totse.com | The Chaos Theroy.

Pandoras Assassin · #1 2008-08-10, 05:21

WTF is that.

enkrypt0r · #2 2008-08-10, 07:29

The Chaos Theory is something that I really have a strong interest in, and really believe in. You may know it as the "Butterfly Effect." It's a bit difficult to explain, and it would probably help if it weren't 2:20am, but I'll give it a shot.

Basically, everything affects everything. That's the simplest way I can put it. In our universe, everything seems random, but it is in fact not. Everything is a cause and everything is an effect. Everything can be calculated mathematically. Not necessarily by us and our understanding, but everything is an equation. As humans, we know some equations, and these equations can be used to predict exactly what will happen. If we were to know every equation for everything in the universe (and enough computing power), we would literally be able to predict what will happen for the rest of time. With our current understanding, everything is random and chaotic, hence the "Chaos Theory."

Well, I hope that made some sense. The Wikipedia article on the Chaos Theory is extremely interesting and well-written. I'm sure you'll be able to find some good books that can explain it better than I can, but just thinking about this theory is mind-boggling. It's extremely compelling... I'll try to post more when I wake up tomorrow afternoon.

Mantikore · #3 2008-08-10, 14:29

i agree. but i am still stumped about how quantum theory and the randomness of the electron fits into this

l33t-haX0r · #4 2008-08-10, 17:34

^ That's not a very good explanation, not all of it is entirely true. I'll give it a go with a little help from wikipedia...

Chaos is the study of non linear complex systems that give rise to seemingly random behaviour dispite being produced by equations which are (obviously) perfectly deterministic. The equations are said to be sensitive to inital conditions. This means whatever numbers are put into the equations, numbers arbitrarily close to these numbers will produce different behaviour in the system.

A good example is Edward Lorenz who used 3 (i think) non linear differential equations to make a simple model of the atmosphere on an very old skool computer which printed the variables off on one of those reels of paper with time along the bottom and the variables graphed up the side. The computer also printed off the values of these variables rounded to so many (say 3) decimel places.

After a run was complete he wanted to carry on the graph where it had left off so he fed in some values from somewhere near the end of the old print out. The graph he got started off the same as the old then started to diverge wildy from what he had before. He realised it was because he hadn't fed in the exact numbers but the rounded ones from the printout. It may seem obvious that these should give different results but no one thought it should make such a huge difference. You expect small variation in input to give small variations in output. This was showing that vary similar numbers gave hugely different results after only a very small time.

It showed that long range weather forecasting was doomed. Even if you had the perfect eqns representing the atmosphere and a grid of sensors mounted every kilometer or so in the atmosphere the slight changes in temperature, wind speed etc.. between the sensors, and the fact that the sensors couldn't measure all variables 100% accuretly, would cause errors to build up in the eqns as they were used to predict into the future.

The guy above me said "If we were to know every equation for everything in the universe (and enough computing power), we would literally be able to predict what will happen for the rest of time". This is exactly what chaos theory is tell us is impossible. Even if we knew the equations you wouldn't be able to measure the variables NOW with enough precision to enter into the equations. Plus the uncertaintly principle in QM tells us that you can never meausure all variables to an arbitrary precision - this is kind of beside the point though as QM also tells us that we can never make concrete predictions, only calculate probabilities.

The flip side to all of this, and I think what the guy above me was getting at, is traditionally you'd see complex phenomena and you'd look for complex causes. However this is telling us that simple equations or causes can create massively complex phenomena. You look at the great red spot on jupiter and you assume that there most be a huge number of factors involved, atmospheric factors but maybe also the shape of the core, gravitational influences, maybe even impacts from previous bodies. Infact fluid dynamicists made a (relatively) simple model of the planet and found the spot can be explained using simple equations and ideas, even though the problem seems to involved a turbulent atmosphere amongst other things.

If you want to read into it futher I recommend 'Chaos' by James Gleick.

Cordova · #5 2008-08-11, 08:10

Quote:

Originally Posted by l33t-haX0r View Post

^ That's not a very good explanation, not all of it is entirely true. I'll give it a go with a little help from wikipedia...

Chaos is the study of non linear complex systems that give rise to seemingly random behaviour dispite being produced by equations which are (obviously) perfectly deterministic. The equations are said to be sensitive to inital conditions. This means whatever numbers are put into the equations, numbers arbitrarily close to these numbers will produce different behaviour in the system.

A good example is Edward Lorenz who used 3 (i think) non linear differential equations to make a simple model of the atmosphere on an very old skool computer which printed the variables off on one of those reels of paper with time along the bottom and the variables graphed up the side. The computer also printed off the values of these variables rounded to so many (say 3) decimel places.

After a run was complete he wanted to carry on the graph where it had left off so he fed in some values from somewhere near the end of the old print out. The graph he got started off the same as the old then started to diverge wildy from what he had before. He realised it was because he hadn't fed in the exact numbers but the rounded ones from the printout. It may seem obvious that these should give different results but no one thought it should make such a huge difference. You expect small variation in input to give small variations in output. This was showing that vary similar numbers gave hugely different results after only a very small time.

It showed that long range weather forecasting was doomed. Even if you had the perfect eqns representing the atmosphere and a grid of sensors mounted every kilometer or so in the atmosphere the slight changes in temperature, wind speed etc.. between the sensors, and the fact that the sensors couldn't measure all variables 100% accuretly, would cause errors to build up in the eqns as they were used to predict into the future.

The guy above me said "If we were to know every equation for everything in the universe (and enough computing power), we would literally be able to predict what will happen for the rest of time". This is exactly what chaos theory is tell us is impossible. Even if we knew the equations you wouldn't be able to measure the variables NOW with enough precision to enter into the equations. Plus the uncertaintly principle in QM tells us that you can never meausure all variables to an arbitrary precision - this is kind of beside the point though as QM also tells us that we can never make concrete predictions, only calculate probabilities.

The flip side to all of this, and I think what the guy above me was getting at, is traditionally you'd see complex phenomena and you'd look for complex causes. However this is telling us that simple equations or causes can create massively complex phenomena. You look at the great red spot on jupiter and you assume that there most be a huge number of factors involved, atmospheric factors but maybe also the shape of the core, gravitational influences, maybe even impacts from previous bodies. Infact fluid dynamicists made a (relatively) simple model of the planet and found the spot can be explained using simple equations and ideas, even though the problem seems to involved a turbulent atmosphere amongst other things.

If you want to read into it futher I recommend 'Chaos' by James Gleick.

This--good explanation. I was going to add something then realized that you already said it, albeit in more complicated language than I would've used.

HeaT · #6 2008-08-11, 16:49

That is good... That was really helpful, thanks man. But do you think that a new system of math or numbers will ever be invented for quantum mechanics? It doesn't really work into calculus or standard math... all the probabilities and variations. If so, we might be able to make better predictions.

Classical mathematics might've died with classical mechanics.

enkrypt0r · #7 2008-08-12, 02:13

You're right, I gave a very poor explanation... What I said is more or less a sort of branch of the theory, which I've kind of added my own thoughts into... The other explanation is better, I merely talked about what I thought it meant. I've really got to stop posting here after midnight, my posts get weird.

danreil · #8 2008-08-12, 05:15

Quote:

Originally Posted by HeaT View Post

That is good... That was really helpful, thanks man. But do you think that a new system of math or numbers will ever be invented for quantum mechanics? It doesn't really work into calculus or standard math... all the probabilities and variations. If so, we might be able to make better predictions.

Classical mathematics might've died with classical mechanics.

This is done somewhat in quantum mechanics already, mainly to make it easier to deal with. Not that it has replaced the traditional system, because the real number system and particularly the complex number system still play a large part in quantum mechanics. However, to deal with the non-commuting aspects of quantum measurements and transformations, there are certain operators and algebras such as Grassman and Clifford algebras and numbers, as well as the Dirac gamma matrices from which you construct a Clifford algebra, and these come up all the time in quantum mechanics.

These have actually been known about for a while, but some physicists, particularly Dirac, noticed they could be applied to quantum mechanics to deal with some aspects of quantum mechanics, particularly with spin and spinors, as well as with many aspects of measurement.

Keep in mind however, this is not some magical new number system, but is built off traditional topics in math such as vector spaces, exterior algebra, and many other things, and mainly facilitate some things in quantum mechanics.

SelfAdjoint · #9 2008-08-19, 23:34

Quote:

Originally Posted by HeaT View Post

That is good... That was really helpful, thanks man. But do you think that a new system of math or numbers will ever be invented for quantum mechanics? It doesn't really work into calculus or standard math... all the probabilities and variations. If so, we might be able to make better predictions.

Classical mathematics might've died with classical mechanics.

That really depends upon what one means by "classical mathematics". Much of the mathematical constructs on which Quantum Theory is based pre-date the theory itself. The mathematics that enabled Einstein to build the General Theory of Relativity might not be termed "classical mathematics" either.

It isn't the sense of "arbitrariness" that causes issues when dealing with Quantum Mechanics, that is how to deal with an element of indeterminacy in a science whose common metaphysic for many centuries contended that nature adhered to a strict set of deterministic laws. Indeed, probability theory has been around since Laplace and the Enlightenment. The problem, from a mathematical perspective, is in viewing this arbitrariness or any other aspect in a superficial way (as with, say, the Bohr Model) and thus destroy the subtly that the theory truly entails and the depth and elegance that now pervades its modern guise as a physical theory.

the achilles heel · #10 2008-08-22, 23:37

Isn't the chaos theory in that one short story.... I forgot what it was called but these guys went back in time to hunt dinosaurs, and one of them stepped on a butterfly. Then when they got back to the present the future had changed a lot. Or something like that...