] https://mitpress.mit.edu/9780262553452/structure-and-interpretation-of-classical-mechanics/ https://mitpress.mit.edu/9780262553452/structure-and-interpr...
    - pontus 2y ago
      Here's how I think about it: 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.
    - deleted 2y ago
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    - Certhas 2y ago
      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?
    - mannykannot 2y ago
      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. https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_the_equations_of_motion https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_t...
    - TeMPOraL 2y ago
      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.
  - chrsig 2y ago
    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.
    - marcosdumay 2y ago
      > 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.
    - naasking 2y ago
      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.
  - cft 2y ago
    I would only call this a thought experiment. We are yet to encounter a physical Norton's Dome: it probably doesn't exist in nature.
    - crazygringo 2y ago
      What do you mean it doesn't exist in nature? 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.
      - cft 2y ago
        >What do you mean it doesn't exist in nature? 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.
        crazygringo 2y ago
        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.
        cft 2y ago
        >I don't know what point you're trying to make 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).
- choxi 2y ago
  Aren’t those examples deterministic? That’s the most interesting aspect of chaotic systems to me, they’re deterministic but still not predictable
- HDThoreaun 2y ago
  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.
xqcgrek2 2y ago
Norton's Dome is only non-deterministic because it assumes perfect spheres exist. One does not need to go to atomic theory for this assumption to be wrong in reality.
- cesaref 2y ago
  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. In the model, you have perfect spheres...
  - xqcgrek2 2y ago
    Newtonian theory has no assumptions or requirements for perfect spheres.
    - munchler 2y ago
      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.
      - wruza 2y ago
        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.
- munchler 2y ago
  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. https://en.wikipedia.org/wiki/Norton%27s_dome https://en.wikipedia.org/wiki/Norton%27s_dome
josh-sematic 2y ago
Here’s an explainer video about Norton’s Dome: https://youtu.be/EjZB81jCGj4?si=VJB5VA1LrvPWMZxz https://youtu.be/EjZB81jCGj4?si=VJB5VA1LrvPWMZxz Also, major shout out to the “Big Picture” book referenced in the question. It is one of my favorite books bar none.
- johnp314 2y ago
  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.
  - ajross 2y ago
    > 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.
    - moralestapia 2y ago
      >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.
dukeofdoom 2y ago
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.
- deleted 2y ago
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- snitzr 2y ago
  Sensitive dependence on initial conditions.
- munchler 2y ago
  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.
- wruza 2y ago
  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.
melenaboija 2y ago
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
- ajross 2y ago
  "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.
- Sharlin 2y ago
  The question is slightly inaccurate, it should be "classical mechanics". Classical thermodynamics is very much stochastic.
  - cft 2y ago
    Stochastic does not mean undeteministc. It means that tiny perturbations to initial conditions lead to huge differences in time evolution. See this gif for example https://gereshes.com/2019/02/18/chaos-and-the-double-pendulum-part-2/ https://gereshes.com/2019/02/18/chaos-and-the-double-pendulu...
- kkylin 2y ago
  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.
pfdietz 2y ago
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.
- kennysoona 2y ago
  > 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. Could you expand on how this is possible?
  - pfdietz 2y ago
    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.
- Sharlin 2y ago
  Of course it could not. As is well known, the speed of causality has a hard upper limit.
  - pfdietz 2y ago
    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).
  - mitthrowaway2 2y ago
    Ah, but does relativity count as classical physics?
wazdra 2y ago
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 ?
- dwattttt 2y ago
  > 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.
gorgoiler 2y ago
Does something to do with gimbal lock fall into this category as well?
swayvil 2y ago
Charge bleeding through a semiconductor
meroes 2y ago
I don’t get these examples. 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).
- nicf 2y ago
  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_theorem 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.
d4v3 2y ago
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
- drpossum 2y ago
  > 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.
  - d4v3 2y ago
    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
deleted 2y ago
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compsciphd 2y ago
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.

57 points by squircle 2y ago

Chaotic dynamics was taught before Norton’s Dome. Three body problem, articulated pendulum, etc. Are those out of vogue as examples of non-determinism?

yorwba 2y ago
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.
- postalrat 2y ago
  I haven't seen a convincing explaination why Norton's dome is non-deterministic other than people wanting it.
  - chrsig 2y ago
    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?
    - kadoban 2y ago
      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?
      - chrsig 2y ago
        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. [0] https://mitpress.mit.edu/9780262553452/structure-and-interpretation-of-classical-mechanics/ https://mitpress.mit.edu/9780262553452/structure-and-interpr...
  - pontus 2y ago
    Here's how I think about it: 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.
  - deleted 2y ago
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  - Certhas 2y ago
    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?
  - mannykannot 2y ago
    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. https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_the_equations_of_motion https://en.wikipedia.org/wiki/Norton%27s_dome#Solutions_to_t...
  - TeMPOraL 2y ago
    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.
- chrsig 2y ago
  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.
  - marcosdumay 2y ago
    > 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.
  - naasking 2y ago
    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.
- cft 2y ago
  I would only call this a thought experiment. We are yet to encounter a physical Norton's Dome: it probably doesn't exist in nature.
  - crazygringo 2y ago
    What do you mean it doesn't exist in nature? 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.
    - cft 2y ago
      >What do you mean it doesn't exist in nature? 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.
      - crazygringo 2y ago
        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.
        cft 2y ago
        >I don't know what point you're trying to make 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).
choxi 2y ago
Aren’t those examples deterministic? That’s the most interesting aspect of chaotic systems to me, they’re deterministic but still not predictable
HDThoreaun 2y ago
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.

xqcgrek2 2y ago

Norton's Dome is only non-deterministic because it assumes perfect spheres exist. One does not need to go to atomic theory for this assumption to be wrong in reality.

cesaref 2y ago
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. In the model, you have perfect spheres...
- xqcgrek2 2y ago
  Newtonian theory has no assumptions or requirements for perfect spheres.
  - munchler 2y ago
    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.
    - wruza 2y ago
      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.
munchler 2y ago
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. https://en.wikipedia.org/wiki/Norton%27s_dome https://en.wikipedia.org/wiki/Norton%27s_dome

josh-sematic 2y ago

Here’s an explainer video about Norton’s Dome: https://youtu.be/EjZB81jCGj4?si=VJB5VA1LrvPWMZxz https://youtu.be/EjZB81jCGj4?si=VJB5VA1LrvPWMZxz Also, major shout out to the “Big Picture” book referenced in the question. It is one of my favorite books bar none.

johnp314 2y ago
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.
- ajross 2y ago
  > 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.
  - moralestapia 2y ago
    >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.

dukeofdoom 2y ago

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.

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snitzr 2y ago
Sensitive dependence on initial conditions.
munchler 2y ago
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.
wruza 2y ago
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.

melenaboija 2y ago

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

ajross 2y ago
"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.
Sharlin 2y ago
The question is slightly inaccurate, it should be "classical mechanics". Classical thermodynamics is very much stochastic.
- cft 2y ago
  Stochastic does not mean undeteministc. It means that tiny perturbations to initial conditions lead to huge differences in time evolution. See this gif for example https://gereshes.com/2019/02/18/chaos-and-the-double-pendulum-part-2/ https://gereshes.com/2019/02/18/chaos-and-the-double-pendulu...
kkylin 2y ago
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.

pfdietz 2y ago

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.

kennysoona 2y ago
> 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. Could you expand on how this is possible?
- pfdietz 2y ago
  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.
Sharlin 2y ago
Of course it could not. As is well known, the speed of causality has a hard upper limit.
- pfdietz 2y ago
  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).
- mitthrowaway2 2y ago
  Ah, but does relativity count as classical physics?

wazdra 2y ago

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 ?

dwattttt 2y ago
> 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.

gorgoiler 2y ago

Does something to do with gimbal lock fall into this category as well?

swayvil 2y ago

Charge bleeding through a semiconductor

meroes 2y ago

I don’t get these examples. 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).

nicf 2y ago
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_theorem 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.

d4v3 2y ago

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

drpossum 2y ago
> 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.
- d4v3 2y ago
  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

deleted 2y ago

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compsciphd 2y ago

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.