A PF Challenge to all you PF'ers

[bobloblaw]

Hi Physics Forumers! I was listenining to the Feynman lectures and something Feynman said got me thinking. He was talking about the indeterminacy that exists in classical physics due to our uncertainty in the initial conditions:

Speaking more precisely, given an arbitrary accuracy, no matter how precise, one can find a time long enough that we cannot make predictions valid for that long a time. Now the point is that this length of time is not very large. The time goes, in fact, logarithmically with the error, and it turns out that in only a very, very tiny time we lose all our information.

I was wondering how he derived this. So I thought I would turn this into a challenge to the people on physics forums: Derive (or refute) what Feynman says in this quote! Make any assumptions you need to and use whatever level of physics you know. I'll post what I came up with after some people (hopefully) post their answer.

 

[xts]

Take a pool billard. You have well defined initial conditions. Even clumsy players are able to shoot cue ball with precision of 1cm. Watch 10000 openings. Then you must see at least two openings differing by less than 1 micrometer.

Have you ever spotted two identical patterns formed by balls after the opening?

 

[bobloblaw]

You've captured the idea perfectly. But what I want you to derive in particular is Feynman's statement that the uncertainty goes as the log of time. So in your case it would be to derive an uncertainty in the pool balls as a function of time. To do that you would probably want to assume the pool balls didn't slow down due to friction.

 

[xts]

Nope. I only assume, that our players don't learn, and in next turn they'll spoil the configuration by the same factor as they did at opening. So the error multiplies with each step.

You: uncertainty goes as the log of time,
R.P.F.: The time goes, in fact, logarithmically with the error

That's not the same, and it is Feynman, who's right ;)
time goes logarithmically with the error means the same, as mine error goes exponentially with time

At the opening the single cue ball inaccuracy (1 micrometer), caused 15 balls to be scattered with some error (let's say: 1cm). In the next step each of those balls collide with others. But its position is not 1 micrometer, but 1 centimeters from its 'ideal' position. In yet next step - it would be 100m apart (if our table were that big...). Etc.

 

[javiermsi]

Nope. I only assume, that our players don't learn, and in next turn they'll spoil the configuration by the same factor as they did at opening. So the error multiplies with each step.

That's not the same, and it is Feynman, who's right ;)
time goes logarithmically with the error means the same, as mine error goes exponentially with time

a = e^t => t = log a

if you have an error on "a" then it will go logarithmically with it

 

[Delta Kilo]

Roger Penrose gives this example in his "Road to Reality" (I can't find the exact quote cos the book is bloody thick so I'll paraphrase): Say you have a qubic meter (or maybe liter, don't remember) of air under normal conditions here on Earth and you know the initial state of all molecules precisely and you want to predict the evolution after 1 second. But if a mass of 1 kilo is displaced by 1 meter somewhere in the vicinity of Sirius, its gravitational influence is sufficient to make the trajectories absolitely unpredictable after 1 second.