Talk:Brownian motion

From Wikipedia, the free encyclopedia

You have new messages (last change).
WikiProject Physics
Portal:Physics
 Physics Portal
This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.
??? This article has not yet received a rating on the assessment scale. [FAQ]
Top This article is on a subject of Top importance within physics.

Please rate this article, and then leave comments here to explain the ratings and/or to identify the strengths and weaknesses of the article.

Contents

[edit] Suggestion

I think the entry would benefit from an example such as:

Informal Discussion

Suppose a computer programmer is assigned the task of simulating Brownian motion in 1-dimension. He writes a program where a particle starts at x = 0 on the x-axis at time t = 0. Every second he picks random numbers to decide whether the particle makes a jump of +1 or -1. Some sample paths of the particle are plotted and show to the boss.

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.)

[edit] The caption

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)

[edit] Merge with Wiener process?

I've heard that Wiener process is simply another name for Brownian motion. Is this true? If so should the articles not be merged? reetep 21:32, 3 Jun 2005 (UTC)

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)

[edit] 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)

[edit] Smoluchowski

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)

[edit] 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

[edit] Too technical

This article may be too technical for a general audience.
Please help improve this article by providing more context and better explanations of technical details to make it more accessible, without removing technical details. See below for more information.

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)

[edit] Historical origins

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)

[edit] Excellent Communication os 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.

[edit] First Image

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)

[edit] Another image (proposed)

Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors.
Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors.

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:

B \left ( t + dt \right ) ~ N \left ( B \left ( t \right ) + \mu dt, \sigma^2 dt \right )

You then write it in terms of a standard (variance of one) normal variable:

X = N \left ( 0, 1 \right )
B \left ( t + dt \right ) = B \left ( t \right ) + \mu dt + X \sigma \sqrt{dt}

When you take the limit as dt\rightarrow 0, B \left ( t + dt \right ) \rightarrow B \left ( t \right ). So B \left ( t \right ) is continuous.

Now you subtract B \left ( t \right ) from both sides and divide by dt to get:

\frac{B \left ( t + dt \right ) - B \left ( t \right )}{dt} = \mu + \frac{X \sigma}{\sqrt{dt}}

At this point, you take the limit as dt goes to 0:

\lim_{dt\rightarrow 0} \frac{B \left ( t + dt \right ) - B \left ( t \right )}{dt} = \lim_{dt\rightarrow 0} \left ( \mu + \frac{X \sigma}{\sqrt{dt}} \right ) = \infty

So the derivative does not exist. In this regard, it is a pathological function.

So should this be added to the page? --Zemylat 00:05, 12 May 2006 (UTC)

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)


I've tried changing the contrast. I can't seem to find a combination that lets you not only see all of the types, but lets you tell the difference. --Zemylat 14:33, 12 May 2006 (UTC)
There. I changed it to have colors. Now the structure should be more visible. --Zemylat 16:48, 25 May 2006 (UTC)
Looks good. Would you like to add it to the page? Stephen B Streater 17:35, 25 May 2006 (UTC)

[edit] Video Section

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)

[edit] removed text

I removed this text from the intro:

All three quoted examples of Brownian motion are cases of this:

  1. It has been argued that Lévy flights are a more accurate, if still imperfect, model of stock-market fluctuations.
  2. The physical Brownian motion can be modelled more accurately by a more general diffusion process.
  3. 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 (nkimble@ufl.edu)

159.178.78.252 19:01, 7 February 2007 (UTC)

[edit] Einstein's article

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)

Retrieved from "http://en.wikipedia.org../../../b/r/o/Talk%7EBrownian_motion_3b44.html"