|WikiProject Systems||(Rated B-class, Mid-importance)|
- 1 Hitchhiker's Guide to the Galaxy
- 2 The caption
- 3 Merge with Wiener process?
- 4 Central Limit theorem
- 5 Smoluchowski
- 6 I can do some work
- 7 Too technical
- 8 Historical origins
- 9 Excellent Communication of Subject
- 10 'Intuitive Metaphor' section: precision?
- 11 First Image
- 12 Another image (proposed)
- 13 Video Section
- 14 removed text
- 15 Einstein's article
- 16 H2G2 Reference
- 17 Mathematical/Physical
- 18 The Boss
- 19 Clarifying the Mathematical / Physical distinction
- 20 This is too funny
- 21 Plagiarism
- 22 Brownian Motion Observed
- 23 Incorrect Deduction
- 24 Caption needed
- 25 Jim Sukwutput's edit
- 26 "Brownian motion has a few real-world applications"
- 27 The time derivative is not everywhere infinite
- 28 Mandebrot
Hitchhiker's Guide to the Galaxy
- Sir! Are you denying that there is no Brownian motion in a very hot cop of tea? Student7 (talk) 01:46, 28 February 2010 (UTC)
- There are all kinds of "In Popular Culture" references throughout Wikipedia far more obscure than this essential element of the Infinite Improbability Drive. I think a reference would be appropriate, especially considering the number of people, myself included, who might look up this article to learn if there is, in fact, a scientific phenomenon called brownian motion and that it is not merely named for the colour of coffee. Besides, if it's that obscure, why is it the first topic in the talk for the article?Titaniumlegs (talk) 14:31, 5 March 2012 (UTC)
The caption to the first picture says the variance is 2. What does that mean when we're talking about a vector-valued random variable, rather than scalar-valued? Often one speaks of a covariance matrix, or of a "variance" that is that matrix or is the associated linear transformation. Michael Hardy 23:42, 23 Jan 2005 (UTC)
Right, I changed the caption to make that clear. Paul Reiser 05:27, 24 Jan 2005 (UTC)
Merge with Wiener process?
- Within mathematics, it is true. In the physical sciences, Brownian motion is the erratic motion of tiny particles suspended in a fluid. The Wiener process is a mathematical model that has been proposed to model that and various other phenomena. Whether the Wiener process adequately models Brownian motion is a question to be decided in part by empirical observation. That they are in some sense the same is hardly an a priori truth. Michael Hardy 21:37, 3 Jun 2005 (UTC)
Can some kind soul please remove the "Suggestion for Merger" caption on the page? It is obvious that we have agreement that a reference to Weiner process from this page and vice versa is the acceptable solution. It is impossible to create a discussion page about the proposed merger therefore surely the proposal should now be deleted. rturus (talk) 04:06, 11 January 2008 (UTC)
- Yes, changed, see Brownian_motion#Mathematical_Brownian_motion —Preceding unsigned comment added by Krauss (talk • contribs) 15:22, 4 February 2008 (UTC)
Central Limit theorem
I think that the mathematical section should make reference to the Central limit theorem, which explains (as far as I know) why the position of a particle at a time t can be considered as normally distributed random variable. Psychofox 01:37, Mar 21, 2005 (UTC)
It is absolutely necessary that Smoluchowski's contributions are discussed here. He worked jointly with Einstein, and derived formulae that are fundamental to the study of stochastic processes. The wikipedia article does not give much information on him, but there are a number of other sources.
- Why not add to the wikipedia article using those sources? PAR 30 June 2005 17:35 (UTC) (PS - type four tildes to sign your name)
I can do some work
I propose myself for writing a section on how to demonstrates the identity between definition 1. and 2 at the macroscopic scale, using the Central Limit Theorem (will write more tomorrow). This leads to a relation between the diffusion coefficient D and the characteristics of the random walk, which I haven't found after a quick search.
- Be bold! Karol 09:05, 21 October 2005 (UTC)
I'll probably start in a couple of days
Finally it appears what I intended to zrite is already covered in the random walk article, though a bit more detail could be useful. If I had something, that will be in that articleuser:ThorinMuglindir
||This article may be too technical for most readers to understand. (September 2010)|
After reading this article, it is extremely hard for the layperson to determine what Brownian motion actually is. —thames 18:44, 6 December 2005 (UTC)
- That is the nature of things. Brownian motion has been found to be rather complex. Can you be at all more specific as to what is confusing? 188.8.131.52 (talk) 05:19, 22 February 2008 (UTC)
- Having just come upon the is article, I can tell you that even the lead is opaque and doesn't make me want to bother to read the rest of the article. I have worked as a university technician for 39 years and have a general interest in all the sciences, so I understand a bit more than the general reader, however, the following sentences are completely incomprehensible:
- "Brownian motion is among the simplest continuous-time stochastic processes, and it is a limit of both simpler and more complicated stochastic processes (see random walk and Donsker's theorem). This universality is closely related to the universality of the normal distribution. In both cases, it is often mathematical convenience rather than the accuracy of the models that motivates their use."
- Do you honestly think that the general reader woulfd get anything fom that? If you read wp:what wikipedia is not#Wikipedia is not a manual, guidebook, or textbook you will find the following statement Wikipedia is an encyclopedic reference, not an instruction manual, guidebook, or textbook. Wikipedia articles should not read like....:Scientific journals and research papers. A Wikipedia article should not be presented on the assumption that the reader is well versed in the topic's field. Introductory language in the lead and initial sections of the article should be written in plain terms and concepts that can be understood by any literate reader of Wikipedia without any knowledge in the given field before advancing to more detailed explanations of the topic. While wikilinks should be provided for advanced terms and concepts in that field, articles should be written on the assumption that the reader will not follow these links, instead attempting to infer their meaning from the text.When I have a bit more time I'll try and plough through the article and attempt to make it a bit clearer, as I have done with a number of other articles, but it would be much better if someone who understands the subject could try to do that first. Richerman (talk) 13:45, 9 December 2008 (UTC)
- Actually I would claim that the quoted part is very descriptive and a necessary because it tells us about very important properties of brownian motion ( with the exception of the donsker's theorem ) - the article would be incomplete with a too dumbed down style as advocated above. The above quote about article structure is also slightly misleading, the same article also notes that For highly specialised topics where it is simply not possible to even give an overview in terms with which a general audience will be familiar, it may be reasonable to assume certain background knowledge. For example, many topics in advanced mathematics fall into this category.. Right now the article suffers a lot from the lack of technical details, which should be a higher priority
- I'm not suggesting that the article should be dumbed down in any way or that any of the information should be removed. I'm saying that, first of all, the lead should be put into language that can be understood by anyone and then more technical concepts should be added later and explained in as simple language as possible. For instance, I added a bracket to say that stochastic means random - I don't call that "dumbing down". Phrases like "This universality is closely related to the universality of the normal distribution" don't belong in the lead unless they are explained properly in a language that can be understood by a non-expert. For instance I found the "intuitive metaphor" section very descriptive but many people wouldn't get that far because they would have been put off by the opacity of the lead section. The whole point of a good article is to engage the reader's interest and explain the concepts as you go along so they are carried along with you. If by the end it gets too technical for them and they give up, then that's fair enough, but you should at least give people a chance learn something new and interesting. I'm a regular reader of New Scientist and find most of their articles quite comprehensible, although some get too technical and I give up before the end. However, with this article I don't even fully understand the lead section. Richerman (talk) 11:03, 23 January 2009 (UTC)
- I must agree with Richerman, (Rich?). The lead is unnecessarily incomprehensible. If the article is properly written, phrases like "..., whose time differential is everywhere infinite,..." can simply be removed. All information in the introduction should already be represented elsewhere in th article, anyhow. BTW, phrases like "dumbed down" border on personal attack and are highly inappropriate. Cliff (talk) 05:33, 5 April 2011 (UTC)
I'm not familiar with the process of demonstrating reliability for historical articles, but the History section of this article sure needs some. I'm sure phrases like "the story goes" are considered unencyclopedic. Could someone come up with some sources that clarify who did what in the process of discovering Brownian motion? BigBlueFish 18:44, 5 March 2006 (UTC)
- [http://www.tau.ac.il/~klafter1/258.pdf] (see page 7)is an academic source and the authors appear to have done their homework on the first known observation by Jan Ingenhousz. It may also be more reliable than other sources because they take the trouble to note that 'coal dust' was used, rather than saying 'carbon dust'. Other references sometimes differ in the use of 'in' and 'on' alcohol, but many seem to have been snatched from Wiki. Davy p 23:58, 14 December 2006 (UTC)
Excellent Communication of Subject
I would like to express my great gratitude and approval for the 'Intuative Metaphor' section in this article. I found it extremely useful in understanding more of this topic. As another commented, the subject is rather complicated, in nature and presentation. This somewhat oblique description really serves to facilitate intelligent reading, especially for mathematical laypersons.
- I personally like Douglas Adams' example for brownian motion, e.g. a nice cup of hot tea.
'Intuitive Metaphor' section: precision?
I'm definitely not knowledgeable on the subject of this article. But at the end of this section I thought the statement, "Considering Brown's pollen particle moving randomly in water: we know that a water molecule is about 0.1 by 0.2 nm in size, whereas a pollen particle is roughly 25 µm in diameter, some 250,000 times larger", to be perhaps too precise while being at the same time too vague. Am I too fussy?
If the average size of the water molecule is 0.15 nm (I assume in diameter, as the size of the pollen particle is listed as such), then the ratio of the diameter is more like 166,667 times larger! The average of the extremes is more like 187,500 times larger. And if the ratio of the volumes is considered rather than the ratio of the diameters, then we're talking a whole different ball game! Assuming, of course, that the particles are assumed to be somewhat spherical!
The first image looks far too discrete to be Brownian Motion. With computers these days, it should be easy to make one with 1 million or more very small steps. This way, as far as the resolution could show, it would appear continuous. --Matthew Carle 07:40, 3 April 2006 (UTC)
- I agree. The lines looks continuous, but it should also look differentiable almost nowhere. Stephen B Streater 05:59, 1 April 2006 (UTC)
- What would be the point of making 1 million tiny steps when not very many people have a monitor with even 2000 pixel widths? --Richard Clegg 12:39, 1 April 2006 (UTC)
- Because a lot of the steps would go back and cross over previous steps. Perhaps 1,000,000 is a bit high, but if the average step was 1 pixel, the average displacement at the end would only be 1,000 pixels ie 707x707 pixels. PS Some of us do have big monitors :-) Stephen B Streater 14:16, 1 April 2006 (UTC)
- You aren't meant to be able to see the steps, that is the whole point. I was just saying the 1 million steps would take a computer only a small amount of time, so why not? The resolution of the image would not have to be changed (and in fact, the higher the resolution, the more steps required for the appearance of continuity). --Matthew Carle 07:40, 3 April 2006 (UTC)
- A million steps is more than any computer could plot. I think you have missed the point of that plot though. If you look at the caption you will see it is not a Weiner process and it does have discrete steps so you should be able to see them. It is debatable whether that is the best image for this article. --Richard Clegg 09:45, 3 April 2006 (UTC)
- A plot with a million steps is too dense to be able to see what is happening (as would be the case with any number of steps sufficient to appear continuous). I assume this is why a discrete approximation was used. I think the only way for a continuous process to be clearly shown is with an animation (like the videos shown at the bottom of the page).--Matthew Carle 02:47, 4 April 2006 (UTC)
Another image (proposed)
I think this image shows how Brownian motion becomes increasingly fuzzy as the time step decreases.
Also, a derivation of why Brownian motion is continuous but not differentiable might help.
Start with the formula for Brownian motion:
You then write it in terms of a standard (variance of one) normal variable:
When you take the limit as , . So is continuous.
Now you subtract from both sides and divide by to get:
At this point, you take the limit as goes to 0:
So the derivative does not exist. In this regard, it is a pathological function.
- I think the picture (with appropriate caption) and derivations are good improvements. Could you add more contrast to the diagram? Perhaps the original diagram could be left in, but captioned "Random Walk" - it is clearly differentiable. Stephen B Streater 06:22, 12 May 2006 (UTC)
- Looks good. Would you like to add it to the page? Stephen B Streater 17:35, 25 May 2006 (UTC)
The quality of the videos are really not up to standard. Any chance these can be upgraded or otherwise removed since they do not enhance the article any way. --Spaztic ming 12:50, 14 June 2006 (UTC)
- I'd rather replace it with something better. Stephen B Streater 13:18, 14 June 2006 (UTC)
- That would be great, but I'm going to remove the current ones for the time being - they're a complete eyesore. --Spaztic ming 12:38, 15 June 2006 (UTC)
I removed this text from the intro:
All three quoted examples of Brownian motion are cases of this:
- It has been argued that Lévy flights are a more accurate, if still imperfect, model of stock-market fluctuations.
- The physical Brownian motion can be modelled more accurately by a more general diffusion process.
- It is not yet known what the best model for the fossil record is, even after correcting for non-Gaussian data.
I did this because the first point is not true. Stock-market fluctuations have finite variance. Lévy flights don't. Therefore, they are not a more accurate model. What has been argued is that certain modifications of the Lévy flight, such as the truncated Lévy flight, are more accurate on short timescales. This is too detailed for the introduction.
The second point is morally true, but not entirely transparent. The "diffusion" must be a diffusion on phase space: the Langevin analysis is different that, say, the diffusion equation which just describes Brownian motion (potentially with drift or rescaling).
I don't have anything to say about three, except to ask what exactly about the fossil record is described by Brownian motion, and to point out that what is meant by "even after correcting for non-Gaussian data" is totally opaque and not really suitable for the intro. –Joke 04:09, 31 October 2006 (UTC)
As per the popular culture reference...
Not to be a nitpicking geek, but:
Brownian motion (as produced by a hot cup of tea) does not power the Heart of Gold. That ship is powered by an Infinite improbability generator. That Generator was itself first created (out of thin air) by the use of a Finite improbability generator. It is, in fact, the finite improbability generator that included "a really hot cup of tea."
This is detailed correctly at the wiki entry for the Infinite Improbability Drive itself, http://en.wikipedia.org/wiki/Infinite_Improbability_Drive
I'm not sure if there is a good and succinct way to accurately express this, perhaps...
In Douglas Adams's The Hitchhiker's Guide to the Galaxy, Brownian motion was an important aspect in the construction of the Infinite Improbability Field Generator which powered the spaceship Heart of Gold. The Brownian motion source was a "really cup of hot tea".
-= Nat Kimble (firstname.lastname@example.org)
184.108.40.206 19:01, 7 February 2007 (UTC)
I'm not sure about what is meant by Einstein's article bringing "the solution of the problem." What problem, the "mathematics of the brownian motion", or the structure of matter???
Also, it is not clear what exactly his article said. I think it even reached interesting concusions, like an estimate of the size of the atoms. But I'm not sure, for example, if Einstein could already rule out the non-atomic theory... It could have been that altought everything fitted, the other theory worked too. A particle released in a force vector field with "white" random forces everywhere would also move in a brownian fashion. Does the article prove that this is not the case?
This is very important. The brownian motion of particles is often mentioned as the first phenomenon that clearly stabilished the veracity of the atomic theory. Doesn't anybody has more details?... -- NIC1138 02:08, 11 March 2007 (UTC)
- Misattribution of ideas?
- A closely related point: the article now says that
The first person to describe the mathematics behind Brownian motion was Thorvald N. Thiele in 1880 in a paper on the method of least squares. This was followed independently by Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets.
- Although the article suggests otherwise, I would be surprised if either of these gentlemen applied their mathematics to Brownian motion per se. Unless someone can confirm that they did, this should be changed, and their names mentioned only in connection with the mathematics. 220.127.116.11 (talk) 07:20, 26 January 2009 (UTC)
This is the relevant piece of text from the radio series, but i really don't know how would be best to word it for the article..
- The principle of generating small amounts of finite probability by simply hooking the logic circuits of a Bambleweeny Fifty-Seven Sub-Meson Brain to an atomic vector plotter suspended in a strong Brownian Motion producer - say a nice hot cup of tea - were, of course, well understood. ...one day, a student ... found himself reasoning this way: “If such a machine is a virtual impossibility, then, it must logically be a finite improbability! So, all I have to do in order to make one, is to work out exactly how improbable it is, then feed that figure into the finite improbability generator, give it a fresh cup of really hot tea… and then turn it on.” He did this and was rather startled to discover that he managed to create the long-sought-after Infinite Improbability Generator out of thin air. —Preceding unsigned comment added by 18.104.22.168 (talk) 18:08, 10 September 2007 (UTC)
The section on modeling Brownian motion is rather confusing and doesn't make the physical/mathematical distinction (assuming something like that exists, that is!) clear enough. -- 22.214.171.124 20:11, 23 September 2007 (UTC)
- I have no idea what the following is supposed to mean, but this is a talk page, so I'm not deleting it.
- It does not belong at the top of the talk page, however, so I have moved it down here. Solemnavalanche (talk) 18:23, 20 January 2008 (UTC)
The boss is delighted. But, being a boss, he says "This is fine, but I want to you to fix your program so the user can select the time interval for the jumps to 1/2 second or 1/3 second or whatever he wants. Some users may want a higher resolution simulation."
So the programmer modifies his program so the user can input the time between jumps.He makes some sample plots using a time interval of 1/10 second and is dismayed to find that the look much more erratic that his original plots, make at intervals of 1 second.
Upon reflection, he understands why. In the original program the position of the particle after, say, 8 seconds was the result of 8 jumps and in the new plots it is the result of (8)(10) = 80 jumps.
He thinks to himself: "If the user inputs a time interval of 1/n, I will change the jump size to 1/n also. That way after 8 seconds the total jumps will be 8n but the max distance the particle can move will be (8n)(1/n) = 8, which is the same as in my original program that used a time interval of 1 second and a jump size of 1."
He makes this change and runs some test plots using a time interval of 1/10. He is dismayed to find that the particle seems to be less variable than it was using the 1 second time interval. He tries using time intervals of 1/100 and 1/1000 seconds and finds that the smaller the time interval, the more the particle tends to stay where it is.
Then he does a mathematical analysis to un-confuse himself. Suppose the time interval is dt = 1/n and the jump size is dx. After 8 seconds there are 8/dt = 8n steps, each is of length +dx or -dx. So the position of the particle at time 8 is the sum of 8n = 8/dt independent random variables, each with value +dx or -dx. The sum of independent random variables can be approximated as a normal distribution, even if the variables themselves are not normally distributed. Let N(m,s) be the normal distribution with mean m and standard deviation s. The approximating normal will have m = the sum of the means of jumps. Each jump, as a single random variable, has mean = 0, so m = 0. The value of s is given by s^2 = the sum of the variances of the jumps. A single jump is a random variable which has value dx with probability 0.5 and value -dx with probability 0.5. The calculation for its variance is (0.5)(dx - 0)^2 + 0.5 (-dx - 0)^2 = dx^2. Since there are 8n = 8/dt such random variables in the sum for s^2, we have that s^2 = 8n (dx^2) = (8/dt)(dx^2) = 8 dx^2/ dt.
It is now clear that if we set the jump size dx to be 1/n when the time interval is dt = 1/n. Then s^2 is 8 dx^2/dt = 8(1/n)^2/(1/n) = 8/n. So as n gets larger (making dt smaller) the standard deviation s, gets smaller and smaller. This explains why the position of the particle after 8 seconds tends to be near x = 0. The probability distribution for its location is concentrated near the mean m = 0.
But how can he fix his program so the variability of the particle approximates what it was in the orginal plots he showed to the boss? In those plots s^2 = 8 1^2/1 = 8. We need to find a dx so 8 dx^2/dt = 8. Solving for dx gives dx^2 = dt and dx = sqrt(dt).
So when the user inputs a time step of (1/n) = dt, we should use a jump size of sqrt(1/n) = sqrt(dt).
He implements this procedure and the resulting plots make the boss happy.
The above story indicates that there are various intellectual problems in going from the verbal description Brownian motion as "the random motion of a small particle" to a precise mathematical theory.
One problem is to determine whether the programmer's algorithm is, in some sense, approximating a process that takes places in continuous time rather than in discrete time steps. This can be established if we precisely define what it means for a sequence of stochastic processes to converge to limiting stochastic process. (This can be done.)
Another question is whether the limit of this sequence of processes is the only kind of continuous random motion that is possible. Apparently it is not. For example, we could allow the user to input a constant factor k and multiply all the jumps of the particle by k. So a better question is whether the limiting process or some constant multiple of it gives describes all possible types of continuous random motions. They don't. However these do constitute an important set of such process that are widely used to analyze phenomena is physics and economics.
(Perhaps we should also mention Wiener and Kholmogorov. I notice that "limit o f a sequence of random variables" is not yet treated in the Wikipedia.)
Clarifying the Mathematical / Physical distinction
Perhaps a simpler way to explain this distinction would be to refer to mathematical fractals and fractals in nature. So for example, a mathematical description of a fractal coastline would have infinite length, because you can "zoom in" on it forever, and never see a "smooth" line. But if you zoom in on an actual coastline, eventually you will be able to see individual atoms, which would appear to describe a smooth (i.e. at least partially differentiable) curve. The same is true of mathematical Brownian motion vs. physical Brownian motion. Right? I'm no expert so someone else ought to verify this intuition before adding it to the article proper. Solemnavalanche (talk) 18:36, 20 January 2008 (UTC)
I don't think this is the only difference. If you consider every atom in a drop of water as undergoing (physical) Brownian motion, the motion is still subject to some rules, such as conservation of momentum. This reduces the degrees of freedom of the motion. True (mathematical) Brownian motion for every particle, even if continuous or if discretized to the level of atoms, will still be an approximation to the physical world.--126.96.36.199 (talk) 00:24, 23 March 2009 (UTC)
This is too funny
I saw almost the same exact opening paragraphs at http://www.absoluteastronomy.com/topics/Brownian_motion but didn't see this site listed under the citations. Because I'm still new to this community, I'm not sure if this is the appropriate place to report something like this or not. Z0wb13 (talk) 12:20, 3 October 2008 (UTC)
Brownian Motion Observed
Hi all, I have no idea on the subject, but I was led here because I happened to read news about this having been observed. Here is a link to the article (http://www.gizmag.com/einsteins-prediction-finally-witnessed/16212/), I figure someone more knowledgeable than me should look it over and add if necessary / applicable
The expression for the root mean square displacement does NOT follow from the expression of the diffusion coefficient. Einstein's formulation of brownian motion is not explained.Bernhlav (talk) 07:36, 7 April 2011 (UTC)
The first video at the top right shows balls bouncing around. It's confusing. Surely a caption should explain that it is the red balls which show Brownian motion, not all of them? If I've understood the video correctly of course (I'm not a scientist so may not have).Malick78 (talk) 11:26, 12 February 2012 (UTC)
Jim Sukwutput's edit
I'm not an expert on the application of Brownian motion to the stock market, but it seems to me that the "unlikely events" in Brownian motion have a predictable probability distribution, while the "unlikely events" that were mentioned in the sentence you deleted may have a different or even unknown probability distribution. Earthquakes, terrorist attacks, etc. The original sentence did not make this point at all, but I think simply deleting it is less helpful than perhaps giving a little detail on the subject. Could you add something after that bare statement about the stock market that gives a little insight into how useful the Brownian motion model is in predicting stock market trends? Thanks. PAR (talk) 06:13, 25 February 2012 (UTC)
"Brownian motion has a few real-world applications"
I'm not sure I like the way this is worded. It seems to imply that science and/or engineering disciplines aren't part of the "real world". Thoughts? (+)H3N-Protein\Chemist-CO2(-) 12:39, 14 January 2013 (UTC)
The time derivative is not everywhere infinite
One of the nice characteristics of Brownian motion is that it is self-similar in the sense that if you zoom in and scale appropriately, you can't tell the difference. This means that one cannot take the derivative; no matter how much you zoom in, Brownian motion won't approximate a straight line in the fashion needed to take a derivative. That's the intuition for why the time derivative of Brown motion does not exist.
However, as the derivative is defined traditionally, the limit does not exist, even in the extended reals, because Brownian motion oscillates arbitrarily fast. The limit does NOT diverge to infinity. That, at least, is my understanding of the matter. — Preceding unsigned comment added by Geekynerdrd (talk • contribs) 20:48, 7 January 2014 (UTC)