arfa's musings about a coin spinning on a tabletop

I don't know if you've tried spinning a coin on its edge on a table surface, but it helps to apply a field-goal technique: you "place" the coin on its edge and hold it there with a finger, then flick the side of the coin and release it at the same time, roughly.

The coin will spin like a top and move around the table surface; how much and how far will depend on how hard a flick you gave it. Eventually, friction slows it down and it tips over, there's a bit of a rattle as it winds down and stops moving. That's all just ordinary old observation.

The question is, does the coin move around the table top because of friction and "drag", or is it because the earth is moving and the coin wants to stay where it is?

 

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Ok. I notice I didn't mention the coin will have an initial forward momentum. Usually you see it start to move in a curve, fairly quickly.

I think you should probably account for the moment of inertia of a spinning disc. Another observation is that the coin stays upright and there isn't any rolling along the edge, at least not until it tips over. Momentum has to be conserved--that table surface has to absorb it so the coin loses "energy" and stops rotating and precessing.

Maybe there's a bit of heat generated by friction, which is lost, maybe the surrounding air carries momentum away too. I think that's all reasonable to assume even if you can't see it. Maybe you could if you set up an experiment to look for it.

What I want, maybe, to do here is show that information is being "written" somewhere, so there's an operator, so there's a Hamiltonian (by one of the principles of information-sharing, or computation as we call it).

I guess I should call it coin-momentum information, say. It's pretty different than what you come across with electronics, because of the dependence on gravity.

 

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Another observation: the coin at all times has a probability of being heads up or tails up, even if I don't do anything to it.

When the coin is spinning, it takes the probability along with it, but it's a global property of coins, locally it's being randomised by the spin it has. A quantum observation here: a coin is too big to tunnel through anything material, like a newpaper or a plank of wood.

But so far, probability is just because of coin geometry and something the coin is doing.

 

So if it hasn't rung yet, the bell is saying that classical information depends on the transfer of momentum from one system to another.

If you consider the coin, spinning around an imaginary line through the centre and the "point" of contact with the table, because of precession the coin transfers momentum to the table and "writes" a path. If you record everything in a video so you can add a visual track of the coin's precessional motion, there you have it; now it's visible--but it doesn't need to be, the information is being written anyway.

The addition of a tracking mechanism that visualises it doesn't make it more real. Because of friction, particles in the table top are interacting with particles in the coin. Sound is generated, because there's an atmosphere in the experiment. If you remove this, the sound has to be carried away by the table.

But we can just approximate this with a constant frictional coefficient, acting on the velocity of the coin relative to the fixed table top. There is transfer to the coin's environment, so a dissipation of information.

But we know how hard it is to record information in materials that aren't solid, like a gas or a liquid, it's just hard to transfer a coherent pattern and keep it coherent, or maybe it's impossible. Solid 'crystalline' matter is just way more useful.

 

Is the question related to the stationary behavior of a gyroscope?
https://interestingengineering.com/what-gyroscopes-are-how-they-work-and-their-importance

 

Gyroscopic motion is apparent in a spinning coin. It's why the coin stays upright when it spins on an edge.

It's also why a coin skips on the surface of water, when it's spinning a different way than on edge.

 

If you want to see a coin transferring momentum to a liquid, make it skip across a pond, or drop it into a column of oil.

If the depth is 'sufficient', the coin will reach a terminal velocity, an equilibrium point at which the phase difference between a pair of vectors is zero--the acceleration and velocity are parallel.

The reason a spinning coin moves around is, as James R notes, because of all the forces acting; gravity is a background field that is acting on the coin and everything else. Because of this and what's known as a "restoring force" the coin "feels" a torque. Torque has the same physical units as Joules (so what?).

Well, now you can think about the minimal amount of energy (as torque), a spinning coin uses to "write" a path, and how much energy is needed to "erase" it (or make it dissipate, or be 'absorbed').

 

--http://verga.cpt.univ-mrs.fr/pdfs/Plenio-2001cz.pdf

The article investigates Landauer's principle in the context of both classical information (in bits), and quantum information.
Landauer's principle states roughly, that storing or copying information requires a minimal amount of energy, but erasure always generates an amount of heat per bit. In short, it costs more to erase classical information than to store it in some kind of memory.

When a spinning coin traces out a path, information is also being erased. So I just thought it would be a reasonable example of this concept of "writing and erasing" information (what information?). The takehome is that information is always physical; but that seems to bump into a problem with quantum information which appears to be able to be transmitted 'non-physically'.

The author does not contend that his paper sheds any light on entanglement or why certain modern quantum protocols seem to work.
We know how to make them work, but that doesn't mean we understand the why.

 

The principle of erasure says something that seems to be a little strange.

Firstly, if information is indeed physical, then every physical object has an information content.
Ah, but this content is very much what we say it is. That is, if you ask yourself how much information is in a coin (how much does it carry around with it, say), it seems you are free to choose from quite a large set. If you could count all the atoms and write down all their positions, that would be some of the information available.

Or I could choose the information to be which face of the coin is visible, and ignore all the rest.

It seems that the concept of work is involved, and this work, since it's a human-defined 'concept' is closely related to what we can know. Work and information are at some remove, the same thing. (??) They are because it costs work to erase information--this isn't what actually happens; what actually happens is the information is transformed into heat and "lost".

Another view is: since you know how to make a coin spin on a table surface, the coin is doing work because of what you know about coins . . . ? Is it useful work? That depends on what you do with the information your brain is recording . . .

How much work is needed to erase the information from your brain?

 

I would like to be able to formulate a Hamiltonian for a spinning coin.

By having a coin spin on a table top, there is a frictional 'energy' acting on the coin to slow it and eventually tip it over. At this point the coin has "chosen" which face will be showing.

The Hamiltonian should include this probability. The friction is being generated by an interaction between a fixed surface, and the edge of a coin with angular momentum (whatever it actually is). So the idea is to see the friction as "measuring" the probability of heads or tails being observed. Eventually the probability, erm . . . "collapses" to one or the other, since it can't physically be both.

All of which is, of course, an heuristic approach. Friction as a measurement.
Similarly, a coin in free-fall in a column of oil will reach a terminal velocity, because friction has measured this (and still is measuring).

 

The Hamiltonian is an operator corresponding to the sum or potential and kinetic energy: https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)
So I imagine the Hamiltonian for a spinning coin would give you the sum of the kinetic energy of rotation plus the gravitational potential energy it has by virtue of being upright on its edge.

But it doesn't have anything to do with probability.

 

Oh, I see.

So if I put on a blindfold, take a coin out of my pocket (i.e. "act" on the coin), and place it on a table, how do I find out which side is facing up? Since whatever I do to the coin has nothing to do with the probability of it showing heads or tails?

How does that work? Say I remove the blindfold, what difference does that make to the physical fact that the coin will be showing one, and only one face?

Or say I pick up the coin, still blindfolded, and turn it over. What does that do to the probability? I know this one: it doesn't do anything, a coin always has the possibility it will be heads up or tails up, if you place it (the right way) on a table top.

If you spin it on edge, nothing changes about the probability. But notice, I'm talking about measuring an outcome which "has" a probability. I should not confuse a probability with a measurement, clearly they're quite different.

p.s. I know what a Hamiltonian is already, but thanks for the link.

 

Ok, I'm going to underline this connection between probability and measurement.

If you have a coin, it can be heads up or tails up; that's because of the geometry of coins.
If instead I have a coin-sized metal disk with identical sides (macroscopically speaking), it has the same probability as a coin of lying on one of its faces, but doesn't have a way to measure/determine an outcome.

Or put another way, a coin has distinguished faces, a blank metal disk doesn't.

I though it was fairly obvious, but hey.

 

Damn this bloody half-rsed kybord!

Ahem.

So what I've read about a friction Hamiltonian is that there are too many degrees of freedom, between the interacting particles, for this to be an easy thing to formulate. But maybe start with forces; a force is something that can be written as the first derivative of momentum.

The coin is oscillating (rotating) about a principal axis, it has angular momentum, it has a mass coupled to g the acceleration vector of gravity.

We need, though, to account for all the forces acting on the coin. Maybe start with something conventional like an equation of motion for a rigid body in SHM, which is damped by friction. Yeah, why the hell not?
The coin is forced to oscillate; moreover you force it to oscillate with left-handed, or right-handed spin.

Right, so write down this initial force as \(F = F_0 \;cos \;\omega t\).
The damping term in whatever equation of motion is used, is acting on the frequency \( \omega\), not the amplitude which is constant.

 

. . . except the "t" in the initial forcing should be set to t = 0; after this "t" parameterizes the "free" oscillations.

And the equation of motion needs \( \omega\) in it because I want \( \frac {d} {dt} \omega\).
I know the torque is given by: \( \tau = \frac {d} {dt} L\), where L is angular momentum.
And since \( L = I\omega\), we have \( \frac {d} {dt} L = \frac {d} {dt} I\omega = I \frac {d} {dt} \omega\); I, the moment of inertia is constant since the coin's spin axis is constant.

Now I just need to find the time to do the rest; I need to formulate how the frictional 'viscosity' component acts on the angular velocity \(\omega\).

 

What's that? Is someone else here . . . ?

 

The angular momentum is precessing; the coin isn't rolling along its edge, is it?
With no rolling, the coin must rotate around the same (constant) central axis, is it not so?

Not, you know, that I'm professing any great knowledge of physics, or that I can observe all that well (I am a bit blind, in fact), so, y'know.