Talk:Maxwell–Boltzmann distribution

Song etc..

I wrote a song about the Maxwell-Boltzmann distribution... and it goes a little like this:

Oh, baby, baby, baby...

We've had some hard times.

You and me.

We've had our differences.

Don't you see?

And maybe... it's just not meant to be. (not meant to be)

Where are we going? And how fast?

How long can this love fast?

Our half-life is approaching rapidly... (ra-apidly)

There's only one solution,

the Maxwell-Boltzmann Distribution.

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What is the definition / expression for γ in the expression for Entropy? In the article I only see expressions for γ1 and γ2.

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The graph there needs some color coding!

Good point. I'll try to add it Pdbailey 15:22, 24 Oct 2004 (UTC)

How is that? Pdbailey 17:53, 24 Oct 2004 (UTC)

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The way it is presented at first it seems to refer to the general Boltzmann distribution? - then it can be used for any, arbitrarily strongly interacting system, as long as the subsystem considered is large enough. user:FlorianMarquardt

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Equation (7) in the article looks positively wrong. The author states Substituting Equation 6 into Equation 4 and using p_i=mv_i for each component of momentum gives:

$f_{p}(p_{x},p_{y},p_{x})=\left({\frac {\pi mkT}{2}}\right)^{3/2}\exp \left[{\frac {-m}{2kT(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}}\right]$ (7)

Equation 4:

$f_{p}(p_{x},p_{y},p_{x})=cq^{-1}\exp \left[{\frac {-1}{2mkT(p_{x}^{2}+p_{y}^{2}+p_{z}^{2})}}\right]$ (4)

Equation 6:

$c=q(2\pi mkT)^{3/2}$ (6)

If I perform the act of substituting 6 into 4 and substituting p_i=mv_i I get:

$f_{p}(p_{x},p_{y},p_{x})=(2\pi mkT)^{3/2}\exp \left[{\frac {-1}{2kT(mv_{x}^{2}+mv_{y}^{2}+mv_{z}^{2})}}\right]$

Please comment. --snoyes 22:17 Mar 9, 2003 (UTC)

You haven't texified (6) correctly. It's supposed to be

$c=q(2\pi mkT)^{-3/2}$

not

$c=q(2\pi mkT)^{3/2}$

-- Derek Ross 22:56 Mar 9, 2003 (UTC)

Thanks a lot for pointing that out ! I really must be more careful. But what about the second part of (7), where one substitutes p_i = mv_i ? --snoyes 23:03 Mar 9, 2003 (UTC)

I'm still lookin at it but I think the (4) is wrongly texified too. It makes much more sense if the sum of p's is on top of the fraction. But I'm just doing a little research to ensure that that's the right thing to do. -- Derek Ross

Which would mean that (3) is also wrongly texified. The problem is that:

exp[-1/2mkT(p_x² + p_y² + p_z²)]

is just so damn interpretable. That is partly the reason I'm going to all the trouble of texifying all these articles; To disambiguate them. --snoyes 23:21 Mar 9, 2003 (UTC)

Yep, (3) should be

exp[-(p_x² + p_y² + p_z²)/(2mkT)]

and the others should changed analogously. -- Derek Ross

Excellent. However, there remain problems with (7); it would now have to be:

$f_{p}(p_{x},p_{y},p_{x})=(2\pi mkT)^{-3/2}\exp \left[{\frac {-(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}\right]$ (7)

(ie. the m's cancel and the stuff in the first bracket is all to the power ^(-3/2)) ? --snoyes 23:42 Mar 9, 2003 (UTC)

And why is cq^(-1) not written as (c/q) in (4) ? (style?)--snoyes 23:44 Mar 9, 2003 (UTC)

Yep, style. They both mean the same thing. -- Derek Ross

Substituting p² with p=mv gives m²v². You can then pull all the m²'s out with the distributive law and cancel the m in the denominator leaving an m in the numerator which is what you want.

As for the other point ...

$(2\pi mkT)^{-3/2}=(1/2\pi mkT)^{3/2}$

... so there's no problem there, just a matter of personal taste about how you want to write the formula. -- Derek Ross 23:52 Mar 9, 2003 (UTC)

I stand corrected again, thank you Derek. As for the personal taste, I don't care which one - do you have a personal preference? I shall use that. --snoyes 23:57 Mar 9, 2003 (UTC)

Some people get confused by the q^-n notation. I think that it's often a better idea to change it to 1/(qⁿ) instead. Also I would separate out the relatively constant parts to make something like... well if I knew Tex I would do it myself. Sadly I don't. Guess I'll have to learn ! -- Derek Ross

I learnt it in a couple of hours solely for changeing all the stuff on wikipedia ;-) --snoyes 00:08 Mar 10, 2003 (UTC)

One thing I would like. exp[x] is actually supposed to be e^x. It would be nice if you could change that -- Derek Ross

Hmm, unfortunately it looks like this:

$e^{\left({\frac {-(p_{x}^{2}+p_{y}^{2}+p_{z}^{2})}{2mkT}}\right)}$--snoyes 00:27 Mar 10, 2003 (UTC)

Too bad, exp it will have to be then. -- Derek Ross

I see someone has learnt some tex. ;-) Couple of small things with (8). Corrected it is:

$f_{v}(v_{x},v_{y},v_{z})=\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left[{\frac {-m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}\right]$ (8) --snoyes 05:34 Mar 10, 2003 (UTC)

Not at all, just monkey see, monkey do, plus good ole cut'n'paste -- hence the mistakes. One other change which I think needs making is that the integral signs should actually be partial integral signs and likewise the differentiation operators should be partial differentiation operators but as I said, if only I knew Tex... -- Derek Ross

Distribution plot

I have to be pedantic regarding the newly added plot: There can not be any units at the axis for "Probability"!! A probability is always without units. This must be fixed. Awolf002 15:07, 26 Aug 2004 (UTC)

Well, I guess that axis should read "Probability density." The probability is obviously unitless, but since you integrate over a certain range on that plot to get the probability, it has to have the inverse units of the x-axis so the integral comes out unitless. If that sounds weird and you don't believe me, do a quick unit analysis on any of the Maxwell-Boltzmann equations on the article page. Ed Sanville 03:31, 27 Aug 2004 (UTC)

I was anticipating that answer. You are correct if you are "binning" the x-axis. Then the y-axis needs to have the inverse units. However, I always thought the M-B distribution function has as value the probability, not its density. If I am correct then it has no units, and that's what should be drawn without binning in order to not confuse people. Awolf002 14:03, 27 Aug 2004 (UTC)

This statement seems confused: "I always thought the M-B distribution function has as value the probability, not its density". Obviously the cumulative probability distribution function has probabilities as its values, and obviously the probability density function does not. Since these graphs are of density functions, their values are not probabilities. But I need to ask you: the probability of what event? If you're talking about the cumulative probability distribution function, the event would be that of being less than or equal to the argument to the function. Is that what you had in mind? Michael Hardy 20:36, 28 Aug 2004 (UTC)

Well, I don't see how you can graph a continuous probability distribution without giving the y axis inverse units to the x axis. Think about it, what's the probability that the particle will have EXACTLY 345.218417823... m/s as its speed. Zero, of course. It's kind of like the little-known fact that the quantum mechanical wavefunction is NOT unitless because in 3d space, it has to have units of $length^{-{\frac {3}{2}}}$. Ed Sanville 17:47, 27 Aug 2004 (UTC)

Okay, let me see if I remember this. The M-B distribution is *in fact* a single probability at one *point*. In real life, you will never ask what the probability of one value might be. Instead you "bin" your x-axis and calculate the integral of the M-B distribution between the boundaries, which is what you did to create the plot in the article. My point is that $\int _{x}^{x+\Delta x}F_{M-B}(\xi )d\xi$ evaluated at certain x is not the same as the M-B function $F_{M-B}(x)$ itself. Right? Also note, that the result of this integration indeed has a unit. Awolf002 18:44, 27 Aug 2004 (UTC)

Since I wanted to create a graph which would display the probability densities of the speeds for those noble gases, I actually graphed the following function which is right above the actual graph in the article:

$F(v)=4\pi v^{2}\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left({\frac {-mv^{2}}{2kT}}\right)\qquad \qquad (11)$

So, as you can see, what I graphed is the probability density (units s/m), which must have units inversely proportional to the x axis (in this case speed). The chart is similar to charts found in many physics and physical chemistry books, for instance see Physical Chemistry by Laidler and Meiser. It's somewhere in the first few chapters. Basically the exact meaning of what I graphed is:

$\lim _{ds\to 0}{\frac {P(s\to s+ds)}{ds}}$

where s is speed, ds is an infinitesimal speed interval, and $P((s\to s+ds)$ represents the probability that a particle will have a speed between s and s+ds. So, the graph I made shows probabilities only as the areas under slices of that curve, (probability density). The only thing wrong with the graph, I think, is that it might be better to change the y-axis label to "Probability Density (s/m)" instead of just "Probability (s/m)". I think I'll do that now... Ed Sanville 20:21, 27 Aug 2004 (UTC)

Okay, the more I think about this, it seems to be the most reasonable thing to do. Thanks for going through this with me! Awolf002 23:19, 27 Aug 2004 (UTC)

speeds density function

just edited the section on speeds to show that the function is actually the pdf, not a "distribution" which is close but not quite right. the graph below should be changed to show that the y-axis is labeled probability density with units (s/m). It should be a graph of $f(v)$ and has these units because $f(v)$ has these units.

Notice that I changed the notation form $F(v)$ to $f(v)dv$. this is because the cpd, pdf pair are defined as

$F(x)=\int _{-\infty }^{x}{f(\xi )d\xi }$

This means that different differentials can have different meanings and different cpds associated with them. Case in point, when you convert from the speed distribution to the energy distribution, you will be looking at a new cpd, pdf pair and will have to convert $dv$ to $dE$. The (often minor) additional clarity does not distract but can prove a very useful reminder. (unsigned comment by User:Pdbailey)

Quantum Mechanics

Why the start with Quantum Mechanic when Boltzmann derived this thing way before Q.M. came to pass? One of the interesting things about the Maxwell-Boltzmann distribution is that it doesn't use Q.M. and can be used to find avagadros number. Also it may be worth pointing out that Maxwell did the first derivation (before we had partials!) with a little hand waving and Boltzmann did the clean-up proof. I don't have the original papers so I can't say much more than that right now. Maybe I'll look into that. (unsigned comment by User:Pdbailey)

Agreed. It looks like the Maxwell-Boltzmann distribution (MBD) derives from Q.M., while, quite the opposite, the first (?) complete demonstration of Planck's law of black body radiation - by Einstein and (?) - uses the MBD as a known fact. Planck himself didn't used it in his proof, but he did derived it again from scratch.-Nabla 13:42, 2004 Aug 30 (UTC)

Okay, the problem is that I don't really want to have to do all of Maxwell's derivation here and this is much shorter. Perhaps we could just say that this newfangled derivation is quick but is not the one used by Maxwell or Boltzmann who derived this before qm. Unless someone wants to do a non-copyrighted derivation of this using clasical physics. --Pdbailey 22:24, 30 Aug 2004 (UTC)

Chart verified to be wrong

I looked at the chart and picked off the maximum of He-4 as about 1500 m/s but i get a different number (with minor rounding)

$v_{p}={\sqrt {\frac {2kT}{m}}}$

$v_{p}={\sqrt {\frac {2*1.38E-23J/K*298}{4amu*1.66E-27Kg/amu}}}$

$v_{p}={\sqrt {\frac {8.22E-21m^{2}s^{-2}}{6.64E-27}}}$

$v_{p}=1113ms^{-1}$

I'll upload a new one. Anyone care to check that I got this right? Why u no doctor? — Preceding unsigned comment added by 93.218.73.181 (talk) 15:12, 20 January 2012 (UTC)

I agree with your numbers (although you mean kg not Kg). You can Google this to get the same answer: sqrt(((2*(Boltzmann constant)* 298 kelvin)/(4*1.66e-27 * kg))). But the graph looks right to me. —Ben FrantzDale (talk) 18:01, 20 January 2012 (UTC)

Spelling in Picture

Gasses should be Gases!

Thanks for the catch. I tried to fix it, I hope I'm just seeing the cache now... Pdbailey 23:45, 18 Nov 2004 (UTC)

Adding energy distribution

I wanted to add the energy distribution because of some things I'm doing on statistics. The main revision besides adding the distribution function for the magnitude of momentum, and the energy, is to take out the quantum "particle in a box" references. We don't need it to go from equation 1 to equation 3, we just need p^2=2mE and theres some complications with a particle in a box, because energy eigenfunctions do not coexist with momentum eigenfunctions, so you can't talk about the momentum of a particle in a box while talking about its energy at the same time. (You can, however, talk about the absolute value of the momentum along any axis.).Paul Reiser 21:55, 22 Dec 2004 (UTC)

Boltzman distribution

Should Boltzmann distribution really redirect here? My memory says that that term describes the simple exponential distribution of energy for each degree of freedom; the Maxwell-Boltzmann distribution consists of three such degrees of freedom. Shimmin 13:28, Feb 17, 2005 (UTC)

Are you thinking of Boltzmann's equation for Entropy? I'm quite sure the Maxwell-Boltzmann distribution has two names, one less flattering to Maxwell. --Pdbailey 00:26, 14 October 2005 (UTC)

Blackbody radiation?

Does this distribution relate to blackbody radiation? That is, is there any relation between the distribution of mollecular energies and the emission spectrum? —BenFrantzDale 17:13, 13 October 2005 (UTC)

Just as the Maxwell-Boltzmann distribution describes the energy distribution for a bunch of massive particles at a particular temperature, the Planck's law of black body radiation specifies the energy distribution for a bunch of photons at a particular temperature. So in that sense, there is a conceptual relationship. However, the two distributions are different because the particles have different statistics. The two are physically related only through the temperature. If you bring a bunch of photons into thermal equilibrium with any other body that is maintained at some temperature T, they will take on the Planck distribution for that temperature and it doesn't matter what the distribution of energies are in the other body. So in that sense there is not a relationship. PAR 18:53, 13 October 2005 (UTC)

PDF Box

Someone should add a Probability distribution template and put the PDF at the beginning of the article. -anon

I actually don't like this because I think (and maybe I'm just wrong here) of the distribution of a mixture of a velocity component distribution, a speed distribution, and an energy distribution. Each of these is a distribution. Granted the last two are related to each other by a change of variables, but they have different summary stastics. --Pdbailey 03:02, 3 March 2006 (UTC)

Order of sections?

I'm not an expert on how technical articles about formulas work, but I think it's preferable to have some of the less-technical explanation earlier in the article. For example, move: "The Maxwell-Boltzmann distribution forms the basis of the kinetic theory of gases, which explains many fundamental gas properties, including pressure and diffusion. The Maxwell-Boltzmann distribution is usually thought of as the distribution of molecular speeds in a gas, but it can also refer to the distribution of velocities, momenta, and magnitude of the momenta of the molecules, each of which will have a different probability distribution function, all of which are related." up before the formula itself, so that non-specialists can get a sense of what the distribution is in a larger context. I'll defer to the people who know the topic. -- Epimetreus 12:43, 16 March 2006 (UTC)

Hi,

There is a problem with a conclusion at the end of the article. The variance of the normal distribution of speed in one direction isn't correct I think. What it is writed is the inverse of the variance. The correct expression is $kT/m$

If you think in units, it is correct, because the variance must be a the square of a speed.

Jonathan

Graph etc of molecular speeds

The graph at the end of the article for molecular speeds of noble gases seems wrong to me based on a maxwell-boltzmann distribution - why is it peeked - using the maxwell boltzmann distribution of energies and v=sqrt(2e/m) surely the graphs should be similar in shape to an exponential decay - like the energy distribution - is there an explanation for this?HappyVR 11:42, 15 April 2006 (UTC)

Yes, the explanation has to do with the number of states, (quantum mechanically), or the volume in phase space, (classically), as the velocity increases. Think of it this way, assuming that the velocity vector of a random particle is random, with the energy weighted as an exponential decay curve, that doesn't mean that the velocity will also follow an exponential decay curve. The reason is that there are far more vectors corresponding to high velocities than there are vectors corresponding to low velocities. In fact the "number," (or volume of space in "velocity space" if you will), increases with the square of the velocity. You can think of this as being the area of the sphere centered at the origin, with a radius equal to the particle's speed. So, when this quadratic increase in the number of states is multiplied by the Gaussian-shaped curve e^(-kv^2), (remember, energy is proportional to velocity squared), you end up with a function which has a non-zero peak velocity. Good eye, though... Ed Sanville 18:19, 15 April 2006 (UTC)

Thanks - so it could be described as the probability of finding a particle in a given volume with a specific velocity (time averaged) - that wasn't totally clear to me from the 'probability density (s/m)' y axis label - still not sure what m stands for but doesn't really matter.
I would have thought (simple model) that the equation would be of the type v x e^(-k x v x v) - also gives a 'bump' or peak - due to a volume proportional to molecular radius squared times velocity ie a cylinder along the direction of travel (very similar to what you said). It's more clear to me what probability density means compared to just probaility.Thanks.HappyVR 19:22, 15 April 2006 (UTC)

I think you have misunderstood my explanation... I was talking about "volume" in the space of velocity vectors, not real space. So the radius and other size characteristics of the moecule don't come into it. Let me try to give a more concrete example... If the particle has a velocity of between 1 and 2 m/s, the velocity vector can be anywhere in the spherical "shell" between radius 1 and 2 m/s in "velocity space." This shell has a volume of ${\frac {4}{3}}\pi (2^{3}-1^{3})=7{\frac {4}{3}}\pi$ in velocity space. However, if it has a velocity between 2 and 3 m/s, the volume of space it can be in is ${\frac {4}{3}}\pi (3^{3}-2^{3})=19{\frac {4}{3}}\pi$. So, my point is that even if all of the energies were equally probable, there would still be a higher chance that the particle has a velocity betwee 2 and 3, than there is that it is between 1 and 2! (This concept is closely related to entropy in fact, but that's another story...) Anyway, the energy of any point in velocity space is proportional to the sum of the squares of the three components $v_{x},v_{y},v_{z}$. According to Boltzmann, the probability that a particle has a velocity occuring in any given volume of velocity space is then proportional to $e^{\frac {-m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}$. But, you have to integrate this probability over the velocity space, so you end up with a speed probability distribution like $\int e^{\frac {-mv^{2}}{2kT}}4\pi v^{2}\,dv$ (where v is the magnitude of the velocity, ${\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}$). If you solve this integral, you will get the cumulative probability with respect to speed, (magnitude of the velocity vector). The graph is actually of $e^{\frac {-mv^{2}}{2kT}}4\pi v^{2}$, the probability density function. Ed Sanville 22:30, 20 April 2006 (UTC) Sorry about all the mistakes before edit... I was tired.

The graph is of a pdf of molecular velocity for particles in a volume with a given energy and given mass. The moledular velocity distribution function is f(v)dv, it has units seconds/meter (s/m) as I've plotted it. This has come up several times, so I'll add a note on the image's page. Pdbailey 02:28, 18 April 2006 (UTC)

Distribution of momentum and velocity vectors are not (yet) M-B distributions

Distribution of momentum and velocity vectors are not (yet) M-B distributions. So formulas (6) and (8) are not M-B distributions. I think this should be more clearly stated, and then the work should be finished in order to actually give the formulas for the M-B distributions of p and v. Otherwise this can get quite confusing.

So I'd happy to see an equation f(p)dp, which would also ease the step between equations 7 and 8, and another equation f(v)dv illustrating the distribution of speeds such as the following: $f_{\mathbf {v} }(v)=N{\sqrt {{\frac {2}{\pi }}\left({\frac {m}{kT}}\right)^{3}}}v^{2}\exp \left[{\frac {-mv^{2}}{2kT}}\right],$ with N, the number of particles you consider in your box (I guess this factor can just be removed to have probabilities instead of number of molecules).

The $v^{2}$ that appears in that formula makes the difference and gives its nature to the MB formula.

B. Roy Frieden's anonymous POV-pushing edits

B. Roy Frieden claims to have developed a "universal method" in physics, based upon Fisher information. He has written a book about this. Unfortunately, while Frieden's ideas initially appear interesting, his claimed method is controversial:

• Binder, Philippe M. (2000). "Physics from Fisher Information: A Unification (a review)". American Journal of Physics. 68: 1064–1065.
• Kibble, T. W. B. (1999). "Physics from Fisher Information: A Unification (a review)". Contemporary Physics. 40: 1999. (the reviewer has some positive comments but concludes that Frieden's work is "misguided")
• Case, James (2000). "An Unexpected Union---Physics and Fisher Information". SIAM News. July 17. eprint (highly favorable)
• Matthews, Robert (1999). "Physics and Fisher Information (a review)". New Scientist. January. unauthorized electronic reprint
• Physics from Fisher Information: A Unification (a review) from Cosma Shalizi (Computer Science, University of Michigan) (highly critical)
• Physics from Fisher Information (a review) from R. F. Streater (Mathematics, Kings College, London) (highly critical)
• Physics from Fisher Information thread from sci.physics.research, May 1999 (mostly critical)
• Fisher Information - Frieden unification Of Physics thread from sci.physics.research, October 1999 (mostly critical)

Note that Frieden is Prof. Em. of Optical Sciences at the University of Arizona. The data.optics.arizona.edu anon has used the following IPs to make a number of questionable edits:

1. 150.135.248.180 (talk · contribs)
   1. 20 May 2005 confesses to being Roy Frieden in real life
   2. 6 June 2006: adds cites of his papers to Extreme physical information
   3. 23 May 2006 adds uncritical description of his own work in Lagrangian and uncritically cites his own controversial book
   4. 22 October 2004 attributes the uncertainty principle to the Cramer-Rao inequality, which is potentially misleading
   5. 21 October 2004 adds uncritical mention of his controversial claim that the Maxwell-Boltzmann distribution can be obtained via his "method"
   6. 21 October 2004 adds uncritical mention of his controversial claim that the Klein-Gordon equation can be "derived" via his "method"
2. 150.135.248.126 (talk · contribs)
   1. 9 September 2004 adds uncritical description of his work to Fisher information
   2. 8 September 2004 adds uncritical description of his highly dubious claim that EPI is a general approach to physics to Physical information
   3. 16 August 2004 confesses IRL identity
   4. 13 August 2004 creates uncritical account of his work in new article, Extreme physical information
   5. 11 August 2004 creates his own wikibiostub, B Roy Frieden

These POV-pushing edits should be modified to more accurately describe the status of Frieden's work.---CH 21:54, 16 June 2006 (UTC)

Too Complex

Hi this page is great but it is very complex - is it possible to give an overview on how this distribution works in simple terms and perhaps with an everyday life example??

Equation 7 (Energy distibution)

Can it be made clearer that $f_{E}dE=f_{p}{\frac {dp}{dE}}dE$ is actually $f_{E}dE=f_{p}{\frac {dp_{x}}{dE}}{\frac {dp_{y}}{dE}}{\frac {dp_{z}}{dE}}d^{3}E$?

While on the subject, is the numerical factor correct? I keep getting 4 (from $d^{3}E=4\pi dE$ instead of 2. Warrickball 11:14, 11 May 2007 (UTC)

boltzmann distribution is not energy distribution???

can anyone explain how one can go from the boltzmann distribution directly to the energy distribution? it is very confusing that the are not identical. —Preceding unsigned comment added by 84.191.237.160 (talk) 00:54, 17 December 2007 (UTC)

Everyone has over-complicated things

Everyone has over-complicated things, at least in the introduction, by going way into applications. In reality, a Maxwell–Boltzmann probability distribution is purely a normalized mathematical function that doesn't have anything to do with the applications. Why not express the Maxwell–Boltzmann probability distribution (and/or Maxwell–Boltzmann probability density) quite simply at first, as a mathematical function. This would be good for the General Reader, a novice, someone who doesn't know much about physics at all. (You people are too busy discussing - or arguing about - things like Classical Mechanica and Quantum Mechanics that the general reader probably doesn't know anything about.) The key notion is "KISS" = "Keep It Simple, Stupid", at least in the beginning, for the General Reader.
To give a specific example, the Gaussian distribution (or Gaussian density) is expressed mathematically as f(x) = Kexp(-ax^2) where the K is a normalizing constant, and "a" is an arbitrary positive constant. This has nothing to do with the physical application; and in fact, it can be used in thousands of different applications.
Likewise, for x greater than or equal to zero, there are many other probability densities that can be considered and applied, where K is a different normalizing constant for each one:
f(x) = Kexp(-ax); f(x) = Kxexp(-ax); f(x) = Kxexp(-ax^2); f(x) = K(x^2)exp(-ax^2); and so forth. These distributions (densities) have various names, such as the Maxwell distribution; the Boltzman distribution; the exponential distribution; and so forth. These can be applied in many other fields than in physics, statistical mechanics, etc.
Once again, to summarize, the Maxwell–Boltzmann distribution should be explained in simple mathematical terms first - before moving on into all of the complications of the applications in physics - and with the recognition that the Maxwell–Boltzmann distribution can conceivably be used in entirely different realms of probability and statistics. 98.67.174.68 (talk) 16:43, 21 June 2009 (UTC)

See the Wikipedia article on the Rayleigh Distribution: http://en.wiki.x.io/wiki/Rayleigh_distribution
For a much more clear expositon of the mathematical properties of a probability distribution, then with the physics explained elsewhere.

I believe that there is a probability distribution in mathematics that is simply called the Maxwell distribution (or Maxwell density), and that is is called the Maxwell distribution. I think that it is expressed as a probability density, for x greater than equal to zero, as f(x) = K(x^2)exp(-ax^2); and f(x) = 0 otherwise. As it stands now, someone who looks for the Maxwell distribution or probability density gets re-directed to the article on the Maxwell-Boltzman distribution - and this article doesn't say anything for the rather-simple Maxwell density (which is purely a mathematical concept, and doesn't have anything to do with all of this Statistical Mechanics and Quantum Mechanics). I always advise putting early a simple statement of what the probability density / distribution is, and then moving on to applications, such as in physics. And how do you know that the Maxwell distribution and the Maxwell-Boltzman distribution don't have applications that don't have anything to do with physics, the mechanics of gasses, and so forth?

Once again, look at the Wikipedia article on the Rayleigh Distribution: http://en.wiki.x.io/wiki/Rayleigh_distribution
for an exposition of the mathematical properties that don't have anything to do with physics.98.67.174.68 (talk) 17:27, 21 June 2009 (UTC)

I have a big problem with equation (3)

Equation (3) has subscript i on the left handside but no i on the other side. How would you fix it? should it be $p_{x,i}^{2}+p_{y,i}^{2}+p_{z,i}^{2}$ instead of $p_{x}^{2}+p_{y}^{2}+p_{z}^{2}$ ???

Besides, what happened to $g_{i}$ in front of (3). $g_{i}$ should have been inherited from equation (1)? —Preceding unsigned comment added by Resonance cascade (talk • contribs) 09:02, 6 February 2008 (UTC)

I've made some changes that perhaps explains where the degeneracy factor goes, and makes the correction that the momentum components are still subscripted with i. 124.171.142.229 (talk) 06:31, 24 December 2009 (UTC)

Merger proposal

Maxwell speed distribution duplicates the information in the section on the speed distribution in Maxwell-Boltzmann distribution. I propose to merge the former into the latter by putting the alternative derivation of the speed distribution into a new section on the target page. Deltasct (talk) 03:34, 13 November 2008 (UTC)

Agreed. While you're at it, how about cleaning up the grammar in Maxwell-Boltzmann distribution? Some of the intro in Maxwell speed distribution is more clear. UncleDouggie (talk) 11:33, 19 April 2009 (UTC)

Agreed —Preceding unsigned comment added by 169.139.217.77 (talk) 01:24, 8 December 2009 (UTC)

I disagree. I had to find something about mean speed distribution and the fact that it's written under Maxwell speed distribution made it very confusing. I suggest separating the two entries.

Agreed. To the anon disagreer: I would think that keeping information about one thing (the Maxwell-Boltzmann speed distribution) in one article, and then making that article findable in all relevant ways, would be the best way to address your concerns, right? DutchCanadian (talk) 20:38, 5 March 2013 (UTC)

Five years and this still hasn't happened. Sigh. I'll put the relevant tags up and hope someone less lazy than me takes care of it.Jess (talk) 20:18, 22 April 2013 (UTC)

about equation4

the paragraph above equation 4 states that f_p is proportional to N_i/N,and we can use ∫∫∫(c/z)exp〔..〕dp_x*dp_y*dp_z=1 to find the constant c.

My question is that would f_E or f_v be also proportional to N_i/N,if not so,why not?if so,why I get a wrong outcome by using the same technique,that is,I

try toset f_v to be proportional to Ni/N,then f_v=(c/z)exp(...),but when I set ∫f_v*4πv^2*dv equal to 1 to solve c,I'll get a wrong answer.Is there a

clearer explanation that why only f_p is proportional to Ni/N and f_v along with f_E is not?And how do we know f_p is proportional to Ni/N directly? f_p is

a "probability density" function,while Ni/N is a "fraction" of the amount of particles with energy ε_i to all of the molecules of the system. How can one

visualize the connection between these two intuitively?

—Preceding unsigned comment added by Yung.cheng (talk • contribs) 13:27, 4 June 2009 (UTC)

Median Speed

We have equations for the mean, mode, and root-mean-square speed, but what about the median? Is there an equation for that that we could put in the article?

68.186.161.80 (talk) 23:19, 21 June 2009 (UTC)

The median is defined implicitly by Eq. 6.7.17 here: [1]. It is quoted as 1.08765 times v_p the most probable speed.

Another related one is the speed of sound in an ideal gas, which (rearranging the formulae on that page a little) is sqrt(5/6)v_p for a monatomic gas, sqrt(7/10)v_p for a diatomic one. Bobathon71 (talk) 22:41, 28 February 2013 (UTC)

Isotropy vs. Independence

The original derivation by Maxwell assumed all three directions would behave in the same fashion, but a later derivation by Boltzmann dropped this assumption using kinetic theory.

I thought that the problematic assumption in Maxwell's derivation was that velocities in the three directions were independent. 72.75.67.226 (talk) 04:04, 8 October 2009 (UTC)

Also, isn't it worth mentioning that the multivariate normal is unique in that it is isotropic and the three velocities are independent? 72.75.67.226 (talk) 04:12, 8 October 2009 (UTC)

Typical speed of nitrogen at room temperature is wrong

It says 600m/s for the most probable speed, which is about what you get if you plug in 300K and 14g/mol. However at room temperature nitrogen forms a molecule with two atoms, so you have to use 28g/mol which gives about 420m/s for the most probable speed. --84.56.21.137 (talk) 21:56, 2 January 2010 (UTC)

The text says, "$v_{p}={\sqrt {\frac {2kT}{m}}}={\sqrt {\frac {2RT}{M}}}$ where R is the gas constant and M = N_A m is the molar mass of the substance. For diatomic nitrogen (N₂, the primary component of air) at room temperature (300 K), this gives $v_{p}=422$m/s "; but, substituting R=8.314, T=300, M=28 into this equation, i get a very different answer: $v_{p}={\sqrt {\frac {2(8.314)(300)}{28}}}=13.35$. This differs by a factor of ${\sqrt {1000}}$ which i think arises from the difference between grams and kg, but the text does not make it clear -- 99.233.186.4 (talk) 20:00, 18 January 2011 (UTC)

Kurtosis correct?

Upon checking this edit by Melcombe with my version of Maple 13, I found that the correction was indeed to the point. O.t.o.h. I also noticed that the infobox value for the kurtosis is different from what Maple says.

Infobox says: $\gamma _{2}=-4{\frac {96-40\pi +3\pi ^{2}}{(3\pi -8)^{2}}}$

Maple says: $\gamma _{2}={\frac {-192+16\pi +15\pi ^{2}}{(3\pi -8)^{2}}}$

Do we have an online source somewhere? Preferably not Mathworld, as it has the skewness wrong as well. DVdm (talk) 17:48, 25 May 2010 (UTC)

Maple's version is the kurtosis, the version in the infobox is the excess kurtosis, so is lower by 3. Both are correct. I must say that as the excess kurtosis is positive (0.1082) i think it would be clearer to rewrite the expression to avoid the minus sign out the front. Qwfp (talk) 20:13, 25 May 2010 (UTC)

Ah yes. I had found the difference of 3 as well. I should have looked the Kurtosis article where the number 3 is sort of staring me in the face. Oops - and thanks. DVdm (talk) 21:07, 25 May 2010 (UTC)

"junk paragraphs"...

I have removed the paragraphs labelled as junk - they are indeed since they add nothing to the article (they were commented out anyway - but why keep them?). The intro has been re-structured to be more concise and to the point. Also made other various edits - such as:

• addition of sources, will slot into article later - very busy right now
• removed the tabs (proposed way back in 2009/10) - there are now sources
• add eqn navbox
• minor bits of clean up
• changed the leading titles - they are not helpful as they are

Hope the intro is now ok. -- F = q(E + v × B) 22:39, 22 February 2012 (UTC)

Lack of unity of articles about the Maxwell-Boltzmann distribution

Maxwell–Boltzmann distribution

“ 1 Physical applications

   2 Distributions (various forms)
       2.1 Distribution for the momentum vector
       2.2 Distribution for the energy
       2.3 Distribution for the velocity vector
       2.4 Distribution for the speed
       2.5 Distribution for relative speed
       2.6 Typical speeds
       2.7 Distribution for relativistic speeds
   3 See also
   4 References
   5 Further reading
   6 External links

”

Boltzmann distribution

“ 1 Definition

   2 Derivation
   3 References
   4 External links
   5 See also

”

Maxwell–Boltzmann statistics

“ 1 A derivation of the Maxwell–Boltzmann distribution

   2 Another derivation (not as fundamental)
       2.1 Comments
   3 Limits of applicability
   4 See also
   5 References
   6 Bibliography

”

Someone needs to untangle this obvious mess.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 19:01, 8 May 2012 (UTC)

Why isn't the derivation for the Maxwell-Boltzmann distribution in the article of the same name? Why are the examples in that article there rather than in the article titled Maxwell-Boltzmann *statistics*?siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 19:45, 8 May 2012 (UTC)

Simple English version

Could anyone please make a simple english version of this? Many thanks in advance! — Preceding unsigned comment added by Mbb1056 (talk • contribs) 19:51, 7 April 2014 (UTC)

Dimensionality (explaining a revert)

@SamuelHon: I'm reverting this edit you made diff which added a dv with the edit summary "Dimensional calculation of the original formula of is wrong. The original function is a probability function, it should be no dimension. But the dimension of the original function is 1/v. Should be amended."

Whilst a lump of probability has no dimensionality, a probability density or probability distribution such as f does have a dimensionality.

One way to write a continuous probability distribution f is

$f={\frac {dP}{dv}}$

where P(v) is the (dimensionless) cumulative probability, so the density f therefore has dimensionality [1/v].

So one can either write an equation for (infinitessimal) dimensionless probabilities,

$fdv=...\,dv$

or (as the article did) an equation for probability densities

$f=...$

But what you can't write is your proposed change $f=...\,dv$

Hope this clarifies things. Jheald (talk) 11:19, 16 July 2015 (UTC)

@Jheald:The probability function of velocity distribution is essentially the mathematical expression of dN(v)/N. It must be absolutly a dimensionless expression. Its summary or integral from 0 to infinity must be 1. You can check the summary or integral of your function and my suggestion for your decision.

SamuelHon

Probability density functions (PDFs) are not probabilities and do not have to be dimensionless. They do have to integrate to 1, which means their dimensions must be inverse of the integration variable. The Maxwell Boltzmann distribution is a PDF with units of (speed )^-1 and integrates over speed to give 1. --Nanite (talk) 15:58, 16 July 2015 (UTC)

Two different version of the velocity distribution.

I am not an expert, and to me it looks like there are two contradictory versions of the velocity distribution. The first equation in the article reads

$f(v)={\sqrt {\left({\frac {m}{2\pi kT}}\right)^{3}}}\,4\pi v^{2}e^{-{\frac {mv^{2}}{2kT}}},$

Later in the article, I see

$f_{\mathbf {v} }(v_{x},v_{y},v_{z})=\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left[-{\frac {m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}\right]$

The difference I am pointing to is the addition of $4\pi v^{2}$ in the first equation. I am not insisting that there must be an error, but I think the difference may need to be commented/adressed, to lessen the confusion of new readers (like me).

(EDIT: Oh, i guess the first one is for the absolute speed $|v|$, and has been integrated across a sphere surface. Perhaps this should be commented/adressed by someone more comfortable with the terminology.

— Preceding unsigned comment added by Eaurs (talk • contribs) 09:54, 19 May 2016 (UTC)

Suggesting to delete a statement

I fail to see how the following is interesting:

An interesting point to be noted is that the Maxwell–Boltzmann distribution will not vary with the value of $m/T$, i.e. the ratio of mass of the molecule to its absolute temperature; mathematically, the derivative of $f(v)$ with respect to $(m/T)$ is equal to $0$ (only when $v^{2}=3kT/m$).

(The section "Distribution function".) The derivative of $f(v)$ with respect to $m/T$ (keeping $v$ fixed) has a factor $(v^{2}{\frac {m}{T}}-3k)$, so surely the function that maps values of $m/T$ to $f(v)$ for a given value of $v$ has a stationary point at ${\frac {m}{T}}={\frac {3k}{v^{2}}}$, but how does this deepen our understanding of the Maxwell-Bolzmann distribution? — Preceding unsigned comment added by Cacadril (talk • contribs) 18:13, 8 June 2018 (UTC)

my question is, is it even true in the first place? The phrase "derivative of $f(v)$ with respect to $(m/T)$" isn't even well formed afaics, if you're going to consider $f$ a function of $m/T$, shouldn't you call it $f(m/T)$, or perhaps more properly $f(v,m,T)$? The claim merely seems to be that if you set $v^{2}=3kT/m$ (which kind of defeats the idea of $f$ being a function of $v$) then the value of $f(v=3kT/m)$ isn't going to depend on $(m/T)$.

Maybe I don't see the sigificance of this, or maybe it just isn't properly phrased, but I have to agree that the statement as it stands doesn't do much to illuminate the properties of the distribution. --dab (𒁳) 08:10, 20 June 2018 (UTC)

Vector notation in introduction

The introduction tries to explain how one transitions from a distribution function wrt the velocity's vector to its absolute value. This should be made clearer. Maybe using arrow notations for vectors would help? HerrHartmuth (talk)

Validity

The validity should be discussed. The distribution for the velocities is a marginal distribution of the canonical distribution function. We obtain it via integrating out all position dependencies from the canonical distribution. the equation is therefore also valid for interacting systems. Compare https://books.google.de/books?id=Hp-TBQAAQBAJ&lpg=PA131&dq=maxwell%20boltzmann%20distribution%20marginal%20distribution&hl=de&pg=PA131#v=onepage&q=maxwell%20boltzmann%20distribution%20marginal%20distribution&f=false Biggerj1 (talk) 09:03, 6 December 2019 (UTC)

The Boltzmann Police

We see that His name has been misspelt as "Boltzman" in a number of places on this talk page. We will let these slip by. However, be warned that The Boltzmann Police will quickly put to justice any criminal who attempts this sacrilege in a Wikipedia article. Boardhead (talk) 16:23, 14 February 2020 (UTC)

Small mistake in the expression for the kurtosis

For some reason the minus sign in the denonimator of the expression for the kurtosis is written in the source, but doesn't actually show up. — Preceding unsigned comment added by TheGam3lerInDebt (talk • contribs) 22:05, 9 December 2020 (UTC)

Energy per degree of freedom has wrong fraction : Mistake in §4.3 right below Equation (9)

In my understanding, the wiki pages has a fault. The energy per degree of freedom, is distributed as a chi-squared distribution with one degree of freedom,

https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0b957722f14b4f35ca02434218927e2ffff4d4

and cites [14] Laurendeau, Normand M. (2005). Statistical thermodynamics: fundamentals and applications. Cambridge University Press. p. 434. ISBN 0-521-84635-8., Appendix N, page 434

I just had a look at [14] , page 434.. I found an error in the denominator of the factor sqrt(epsilon * kT).

$${\frac {1}{\sqrt {\pi }}}{\sqrt {\frac {\epsilon }{kT}}}$$

That makes sense, as $f_{\epsilon }(\epsilon )$ should be unitless. so the fraction ${\frac {1}{\epsilon kT}}$ has the unit (1 by Energy *energy) which is definitly a fault.

I have double checked it with [14] Laurendeau, Normand M.

The formula for each energy per degree of freedom $\epsilon ={\frac {1}{2}}mv^{2}$ is:

$$f_{\epsilon }({\frac {\epsilon }{kT}})={\frac {2}{\sqrt {\pi }}}~{\sqrt {\frac {\epsilon }{kT}}}~\exp \left[-{\frac {\epsilon }{kT}}\right]$$

I found in reference [14] at another page:

[15] Laurendeau, Normand M. (2005). Statistical thermodynamics: fundamentals and applications. Cambridge University Press. p. 434. ISBN 0-521-84635-8., Section 15.3 Equation (15.25) , page 294

$$f_{\epsilon }(\epsilon )\,d\epsilon ={\frac {2}{\pi }}~{\sqrt {\frac {\epsilon }{~kT}}}~\exp \left[-{\frac {\epsilon }{kT}}\right]\,d({\frac {\epsilon }{kT}})$$

REMARK: Above: There is a artefact in the parameter of a (missing?) constant ${\frac {1}{kT}}$ in the function $f_{\epsilon }(\epsilon )$. The factor kT is a constant, so it might been omitted.

I will put my nose deeper in [14] Laurendeau, Normand. Willyfoobar (talk) 01:53, 1 January 2022 (UTC)

You just rewrote Eq.(9) in the dimensionless units (E/kT). In the following equation, $\epsilon$ is not E/kT! Evgeny (talk) 10:02, 2 January 2022 (UTC)

Discussion part B

discussing the equation below eq (9), please discuss if the requested change of the formula below (9) is valid and if it is possible to give this equation a label (likewise : 9b)

It is about the distribution of a single degree of freedom using the equilibrium theorem

I certainly know, that $\epsilon$ is not $E/kT$!

But in combination with the suggested change 8below) the dependency is indeed $f_{\epsilon }({\frac {\epsilon }{kT}})$.

This makes sense, in order to apply a chi-square distribution.

ARGUMENT B.1: used citation [14] differs to equation on wikipedia

The cited [14] Laurendeau, Normand. p. 434 , (taken as reference for this equation), states another dependency.

I just bought the eBook. , It says:

$$f_{\epsilon }({\frac {\epsilon }{kT}})={\frac {2}{\sqrt {\pi }}}~{\sqrt {\frac {\epsilon }{kT}}}~\exp \left[-{\frac {\epsilon }{kT}}\right]$$

THIS FORMULA should be in the WIKIPEDIA page

The cited Laurendeau, Normand. ([14] ) also defines at (another page) p.294 in equation (15.24) the expected:

$$\epsilon ={\frac {1}{2}}mv^{2}$$,

ARGUMENT B.2: Probabilty Density Function f(...) must be unitless

so $\epsilon$ has the UNIT of ENERGY.

Do we agree that $f_{\epsilon }(...)$ is a unitless PDF ?

BUT the current factor ${\sqrt {\frac {1}{\epsilon ~kT}}}$ is NOT unitless! It has the unit 1 by Energy.

This supports argument B.1

ARGMENT B.3 : parameter of f(....)

I believe we agree: $f_{\epsilon }(...)$ depends on $\epsilon$ and $kT$ ?

If we have consent in ARGUMENTs B.1 and B.2 , the factor in the equation from citation [14] is: ${\sqrt {\frac {\epsilon }{kT}}}$

So the overall parameter dependency is : ${\frac {\epsilon }{kT}}$

use as a Chi-squared distribution If we have consent in ARGUMENTs B.1 and B.2:

In order to make use of a Chi-squared distribution an argument substitution the dependency must be : $f_{\epsilon }({\frac {\epsilon }{kT}})$

Summary of Arguments

• As f(...) must be a unitless probability the factor before exp() is unitless, too.
• When using citation [14], it has to be cited correctly
• Certainly ONE can argue that $kT$ is constant and may an be striped off.

If you agree the ARGUMENT B.1, and B.2

• both parameter dependencies with and without the fraction kT are valid options.

The formula should have a label e.g. (9b) and should be

$$f_{\epsilon }({\frac {\epsilon }{kT}})={\frac {2}{\sqrt {\pi }}}~{\sqrt {\frac {\epsilon }{kT}}}~\exp \left[-{\frac {\epsilon }{kT}}\right]$$

I know, that this differs by a fraction of ${\frac {1}{\epsilon }}$ from the formula of this wikipage. Willyfoobar (talk) 16:09, 2 January 2022 (UTC)

I can only repeat that "your" equation is absolutely valid, but it is a trivial rewrite of Eq.(9). The dimension of any PDF is that of its argument in the power of minus one. So when you integrate over it, the result is the dimensionless unity. Hence f(v), f(E), and f(E/kT) all are in different units; only f(E/kT) is dimensionless – because E/kT is. The original equation has a different meaning. It can be derived from the one-dimensional M–B distribution for velocity. Evgeny (talk) 21:55, 2 January 2022 (UTC)

Discussion part C

So we agree that my formula is valid?

It might be that the equation above is a rewrite of Eq.(9).

But there is no wikipage dealing with the energy distribution of PDF of one degree of freedom.

I need to have a derivation for the CDF of energy distribution of a (ideal) diatomic gas.

Is it apropiate to create an extra wikipage or is it possible to add it in the page "M-B-distribution" ?

Can you decide how to proceed

I will start writing some content on my SANDBOX: http://en.wiki.x.io/wiki/User:Willyfoobar/sandbox — Preceding unsigned comment added by Willyfoobar (talk • contribs) 11:22, 3 January 2022 (UTC)

So we agree that my formula is valid? Yes, but trivial and so unnecessary.

It might be that the equation above is a rewrite of Eq.(9). Exactly.

But there is no wikipage dealing with the energy distribution of PDF of one degree of freedom. There is. This page exactly (not that it's well organized, though). Including the formula you tried to erase...

There is some information in this article about dealing with extra degrees of freedom. As I mentioned above, it does need some reshuffling of the contents, IMO. Evgeny (talk) 12:43, 3 January 2022 (UTC)

Unfortunate domination by constants

It is certainly useful to understand the Maxwell-Boltzmann distribution in terms of all the physical constants that affect it.

But to appreciate what type of distribution this is among probability distributions, it would be extremely valuable if all the constants could be consolidated into as few as possible so one can view the form that the distribution takes in terms of velocity or in terms of speed.

I hope someone knowledgeable about this subject can add some summary lines that express these distributions as simply as possible. 2601:200:C000:1A0:3C4F:AE96:D771:5385 (talk) 00:50, 29 August 2022 (UTC)

Inconsistent equations for distribution function of speeds

Near the beginning of the article under 'Distribution function' it reads

$$f(v)~d^{3}v={\biggl [}{\frac {m}{2\pi kT}}{\biggr ]}^{\frac {3}{2}}\,\exp \left(-{\frac {mv^{2}}{2kT}}\right)~d^{3}v,$$

i.e.

$$f(v)={\biggl [}{\frac {m}{2\pi kT}}{\biggr ]}^{\frac {3}{2}}\,\exp \left(-{\frac {mv^{2}}{2kT}}\right)~$$

whereas further down under 'Distribution for the Speed' ( http://en.wiki.x.io/wiki/Maxwell%E2%80%93Boltzmann_distribution#Distribution_for_the_speed ) it reads

$f(v)={\sqrt {\frac {2}{\pi }}}\,{\biggl [}{\frac {m}{kT}}{\biggr ]}^{\frac {3}{2}}v^{2}\exp \left(-{\frac {mv^{2}}{2kT}}\right)$ 140.228.40.161 (talk) 16:12, 21 April 2024 (UTC)