three-body problem

Yeah, I've never thought hard about superregens, and it would be interesting to explore one rainy weekend. After it quenches off and the bias is creeping back towards the gain+oscillation region, the shape of the bias exponential will hugely affect the sensitivity. I don't know if the old 1-tube circuits controlled that well.

Of course, the quench bias mechanism could be a DAC. Random number generator?

Optically: we see mode jumps in fiber-coupled lasers, as a result of minor reflections in the external fiber plumbing, so they are sensitive to input. That causes picoseconds of jitter, and isolators fix that nicely.

One could drive a cheap laser diode with a sine wave, so that it's only lasing near the current peak, and then shine light into the laser and see if that kicks it off sooner. Play with the sine frequency. I think lasing is visible in the voltage across the laser but certainly in the usual monitor photodiode. That's an externally quenched superregen.

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John Larkin         Highland Technology, Inc 

lunatic fringe electronics

(Which is a good thing.) Still you can figure out what the average speed of a ball would be.

George h.

People are doing all sorts of fun stuff with coupled optical fibers and four wave mixing. It's not something I understand that well.

George H.

If energy is conserved in the system, the sum of the squares of the velocities is constant.

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John Larkin         Highland Technology, Inc 

lunatic fringe electronics

Sure that's easy, we should also be able to predict the distribution of the energy amongst the 16 balls... though at the moment I'm not seeing how that comes out.

George H.

If they bounce around long enough, one of the balls might wind up with almost all of the system energy. Zooooommmmm. I think.

Of course, the que ball started at break time with all the energy, so no ball can ever go faster than that.

--

John Larkin         Highland Technology, Inc 
picosecond timing   precision measurement  

jlarkin att highlandtechnology dott com 
http://www.highlandtechnology.com

The cue ball is slightly bigger than all the others.

To get the velocity distribution I think you have to define a pool ball temperature. And then use the Boltzmann factor to calculate the probability. Prob =~ exp(-energy/kT)

Maybe we can put the pool table on a big shake table, then the larger the shake amplitude the higher the temperature. Put the table on an angle and the height (that a ball rises to) will relate to the balls energy.

George H.

only on pay-to-play tables so the machine can tell it rom the other balls

Seems something like brownian motion in 2 dimensions. so Maxwell distribution?

On a lossless table (given that collisions between balls conserve momentum, and collisions with the border conserve speed) average speed is conserved.

If the table has pockets, given that faster balls are likely to find pockets sooner a downwards trend in average speed and average energy is likely.

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Sure, that's the energy integral that I was talking about.

Cheers

Phil Hobbs

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Dr Philip C D Hobbs 
Principal Consultant 
ElectroOptical Innovations LLC 
Optics, Electro-optics, Photonics, Analog Electronics 

160 North State Road #203 
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hobbs at electrooptical dot net 
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I think it is on all pool tables.. though I've never measured. I heard it was so when you crashed the cue into the stack, it would first have a lateral force down into the first ball. That could very well be an urban myth.

George H. (in college we had a pool table in the living room, it came with the apartment.)

Where's the noise coming from? I picked all the initial conditions down to the last bit, yeah? In fix 16 how about 10101010.10101010. That's a perfectly reasonable starting position in this universe - all other binary digits to the left and right out to infinity are implied to be zero. This is a unique set of initial conditions, there are no others like it.

The only place that noise can come from is via truncation/roundoff error in the calculations, but I don't see it written anywhere that noise power density can or must increase exponentially without bound as time advances. I don't think it can, it finally requires time and energy to flip bits on your simulating computer.

There's no Heisenberg uncertainty in a computer simulation of Newtonian dynamics.

I'm watching an idealized three body simulation like that evolve right now. It looks kinda chaotic! Does it represent the real world? Idk find me some point masses and an empty universe and let's find out. Hey look I just code the stuff it's not my problem if the model I'm given either doesn't represent or is irrelevant to the real world.

Splitting hairs - there aren't any lossless pool tables. The uncertainty principle plays no role at all in how the planets and billiard balls we're familiar with behave (at least at this moment in time, I'm sure it was a pretty big deal around the time of the Big Bang but I'm not a cosmologist/string theorist/etc)

Just because there isn't a closed-form "solution" to some set of equations doesn't mean you can't learn plenty from manipulating the symbols. That's pretty much what say quantum field theorists do with most of their thinking time - with a pen and paper. They don't spend it in front of a supercomputer fudging numbers.

You were the one talking about how silly mathematicians were, in wanting closed-form exact solutions. Now you're doing it.

You're being thicker than usual this morning. It's the physics of multi-body orbits that does the exponential noise multiplying, for the reasons I already gave. If you thought about it for five minutes, you'd see that that's the case. A slightly too-close periapsis changes the angle at which the bodies separate, and the angular error grows steadily, causing the next periapsis to be further off, and so on. Iterated, you get exponential growth, which is why chaotic systems are unpredictable even in principle.

You can also solve the whole problem by multiplying by zero and adding in what you want it to look like.

Every floating-point multiplication introduces LSB/sqrt(12) RMS noise. Widrow's theorem. YCLIU.

That's not the case. There are other sources of error in our estimates, but epistomologically, new information is appearing in the universe all the time, on a gigantic scale.

(at least at this moment in time, I'm sure it

Or a physicist or mathematician of any stripe, apparently.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs 
Principal Consultant 
ElectroOptical Innovations LLC 
Optics, Electro-optics, Photonics, Analog Electronics 

160 North State Road #203 
Briarcliff Manor NY 10510 

hobbs at electrooptical dot net 
http://electrooptical.net

"Coin-operated pool tables such as those found at bars historically have often used either a larger ("grapefruit") or denser ("rock", typically ceramic) cue ball, such that its extra weight makes it easy for the cue ball return mechanism to separate it from object balls (which are captured until the game ends and the ..."

What you are claiming is that if you run a bad computer program twice, you will get the same bad answer. OK, I'll concede that.

--

John Larkin         Highland Technology, Inc 

lunatic fringe electronics

It isn't a closed-form solution (at least in the definition I'm familiar with); a closed-form solution would mean I could plug in the initial conditions and hypothetically have the positions and velocities for t + X for any X in one iteration.

Also I don't think I was the one who said that mathematicians were "silly", I just said that in hindsight it wasn't surprising that the search for one was confounding because it is that way, more often than not.

Error compared to _what_, though? To have an error you need to have some kind of reference to compare it against. What am I to compare the results of my simulation against? I can't compare it against what the closed-form solution says the answer should be at t + X, there is none. I can't compare it against the real world, I don't have any empty universes nor point particles.

If I compare it against a second run of my own simulation running on the same (deterministic) finite-state machine running the same code I should very much hope it plays out the same way each time, or there's something wrong with my computer. If I compare it against some other computer running a different set of code maybe it's the same, maybe it isn't, but that's to be expected as that one isn't "my universe" anyway.

Yeah, that's pretty much the "joke" I was making originally, jeez!

So the universe as a whole doesn't obey conservation of energy? Locally, sure, but globally you supposedly can't create information out of nothing, any more than you can create mass or energy out of nothing. IIRC it's actually an open question whether the universe obeys global conservation of energy, rather than just locally. i.e. there are serious problems re: singularities at infinity for topologies that are both unbounded and infinite.

I think you missed my point in all the discussion about "error". Newtonian dynamics is deterministic. It's perfectly OK for a system to be deterministic and chaotic at the same time, because the definition of a chaotic system doesn't say anything about statistical notions of randomness. There aren't any random variables or probably distributions or anything of the kind in the ODEs for the three body problem.

Can you use a random variable with a uniform distribution between 1 and

6 in a statistics problem and call it "a dice roll"? Sure. Does that mean a physical dice roll is actually a mathematical random variable? No, definitely not, once you launch the die out of your hand at the point in spacetime you do the outcome is fixed. That you can sometimes use statistics to make money gambling is nice, but the math doesn't care a bit whether dice or cards or casinos actually exist, or not.

Conversely, the three body problem is also deterministic but has no closed-form solution. But the existence and uniqueness theorems for ODEs say that for any set of initial conditions there exists a unique solution at some point t in the future. And it sounds like you believe that even though the closed-form solution is not known to exist, there exists some "Platonic ideal" of a closed-form solution known only to the mind of God that computer simulations can only approximate, and that all these errors will accumulate that will throw everything "off."

This ideal solution doesn't exist. With no point of reference, as you say just about any answer is as good as any other. Solutions "corrupted" by a universe with "noisy spacetime" i.e. full of floating-point multiplication noise are perfectly acceptable, and who says you have to deal with that anyway? You can write a fixed-point version of the equations, too. Is that version "wrong"?

THAT'S THE JOKE

Also that it's meaningless to talk about "bad" solutions to non-physical problems. Bad compared to _what_?!

It's another way of saying: "Entropy is always rising."

It sounds less surprising when it's put in a familiar form like this. So, uh, what is entropy?

S = k_B ln W

When entropy is rising, that means there are more and more configurations available to the system. Any given configuration is an element of information, so the available information also rises.

Another thing to remember: a maximal entropy state is also one of maximal information, not minimum information. The key to /useful/ information is just that: some order (lack of entropy), to make sense of it. Pure noise is, at once, everything, and nothing!

Indeed, all that "waste" energy, that flows from more-orderly things to entropic sinks, is exactly the energy and therefore information giving rise to the ever-increasing number of available configurations! (I think? I haven't heard it put in this term before, but it sounds like the logical conclusion.)

Tim

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Seven Transistor Labs, LLC 
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