Adrian’s Game of Bagatelle (named after its creator - AJ)

[BanjoPaterson]

Bagatelle Board

One has a square board with nails that one rolls a steel ball down from top to bottom. At the bottom there is a box as long as the board that has a number of receptacles for the ball to fall into.

The board is held upright and slanted at 45 degrees, so that it rolls down the surface of the board, hitting the nails as it rolls, and ends up in a receptacle at the bottom. It is assumed the board is enclosed by glass so that the ball cannot bounce from it and will land at the bottom. The each nail must have from its neighbour must wider than the steel ball (even if it is only by a fraction). The nails must also end greater than the diameter of the ball from the top of the receptacle. Given the restrictions on the position of the nails, the maximum number of nails must be present on the board (although the configurations can be different). The board must be at least an order of magnitude bigger than the ball and the size of each receptacle must be slightly bigger than the size of the ball. There must be more than one receptacle. Finally, from any starting position the ball should hit at least one nail (on any part of the ball) if it were to roll straight down.

If we roll the ball from a set position repeatedly we might expect there to be probabilities associated with each receptacle, with some highly likely for the ball to fall into; and some less so.

Question

If we could perform the experiment of rolling the ball from any starting position down the board twice by going backwards in time (i.e. the initial conditions are exactly the same), would the ball always land in the same receptacle? If this might not be the case, what would imply? (e.g. lack of causality? That the act of going back in time had altered the experiment? That when two atoms collide there is an area of uncertainty where they will end up – even if the conditions can be exactly repeated?) Note: the question is framed in terms of time travel (although impossible) so as to reproduce the exact starting conditions in order to repeat the same experiment twice.

The question can be expanded as: if the universe were to be rewound to its earliest possible point with the exact starting conditions it had – would it become exactly as it is now?

Adjunct: If we perform this experiment repeatedly (without the time travel) and we find the ball always going into one hole does this mean there is no probability of it going anywhere else?

 

[malty]

An interesting thought experiment, I must say. I certainly hope that it proves unpredictable because otherwise it means that the movement of every atom has been predetermined, and I've no control over my life which I'd like to believe otherwise!

However I don't think, that we can fully appreciate just how unpredictable the world really is, for example your thought experiment assumes that its the ball's outcome we have to predict, yet in reality it's every part of the table and its environment we have to predict, and we have to release the ball at the exact same time interval with the exact same properties for each particle present at that moment. Ok let's assume this was achieved, then you'd have to make sure the ball is launched with the same quantum spin quantum particle each time. Let's assume this is also achieved. Then we must consider which "picture" of quantum mechanics that we want to use to model the outcomes of the ball rolling.

If say the shrodinger one was used then we can assume that the ball will fall into the hole, or that it may not. If let we the ball go repeatedly though how many times is enough before we can say for certain that the ball will ball the same hole each time, 10, 20 100 1000 1 million, 2 million, remember we're dealing with the infinite and unpredictable here.

Now what would happen if we were to use the "many worlds" picture we would have the ball continously dropped in the same hole, but it also dropped into a different hole each time, where does that leave us? Well the only thing we can say for certain is that the ball will drop into a hole, which one depends on the "world" we are living in. So now let's assume we can re-enter the same world each time and carry out this experiment, then for sure the ball should drop in the same hole, otherwise this "picture" is wrong. Does that mean however that this "world" is completely predictable? Hardly because the theory assumes that whenever the two spins are present then both possibilities are realized and actually happens. What happens next therefore depends on the spin at that particular instant, if we could somehow isolate the spins so that we could choose whether the quantum spin is clockwise or anticlockwise, then we should be able to control what happens in our world, shouldn't we? However if we were to do that, we would in turn effect the spin that happens in another "world", so that would yet again cause to our dismay another possibility to be realized in our "world" and where would that leave us. Lost is my answer.. Maybe, changing one spin wouldn't affect another but one would have to assume it would.

 

[BanjoPaterson]

Thanks for looking commenting Malty - I did some undergraduate physics, but certainly not enough (and not enough quantum mechanics) to know the answer. I guess the thought experiment can be restated as: If we could reproduce two atoms colliding with the exact same initial conditions; would the angle of incidence be the same each time.

We know that if we do the experiment without going backwards in time, i.e. the initial conditions are similar but not the same, that we have a measure of uncertainty; but is this uncertainty because the initial conditions are not the same or is it intrinsic to two atoms colliding and being observed (even if the initial conditions are the same?)

The "many universes" view would probably state that if we were to reproduce the game again, exactly, then we might get a different outcome. However, those among my colleagues (computer scientists - not physicists) were in general agreement that the outcome would be identical "because nothing had changed and you have causality"; but were less certain if you rewound the universe and replayed it, whether it would be the same (the natural consequence of their arguments).

For my part, the one of the more amusing outcomes I have found is I, who have a Christian faith, should think the outcome would be different (i.e. no determinism) whereas my non-God believing colleagues believe the outcome would be identical (i.e. determinism).

A final comment - if you believe the outcome can be different it would mean if you were to rewind your life and replay it, then you could get a different result.

I had hoped this post did not break any of the terms and conditions of this forum. I am a naive physicist (or perhaps "more ignorant in physics than real physicists" may be appropriate); but I found the question intriguing. I hope others did as well.

 

[Loren Booda]

Is your game similar to pachinko? See http://en.wikipedia.org/wiki/Pachinko

Classically speaking, under perfect conditions, your setup is time symmetric, albeit nonlinear. With quantum uncertainty, observation itself skews any temporal similarity although the wavefunction is generally time symmetric.

 

[BanjoPaterson]

Thank you Loren - Bagatelle is similar enough to Pachinko (as far as I could tell) - it's a game played in fairs in Britain.

Your response re the wave function is intriguing and I would have to do some research before I could intelligently comment on it. One thing I believe you are saying is the thought experiment assumes an observer that is outside the game's time reference so this observer could see both events and record if there are any differences. I believe you mean that this observer could effect the results so that they different (by his act of observation).

 

[kalle437]

I believe that if we rewound to the early years of universe and let it all happen again, with the same starting conditions, it would all turn out the same.

You could look into it in another perspective. If there were another planet with the exact same conditions as earth, which had the same age, would everything that happened on these planets be exactly the same? Would a person on that planet have my name, live in my house, and look like I do? Would that person live in the same town, have the same family? Would all the mountains be exactly the same, and every ocean as deep? Would there be a book called Robinson Crusoe, written by Daniel Defoe there too?

I believe that would be the case.

 

[Loren Booda]

Only the observer's action is unidirectional in time; its wavefunction (description of microscopic probability) collapse marks the difference between between our having interacted with a particle or not. Classical entropy, however, statistically allows for a time-symmetric universe, no matter how improbable it may seem.

If the universe is infinite, then all events within it would somewhere be repeating an infinite number of times, e. g., the solar system duplicated, my computer minus one electron, etc. Only macroscopic or microscopic infinite events would not necessarily be replicated.

 

[BanjoPaterson]

I believe that if we rewound to the early years of universe and let it all happen again, with the same starting conditions, it would all turn out the same.

An interesting perspective - deterministic in its approach (i.e. from the beginning of the universe everything that happens henceforth is determined). From the people I have spoken to, many would agree with you. I would put that in the same vein as classical physics where La Grange once thought that if we could know the position and momentum of every particle then all history would be known to us (in both directions in time).

Strangely my opinion (and I do not have enough of a theoretical or mathematical background to argue in any stronger mode) is that there is enough degree of uncertainty that if you rewound the universe and let it all happen again, with exactly the same starting conditions, that it could (possibly) turn out differently. And if you did this again and again then the differences would start to diverge significantly.

My 2 pence :-)

 

[BanjoPaterson]

Only the observer's action is unidirectional in time; its wavefunction (description of microscopic probability) collapse marks the difference between between our having interacted with a particle or not. Classical entropy, however, statistically allows for a time-symmetric universe, no matter how improbable it may seem.

If the universe is infinite, then all events within it would somewhere be repeating an infinite number of times, e. g., the solar system duplicated, my computer minus one electron, etc. Only macroscopic or microscopic infinite events would not necessarily be replicated.

Darn I wish I was smart enough to know what you are saying. I think I get the gist of some of it (I went back to Schrödinger's equation, but years of not doing any maths or physics got the better of me), but not all of it. I think you mean that a time-symmetric universe could be perfectly rewound back to starting conditions or end conditions regardless of which time t we took in-between these end points; and that there is a non-zero possibility that such a universe exists regardless have how "improbably it may seem."

I would say if the universe did repeat from the above conditions I (for one) would consider such an event improbable

 

[kalle437]

An interesting perspective - deterministic in its approach (i.e. from the beginning of the universe everything that happens henceforth is determined). From the people I have spoken to, many would agree with you. I would put that in the same vein as classical physics where La Grange once thought that if we could know the position and momentum of every particle then all history would be known to us (in both directions in time).

Strangely my opinion (and I do not have enough of a theoretical or mathematical background to argue in any stronger mode) is that there is enough degree of uncertainty that if you rewound the universe and let it all happen again, with exactly the same starting conditions, that it could (possibly) turn out differently. And if you did this again and again then the differences would start to diverge significantly.

My 2 pence :-)

Well I believe it would turn out the same every single time. It's really impossible to get exactly the same conditions, but if it would be possible, there are no reasons why it shouldn't have been the same time after time after time.

I don't know how I could explain it, but every action we do, everything that happens have a reason. There are no coincidences, just things that are (by now) too complicated to explain.