Chade on 1/10/2007 at 21:38
Exactly the same sort of thing. :)
In fact, it was pascal's triangle that first gave my an intuitive understanding of the central limit theorem.
heywood on 1/10/2007 at 23:00
Quote Posted by addink
1. What Fafhrd said
Also (
http://en.wikipedia.org/wiki/Photon#Photons_in_matter) this and the concept that light traveling through matter is not a fixed set of photons, but rather energy in the form of photons being absorbed and then emitted by the matter's particles over and over again.
Any photon travels at lightspeed all the time.
I'd like to clarify that absorption & re-radiation of photons is not the reason why light travels slower in matter. For example, glass is transparent over the visible spectrum because the vast majority of photons are not being absorbed and re-emitted by the atoms of the glass. But the speed of light in glass is still significantly lower than in a vacuum.
According to Maxwell's equations, the propagation velocity of an electromagnetic wave depends on the dielectric permittivity and magnetic permeability of the medium. In glass, which is a dielectric material, the permittivity is higher than in a vacuum, so the wave travels slower. What happens in the material is that the electric field of the light wave polarizes the atoms, the polarized atoms align to produce a phase delayed opposing electric field, and the net summed electric field traveling through the medium is delayed relative to free space propagation.
Absorption and re-emission of photons by atoms are narrow band effects. They can only occur only at specific frequencies where the photon's energy equals the change in energy between two orbital states of an electron. When this occurs, you will get absorption lines in the spectrum of light after it passes through the medium.
Quote Posted by RocketMan
Martek:
Thanks for the alternate viewpoint. I hadn't pictured it quite that way. You are essentially saying that the most orderly outcomes have the fewest permutations, the least orderly having the most permutations and therefore, entropy is just a consequence of there being more disorderly states than orderly ones? I had been trying to incorporate that into my view but you presented it more effectively.
Forget about defining an "orderly" outcome. The key point is that because each trial is independent, any particular sequence of coin flips is as likely as any other. This is important because it means that the set of possible outcomes follows a uniform probability distribution, and a uniform probability distribution by definition has the maximum entropy possible. If you repeat the experiment over and over, the distribution of outcomes will tend toward the uniform distribution, which is the maximum entropy probability distribution.
So no invisible hand is necessary to guide the process. The simple mathematics of independent random trials will yield a probability distribution consistent with the second law of thermodynamics.
Quote:
I'd like to offer a thought experiment with no particular bias:
Imagine you are an omnipresent being looking down on the universe in its entirety from the beginning of time to the end of time. To simplify the scenario lets stick with coin tosses as they only have 2 outcomes. If you could look at every coin toss in the history of the universe and count them all up...what might the distribution look like? Would it be skewed severely? Would it be fairly even? If by the end of the universe the results are highly skewed, what might you say to account for this behaviour. If even, what can be said for the symmetry you see? If the universe had been abruptly terminated at any random point in time prior to its true end, would the distribution look any different? Would you be catching the universe with its pants down before it could even the coin count or would it never skew enough to really matter anyway? Assume that somewhere there's ppl flipping coins for as long as the universe exists.
The probability of getting K heads or tails in a sequence of N coin tosses is given by the binomial probability distribution B(N,P) with P=0.5. The expected value of this distribution is N/2. The variance of this distribution is N/4 and the standard deviation is sqrt(N/4). If you divide the mean and standard deviation by N to obtain the distribution in terms of a percentage of trials, you can see that the standard deviation becomes narrower in terms of percentage as N increases. So, for example, the probability of getting 55 heads out of 100 is equal to the probability of getting 5050 heads out of 10000. Again, the mathematics of independent random trials yields the outcome you expect.
RocketMan on 1/10/2007 at 23:53
Quote Posted by heywood
I'd like to clarify that absorption & re-radiation of photons is not the reason why light travels slower in matter. For example, glass is transparent over the visible spectrum because the vast majority of photons are not being absorbed and re-emitted by the atoms of the glass. But the speed of light in glass is still significantly lower than in a vacuum.
According to Maxwell's equations, the propagation velocity of an electromagnetic wave depends on the dielectric permittivity and magnetic permeability of the medium. In glass, which is a dielectric material, the permittivity is higher than in a vacuum, so the wave travels slower. What happens in the material is that the electric field of the light wave polarizes the atoms, the polarized atoms align to produce a phase delayed opposing electric field, and the net summed electric field traveling through the medium is delayed relative to free space propagation.
I understand much better now. It seems to me in this case, that the light is "virtual" so long as it's inside the glass because its a product of interference or superposition or whatever you want to call it. It certainly doesn't behave like it did prior to entering the glass.
Quote Posted by heywood
Absorption and re-emission of photons by atoms are narrow band effects. They can only occur only at specific frequencies where the photon's energy equals the change in energy between two orbital states of an electron. When this occurs, you will get absorption lines in the spectrum of light after it passes through the medium.
Indeed, I know about this...its used in spectroscopic analysis of stars to figure out what they're made of, among other things....but what confuses me is why a black body (the sun or an oven element for example) emits all wavelengths up to a maximum frequency determined by its temperature. How is it that every single wavelength over a continuous range can be produced by matter that has discrete quantum states?
Quote Posted by heywood
Forget about defining an "orderly" outcome. The key point is that because each trial is independent, any particular sequence of coin flips is as likely as any other. This is important because it means that the set of possible outcomes follows a uniform probability distribution, and a uniform probability distribution by definition has the maximum entropy possible. If you repeat the experiment over and over, the distribution of outcomes will tend toward the uniform distribution, which is the maximum entropy probability distribution.
So no invisible hand is necessary to guide the process. The simple mathematics of independent random trials will yield a probability distribution consistent with the second law of thermodynamics.
I have trouble seeing this. I can't figure how every set of outcomes has an equal probability of occurring. If you flip 50 heads in a row and then flip 50 tails in a row compared so some other 101101100010101001110001....sort of distribution that also has 50 heads and 50 tails, how can the former be just as likely as the latter? The former resulted from 50 trials that have a 0.5^50 chance of occurring followed by another 50 trials with the same low chance. The final outcome is what we expect, yes but the way we got there is highly unusual. The whole notion of independent trials confuses me too because I don't like the idea of saying a trial is independent when there are metrics which can be used to describe a set of trials, as though they are dependent.
When you say that its unlikely to flip 10 heads in a row and that the chance of such an occurance is 0.5^10 or a tenth of 1% that's a statement about trials that have a very real link to each other. The fact that they are consecutive is critical to this calculation....YET each trial is supposed to be independent and have the same 50-50 probability. Its like the story changes depending on whether you're looking at one flip like the others don't exist or if you're looking at all 10 flips.
Say I find a coin on the table. I flip it and it turns up heads. I am totally unimpressed. The chances of this happening are 50%. Unknown to me, the same coin had just been flipped 5 minutes earlier by a different person 100 times and every single time it came up heads. To me, everything seems very normal. I flipped 1 coin representing an independent trial and it came up heads. To the guy before me, I just did something that had a 0.5^101 chance of ocurring.
Chade on 2/10/2007 at 01:25
RocketMan, most people have trouble accepting that a "random looking" sequence of numbers is just as likely as an "ordered looking" sequence of numbers.
Keep in mind that "random looking" and "ordered looking" are not real physical attributes of a sequence, they are just a consequence of the way our brain picks up patterns.
There are more "random looking" then "ordered looking" sequences, because our brains can't find a pattern in most of the possible sequences. So there is more chance that a "random looking" sequence will occur. So you intuitively think that each "random looking" sequence is more likely then an "ordered looking" sequence. But this is incorrect. Each sequence has an equal chance of occurring.
Flip three coins in a row. Write down how many times you get all heads, and how many times you just get 1 head. If you do this enough times, you should get 1 head more often then you get all heads. But then write down how many times you get the exact sequence: head, tail, tail. If you do this enough times, you will find that you get head, tail, tail, about as many times as you get all heads.
The trouble with this experiment is that it will take you a lot of trials to get reliable results, so it might be better to whack something up on a computer.
~~~~~~~~~~~~~~~~~~
Now, let's take your last example.
The probability of somebody tossing a head is 0.5.
The probability of somebody tossing 100 heads in a row is 0.5^100.
The probability of tossing 101 heads in a row is 0.5^100 * 0.5 = 0.5^101.
The probability that the guy just tossed 100 heads, if he has already done so, is 1! It was unlikely to happen, but it did. So that chance that it occurred is 1. There is no chance that it won't happen, because it already has. This is the key point that you need to understand here.
So the probability of tossing 101 heads in a row, given that the first hundred heads have already been rolled, is 1 * 0.5 = 0.5.
So it doesn't depend on whether you are aware of what occurred previously. :)
Keeper_Andrus on 2/10/2007 at 02:41
I didn't read all this, but if you're really interested in your first question, I recommend QED by Richard Feynman
RocketMan on 2/10/2007 at 02:51
Those last 2 sentences there....that's the shit that I can't deal with. I'm not saying you're wrong. I just can't deal with it. Its a retroactive piece of causal bullshit to me. In fact its starting to look more like quantum mechanics by the second....ie. knowledge of something affects the outcome. Ok knowledge isn't affecting anything here physically but it changes the way the math works. I figure just because we know the 100 heads actually occurred, doesn't change the fact that it was really really rare for that to happen and something should make it change back. I don't think we're going to resolve this from the available data because there's nothing in any book quantifying what I'm saying here. Its just that I don't think rare occurrences should perpetuate. They should not do so. 100 heads is bad. 1000 heads is worse. Its gotta stop some time. Just because we know about it shouldn't mean all is forgiven, lets go back to 50% odds....it means we know there's a huge imbalance that's just taken place, it should fix itself soon, so that the universe continues to work in a sensical way and coffee mugs don't go un-shattering all around us. Damn it!:mad:
Keeper: Read that..lol...a lonnnng time ago. So maybe its time for another read.
Chade on 2/10/2007 at 03:37
Well, these are all hypothetical examples. I highly doubt anyone has ever actually tossed 100 heads in a row in the entire history of humanity, without cheating somehow.
*Zaccheus* on 2/10/2007 at 12:26
Quote Posted by Chade
So the probability of tossing 101 heads in a row,
given that the first hundred heads have already been rolled, is 1 * 0.5 = 0.5.
That is correct.
:)
Vernon on 2/10/2007 at 15:32
Quote Posted by Chade
Well, these are all hypothetical examples. I highly doubt anyone has ever actually tossed 100 heads in a row in the entire history of humanity, without cheating somehow.
The "great" playwright Tom Stoppard had his bastardised versions of Shakespeare's characters tossing heads constantly in his incredibly retarded and interminably pretentious play/Beckett ripoff 'Rosencrantz & Guildenstern Are Dead.' I was forced to read that book for my HSC :(
heywood on 2/10/2007 at 23:07
Quote Posted by RocketMan
I understand much better now. It seems to me in this case, that the light is "virtual" so long as it's inside the glass because its a product of interference or superposition or whatever you want to call it. It certainly doesn't behave like it did prior to entering the glass.
Virtual is probably the wrong word. It's still the same light wave, it's just interacting with the glass as it passes through.
I think you may have been struggling to understand light propagation in a medium because you were thinking only in terms of the classical particle model. Because of the wave properties of light, you don't need to have photons actually hitting atoms in order for the light to interact with the medium.
Neither the wave model nor the particle model can fully explain all interactions between light and matter. For that, you need to learn quantum electrodynamics.
Quote:
Indeed, I know about this...its used in spectroscopic analysis of stars to figure out what they're made of, among other things....but what confuses me is why a black body (the sun or an oven element for example) emits all wavelengths up to a maximum frequency determined by its temperature. How is it that every single wavelength over a continuous range can be produced by matter that has discrete quantum states?
Because electron orbital transitions are not the only mechanism by which energy is absorbed or emitted in the form of photons.
The radiation spectrum of a black body is a consequence of applying the relevant electromagnetic boundary conditions of a black body and assuming it is in thermal equilibrium. To derive the spectrum, Planck started with an idealized reflectionless cavity bounding an electromagnetic field. The total field is a superposition of an infinite number of discrete modes which are dependent on the shape & dimensions of the cavity. Planck made a big assumption that the energy stored in each mode was a quantized multiple of a constant times the frequency. He then applied statistical mechanics to enforce the conditions of thermal equilibrium in the limit as the size of the cavity approached infinite. He derived a function for the spectral energy density of the modes inside the cavity, which also describes the radiation spectrum of the black body. Planck's work predated the concept of a photon, but an alternate derivation of Planck's law by Einstein treats the cavity as if it were filled with a massless quantum ideal gas consisting of a finite number of photons, in which the photon gas is in thermal equilibrium with the cavity walls. Because the cavity behaves as a black body, this interpretation implies there is a constant exchange of photons between the cavity walls and the photon gas.
Now here is where things get murky for me. Because Planck's law is based on the statistics of thermodynamics, none the discussions of black body radiation seem to deal with the specific mechanism of emission at the atomic level. So I'm still a little unsure of how a single photon is absorbed or emitted by an atom due to thermal energy transfer. We know that thermal energy is stored in the mechanical vibration of atoms, and that atomic nuclei and electrons are charged particles, and that the periodic motion of charges produces electromagnetic waves. So I suppose there is a process through which vibrating atoms can transform thermal energy into electromagnetic energy, but I don't know if it's that simple.
