[Experts only] What are probabilties of heads / tails / edge of the coin?

Question

Recently discovered intelligent ants mint and circulate perfect
silver coins d = 1.483 μm in diameter and h = d = 1.483 μm thick.
Such coins placed on flat level surface flip from time to time
due to Brownian motion.

What portions of time Th /Tt /Te does such coin spend in
heads / tails / edge states?


k = 1.38065 × 10e-23 J/K
ρ(Ag) = 1.049 × 10e4 kg/m³
T = 300 K
g = 9.80 m/s²

frictionless coin?

Friction or not in the end it's made of electrons and nucleii which are described by certain hamiltonian.

This one requires some elementary calculus, though.

'The coins would abosorb more energy through their larger side areas than through the narrow surface contact areas of their edges'

This note is a very good one. My fault.
Clarification:
Consider the surface to repulse silver
atoms sharply. Say, impressing a single
atom of silver by 0.5 nm into the sustrate will increase the energy by 1eV >> 300K.

This problem does not come from the book, so in the end I may be wrong.
Your input is most welcome.

More about friction:
there is some macroscopic friction,
needed to destroy conservation of
angular momentum Mz, but no friction
as microscopic constraint.

Here is my point of view on friction:

Friction is always macroscopic, and can
never happen in real world as microscopic
constraint. In this particular case I
assume that the substrate provides a sharp inpenetrable potential wall.
This potential, is however non-static;
it 'vibrates' becuse of thermal motion
of the substrate, destroying conservation
of energy of the coin.

In real world this potential is never smooth
(i.e. dE/dx != 0), which means macroscopic friction. Microscopically, however you do not have new non-integrable constraint, it only ergodizes the
motion of the coin (in particular it kills
conservation of Mz). All of this would mean
that coin is weakly interacting, fully ergodic
hamiltonian sybsystem, and Boltzman
distribution does apply.

Finally: de Broglie wavelength of the coin
is about 10-19m, much smaller than
both kT/mg ~ 10-7m and the coin itself, making the system classic.

'the temperature would actually matter'

The temperature is present in the answer.

Answer 1

I think you should be able to get the probabilities from the Boltzmann distribution.

They are proportional to exp(-E/kT)

To get the energies, just use the gravitational potential, mgh, where h is the height of the center of mass of the coin.

For heads and tails, that's half the thickness
For edge, that's half the diameter

Plug and chug.

Edit: the answer below is correct in that realistically there is a lot more you'd have to consider. If this is a question in your intro to stat mech course, however, I think that they're just looking for the simple analysis I provided--3 states of different energy levels and no motion beyond flipping between the three levels.

This isn't "realistic" in terms of ant-size coins, but the method is exactly what you do in truly microscopic systems which may have discrete quantum levels just like this.

Edit--@eyes on the screen. The boltzmann distribution applies to any system in thermal equilibrium with its surroundings. In this case, the coin is in thermal equilibrium with the gas surrounding it. Gasses (which are discussed in the wiki article) are just one example of a system which obeys boltzmann statistics. Check a stat mech book (I used Pathria).

Also, you would have to consider rotational energy. Fortunately, we find out that whoever designed the problem is including just enough friction to supress rotation. I'm still leaning heavily towards my original answer.

This would be a much more interesting problem if the three modes were NOT at equal energy levels. Then the temperature would actually matter.

I've done enough textbook stat mech problems to know a Boltzmann distribution problem when I see it. Discrete energy levels, equilibrium with surroundings, supression of friction of any kinetic energy to confuse the issue (as they would in reality)

Answer 2

What complicates this problem is that there's actually many more degrees of freedom than just the position of the center of gravity of the coin. You've got the different axes of rotation of the coin to consider. This is a fascinating problem, but way too complicated for me to thoroughly investigate at this time. For example, is this a frictionless coin? It wasn't specified. If it's frictionless, maybe Boltzman statistics can be used, but if there's friction, forgeddabboudit.

Addendum: Friction does matter, because if were were thinking about some tiny particle that happens to have a shape of a coin, but where friction is meaningless (do electrons have friction?), then it might be possible to analyze this along the lines of Bekki's suggestion. Imagine that we have a large ensemble of such coins bouncing around and having any one of the states 1, 2, 3, which are further split on account of rotations which may or may not be quantized. Then we have something not very different from what solid state physicists work with all the time. But does this problem involve a real coin, with other considerations far beyond nanoscale physics? A very important assumption in Boltzman statistics is that the different degrees of freedom are at most only statistically related, but I don't see that with a real coin that happens to be spinning when it lands on its edge, and there's friction involved. I cannot expect this "friction factor" to simply drop out of equations leading to probability distributions of the outcome.

Answer 3

Given that the center of mass is the same no matter what it's orientation on the surface and the coin appears to be a fair coin, I believe each orientation is equally likely. Therefore, p(H) = p(T) = p(E) = 1/3 = n(s)/N; where n(s) is the number of ways to succeed (heads, tails, edges) and N is the total number of possible outcomes (three possibilities) per trial.

A trial is simply a jump (change in velocity, orientation, energy) due to supposed energy source that causes Brownian motion. Because you specified Brownian motion (which is stochastic), the sum of velocities for all fair coins over time would be zero. This implies that all possible outcomes of velocity and momentum are equally likely and cancel each other out. This is consistent with the equally likely orientations.

(OH...just thought of something...do mean the forcing function for the motion stems from the surface energy? In which case, the coins would abosorb more energy through their larger side areas than through the narrow surface contact areas of their edges. In which case, the coins on their sides would be more likely to jump than the ones on their edges. Consequently, the coins on their edges would spend more time there than their counterparts lying on their sides. Therefore Te > Th or Tt. Sorry, but I think you need more specification in your assumptions.)

By the way, I don't believe the Boltzmann equation applies here. Check this out:

"It corresponds to the most probable speed distribution in a collisionally-dominated system consisting of a large number of non-interacting particles in which quantum effects are negligible. " [See source.]

But, again, are these coins close enough to collide? More info is still required to model this problem.

Answer 4

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