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Norton's dome is a thought expriment that shows Newtonian mechanics is non-deterministic. It has the shape of a dome (see exact details of its construction on the linked page) with a rather peculiar property, if you place a ball at the very top of the dome there is a whole class of solutions to its spatial evolution, one solution is that the ball stays at the top forever but for any time $t_0$ there is also a solution where the ball stays stationary at the top until $t_0$, after which it starts to roll down the dome in a random direction, all without any external force or action applied to the ball. Since all of these are valid solutions for the starting condition where the ball is stationary at the top of the dome we see that Newtonian mechanics is non-deterministic.

One common way to see why this is reasonable to to run the process in reverse. It's possible to verify that you can set the ball to roll up the hill with a precise amount of energy so that it reaches the top of the dome in finite time and then stops. Since the laws of physics are reversible, the alternative where the ball spontaneously starts rolling down the hill is also valid (this argument doesn't work for other dome shapes like a hemisphere because the amount of energy needed to make the ball reach the top and then stop requires infinite time for the ball to reach the top).

Norton's dome

While Norton's dome is very interesting to think about for multiple reasons, the important point to my question later down is that it's possible to launch the ball from any direction up the slope so that it reaches the top in finite time and then stops there, and that once it's gotten to the top and stopped there is no way of telling exactly which direction the ball originally came from (since e.g. in the ball falling down version which is the time reversed version of this there is no way to predict whether the ball will fall to the left or to the right while it's still up there, even conditional on knowing that the ball will eventually fall).

On to my actual question: The standard reasoning given behind why Maxwell's Demon doesn't reduce the entropy of the whole universe is that to decide how fast a molecule is moving (and whether to open the door and let it through) it first has to measure the speed of it, which generates information. Initially it was argued that measuring the state of the system necessarily required increasing entropy, and the entropy increase here would counteract the decrease in the box.

However Landauer showed in 1960 that using the principles of reversable computing (such as the Billiard ball computer) you could perform this measurement without increasing entropy as long as the process is thermodynamically reversible. He then proposed that it was not the measuring of information which incurs a thermodynamic cost but rather the erasing of it, see Landauer's Principle.

This fixes the issue in the standard argument because once the information is generated the demon then has to either store this information or discard it. If it stores the information then because the demon is a finite entity eventually it will run out of memory and then have to discard the accumulated info by erasing it. Hence eventually the demon (if it wants to continue operating indefinitely) will have to erase its accumulated information and the total entropy increase of the world caused by this erasure would counteract the entropy decrease inside the box.

I don't fully understand why must erasing information necessarily have an entropic cost and I have a construction using Norton's dome where I can't see the point at which information is being sent to the environment. Looking on StackExchange I found this answer https://physics.stackexchange.com/a/325563/141472 and this answer https://physics.stackexchange.com/a/151099/141472 which say that it's not possible for an operator to map two distinct physical states to the same one, hence information can't be destroyed, only exchanged, therefore "erasing" the demon's info necessarily requires sending this info elsewhere into the world where it will increase the number of possible microstates and therefore entropy.

In the below construction I don't see where exactly the increase in entropy happens, even though it does end up erasing the information: Imagine a ballistic billiard ball computer demon which measures the speed of an incoming particle in the box to decide whether to let it through or not. This is going to generate some bits of information. After making the decision of whether to open the door or not the demon is left with the information generated during the decision process that it now has no use for and would like to erase.

The demon decides to store this information by shooting a ball from the left up Norton's dome if the bit is a 0 and shooting a ball up from the right if the bit is a 1 (it might previously have stored this information in a different format when it had use for it inside the billiard ball computer, but now when it has no use for the info it decides to get rid of it in the below manner).

The energy given to these balls in both cases is just enough for them to rise up the dome and reach the top in finite time and the stop there. You can construct something like this with a billiard ball computer where you have a hole in the surface of the computer where a ball can fall through which corresponds to a 0 and another hole which corresponds to 1. These holes are smooth and connect to the left and right sides of the Norton's dome respectively. Furthermore the depth of these holes is carefully chosen so that the kinetic energy of a ball which falls down the hole when it reaches the bottom is just enough to climb up the dome and and stop at the top.

In this case the demon stores each bit of its information in a Norton's dome "bit", however eventually the ball will get to the top of the dome in finite time and come to a stop there, regardless of whether it was sent from the left or sent from the right. Note that no heat has been dissipated during this process (which e.g. wouldn't be the case if we just had a device which stopped the ball in its tracks regardless of the direction it came from, that heat dissipation would increase the entropy of the world) and also that this process happens in finite time (which would not be the case if e.g. we were using a spherical dome instead); this is where we specifically need a Norton's dome and not some other object.

Once the ball has come to the top of the dome and stopped there the information has been successfully erased. This is because there is one final end state (ball at the top of the dome and stationary) for both of the two start states (ball moving up from the left/ball moving up from the right) created by applying the time evolution operator.

We haven't lost any useful energy to heat (the energy is now all stored as potential energy in the ball at the top, where it can be taken and used for other purposes), so there can't have been any increase in the total entropy of the universe through that and other than the Norton's dome everything else is a reversible process which doesn't generate entropy in our idealised world. The time evolution operator here really seems to have mapped two different states of the universe to one single state, which shouldn't be possible according to the first answer.

If everything here works as I think it should then the demon has decreased entropy in the box by sorting fast/slow particles while it was able to erase the information it gained from this process for free, which gives us a repeatable process that leads to a reduction in the total entropy of the universe, which shouldn't be possible as it violates the second law of thermodynamics. What am I missing here?

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    $\begingroup$ The argument is flawed because it depends on a null-set of trajectories that is practically non-existent (we can only sample a finite number of times). There is no need for it, to begin with. The actual physical mechanism by which irreversibility is created is much more simple: energy is lost to the environment. There are no perfectly closed systems in nature. That's just an approximation and it turns out to be a flawed one. $\endgroup$ Jan 27 at 18:19
  • $\begingroup$ I agree the null-set of trajectories is measure 0 and we can't do this in the real universe we live in (there are also arguments you can't construct a Norton's dome in the real world). However as a theoretical model I don't immediately see the issue with it. Indeed Landauer's principle provides a formula for how much energy a bit erasure needs as $k_BT\ln(2)$ while my description is indep. of temperature, so at the very least it would violate that bound for high enough $T$ regardless of a small amount of dissipation and so the entropy reduction in the box caused by the demon would be higher. $\endgroup$
    – Hadi Khan
    Jan 27 at 19:28
  • $\begingroup$ It doesn't take any energy to erase a bit in an open system. The universe keeps erasing local observer state all the time in a way that makes it completely impossible to reconstruct or retrieve it. The entire conundrum stems entirely from the assumption of perfectly closed systems. That's why temperature is in play, as well. $\endgroup$ Jan 27 at 19:36
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    $\begingroup$ Did you check that the ball and dome mechanism used by the demon does not consume more energy than what could be produced by the temperature difference generated by sorting the molecules? Considering that you need one dome and one ball for each molecule, it is hard to imagine that the balance can be right. $\endgroup$ Jan 27 at 19:47
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    $\begingroup$ Once the ball is at the top and stationary to get it back you can push it down the slope with an arbitrarily small force, lets say coming out of the page (so not going left or right where we have the ball come from), which requires an arbitrarily small amount of energy dissipation since once the ball starts rolling it will roll down all the way. This arbitrarily small amount of energy disippation leads to an arbitrarily small universe entropy increase which you can choose to make smaller than the decrease the demon gets from letting the high energy molecule through. The problem remains. $\endgroup$
    – Hadi Khan
    Jan 27 at 23:43

2 Answers 2

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which gives us a repeatable process that leads to a reduction in the total entropy of the universe, which shouldn't be possible as it violates the second law of thermodynamics. What am I missing here?

Second law of thermodynamics, like all physics laws, is generalization of experience. Its acceptance is contingent on agreement with observations. As with all physics laws, 2nd law may be experimentally discovered to be violated tomorrow. The violation need not cause societal shift, it may be subtle, hard-to-achieve, numerically small, with little practical consequence. Thus 2nd law may not actually hold in the real world - certainly, all physics laws are expected to fail somewhere, some of the time. Physics laws are understood to be approximate, with loopholes.

Second law is also not a theorem of classical mechanics, at least not without additional assumptions about experimental consistency of evolution of certain macroscopic states.

Classical mechanics alone, in the usual deterministic and mathematically reversible systems, is compatible with both 2nd law being satisfied, and 2nd law being violated. For example, a model of expanding gas, a set of particles or solid balls escaping confined space through a hole, to which all velocities are reversed at single time, obeys 2nd before the reversal, and violates 2nd law after the reversal. It is not just the act of reversal violating the 2nd law; it is the whole model with the initial condition, because after the reversal, it autonomously decreases thermodynamic entropy. So mechanics without further assumptions does not imply or disprove 2nd law.

Thus when you find a mechanical system which contradicts 2nd law, this is not notable.

The notable thing about Norton's dome is this is mathematically non-deterministic system with one particle only, under action of a constraint force (the dome).

Other non-deterministic, irreversible systems are known in mechanics, e.g. a system of two or more point particles which all move towards single point of collision in time and space (two or more dimensional). One particle moving freely is deterministic, but collision of two points is underdetermined in 2D or 3D space. Or if you prefer solid bodies, collisions where three or more perfectly solid balls touch at single time is non-deterministic (collision of two balls is deterministic).

Using the Norton dome or two point particle collisions to realize violation of 2nd law in a lab is not likely to succeed, though. Their "no-deterministic moment" is too sensitive to their ideal nature, or special tuning of their properties. One can't really find and experiment with Norton's dome, and one can't send two point particles to a single point - these things are idealized concepts, not real objects.

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  • $\begingroup$ I agree with your post, but I have a question about the system and the definition of determinism here. If the dome is finite, then the ball has to hit the bottom and bounce back up. Now it encounters the tip of the dome, again. Will it behave the same way and fall in the opposite direction of the first fall or can it change direction randomly? How do we define determinism/non-determinism for these systems? I don't know if the same argument applies to n-particle collisions because of the larger phase space. Are there some mathematical details in e.g. Liouville's theorem for bifurcations? $\endgroup$ Jan 28 at 3:21
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    $\begingroup$ @FlatterMann The dome top is the place where determinism is violated, the particle may stay there indefinitely, or start moving at any time, in any direction (the dome is three-dimensional). When the particle moves up and gets to the top, another violation of determinism happens when reaching the top. The model is such that past state implies future state only until the special point is reached; then, there is no unique prediction beyond that, motion in any direction obeys the initial condition and equations of motion. $\endgroup$ Jan 28 at 23:24
  • $\begingroup$ After thinking about this for a little bit this looks like an ad-hoc extension of Newtonian physics to me. We seem to be adding a rule like "If, for any reason, the equations of motion don't predict a unique evolution, use a pair of dice.". Personally I would prefer a different resolution like: "If, for any reason, the equations of motion don't predict a unique evolution, use a better theory of motion in the first place.". I am beginning to wonder what the Schroedinger equation has to say about this potential. $\endgroup$ Jan 29 at 3:28
  • $\begingroup$ @FlatterMann I didn't suggest using dice when the top point is reached would be improvement of the model. The model obeys mathematical formulation of Newton's laws, yet it is not well-posed. Thus mathematical formulation of Newton's laws need not imply well-posed problem (causal model) in general. They're constraints on the possible motions, not laws determining the future (except when only single motion obeys the constraints, in which case they do determine the future). $\endgroup$ Jan 29 at 12:37
  • $\begingroup$ I know that you didn't, but it seems to me the OP did, or maybe I just misunderstood what he was trying to say. In any case, my suggestion stands... since we already know that Newton is wrong, and more importantly, HOW it is wrong at the scale of the tip, we might actually try to analyze the system "the right way". $\endgroup$ Jan 29 at 14:50
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This is perhaps more a comment than an answer, but since it might be an answer and there are already a lot comments I will place it here.

I think the answer might be that Landauer's principle is the kind of 'law' for which a careful statement has to take the form "with the exception of certain very special cases ...". I note that some of the 'laws' concerning black holes are like this. For example, Penrose's proof that if there is a trapped surface then there must be a closed timelike curve or a singularity holds under very general conditions but not quite all conditions. But the special cases are a set with measure zero (or something like that) so although the mathematicians should worry about it, the physicists should not. That is, the idealisation required to say that a given physical system realises the special case is not an idealisation that ever holds in practice.

On this kind of answer, then, the Norton's dome presents a very interesting case, but not one that ever occurs in the physical universe. I mean, all physical treatments involve an idealization, such as the idealization of the isolated system (where we ignore van der Waals forces and gravitational effects from distant bodies, for example). For many cases such an idealization gives a good first approximation to the behaviour. For the Norton's dome with a body coming exactly to rest at the top, the idealization does not give a good first approximation. The perturbing effects of the rest of the universe cannot be ignored.

Similar comments can be made about classical chaotic systems.

It remains perfectly valid to say that Newtonian mechanics is not deterministic and it would be good for us to bring this to students' attention when we teach mechanics. A review of this and other questions is:

https://iopscience.iop.org/article/10.1088/1361-6633/aa8731

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