# Applying the Maxwell–Boltzmann statistics to astrophysical objects

In statistical mechanics, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in thermal equilibrium, when the temperature is high enough and density is low enough to render quantum effects negligible.

1. Is it possible to apply Maxwell–Boltzmann statistics to objects as large as nebulae; globular clusters or galaxies, that is, treating stars as Maxwell-Boltzmann particles; or even the universe as as whole, treating galaxies or clusters of galaxies as Maxwell-Boltzmann particles?

2. Can the Universe be considered in thermal equilibrium? Or does an expanding Universe imply non-equilibrium?

• 1) and 2) are really different topics. You should consider posting this as two different questions. Jul 2, 2011 at 1:05
• Jul 2, 2011 at 1:13
• metzgeer, if my edit of the question wasn't what you meant, I can roll it back. Jul 2, 2011 at 15:33
• @andrew - well maybe, I have no idea what a maxwellion is, is it a noun for a maxwell-boltzmann distribution? Jul 3, 2011 at 4:12
• Re: "Can the Universe be considered in thermal equilibria?". Why equilibria? Why isn't it equilibrium? Jul 3, 2011 at 8:01

It takes a lengthy proof, but Lyman Spitzer shows in the second chapter of Physical Processes in the Interstellar Medium (the standard text in interstellar matter studies) that the velocity distribution of interstellar gas particles (which is what forms nebulae) is very nearly Maxwellian - the deviation is less than 1%.

Other larger systems, probably not so much - Maxwell-Boltzmann statistics work best when kinetic energy is dominant in a system. But I don't know much about the topic, so that is a guess.

• Thanks Spencer, this was the information I was looking for, I now have a copy of Spitzers work. Jul 2, 2011 at 13:03

This is well outside my area of expertise but I believe ou can apply Maxwell–Boltzmann statistics, at least loosely to clusters of galaxies, clusters of stars and in many cases the gas molecules in nebulae. For clusters this is known as the Virial Theroem and Andrew described it quite well in the question Stellar Viscosity in Galaxies.

For the nebular gases, they are many times in thermodynamic equalibrium and so Maxwell–Boltzmann statistics apply directly. (Of course there are times that they aren't, the trick is deciding which case you are in.). If you were asking about applying to collections of nebula, the answer is, I believe, no.

As for applying it to the Universe, I think that is out as well. As you stated in your question, I believe the expansion of the Universe precludes thermodynamic equalibrium for the entire system.

Ok, I dug out our old stat mech/thermo textbook. YES, Maxwell-Boltzmann statistics definitely apply to stars in a globular cluster or galaxy, but you have have to pare back the results to the absolute most general.

Sears and Salinger go through an excellent derivation of Maxwell-Boltzmann statistics as well as the Maxwell-Boltzmann distribution function. The most general results leave the distribution function as a function of completely unspecified energy levels, and everything I saw looks like it is absolutely applicable to big astrophysical things (e.g. stars) clustering together into even bigger astrophysical things (e.g. globular clusters, galaxies).

I had some qualms, though. The energy levels are quantized in their treatment. And the particles are non-interacting. However, it looks like M-B stats are still applicable. To put the icing on the cake, later in the text they go through a derivation of the usual ideal thermodynamic piston, except this one is IN A GRAVITATIONAL FIELD, BABY! They assume a uniform field, and use that to derive the Newtonian hydrostatic equation (my link is for the general relativistic generalization of the Newtonian; the Wikipedia link for Newton's version was unsatisfactory) from a pure thermodynamic point of view, which, as a professional physicist, made me nearly literally stand up and clap.

So- at the very least, I can see someone assuming that stars are floating around passively, i.e. not interacting gravitationally with each other, but subject to a magical, arbitrary gravitational field matching the real-life solution. They would then derive a purely thermodynamic equation showing the distribution of stars in that magical gravitational field. Then, they would calculate the gravitational field of stars distributed how they just calculated. Then, they would show that the generated gravitational field matches the original, magical, arbitrary field.

This is called generating a self-consistent solution. I'd have to check with a mathematician, but I believe the equations you would use have only a single solution, so even if you kinda-sorta cheated in solving them, your solution is still The solution.

If you were even smarter, you might be able to generalize Sears and Salinger's gravity piston to self-gravitating particles and derive the solution directly. Not sure that's possible, but maybe.

I don't think that an expanding Universe can be considered in thermal equilibrium, except on short time scales. I mean, the CMB started out in short-term equilibrium, then just look at what happened!

• My original confusion in whether M-B was applicable came from the traditional treatment of the subject moving directly into the ideal gas assumption after the general results, and finding specific results that would not apply to stars in a cluster, but that's just a special case. Jul 3, 2011 at 21:45

You can apply Maxwell-Boltzmann statistics to Jeans escape - the escape of low-weight atmospheric volatiles. Basically, there is a timescale for which any molecule can expect to reach the right-end tail of the Maxwell distribution and escape the atmosphere of any astrophysical object (so you can describe a timescale for each set of [molecule,planet] pairings). For gases such as hydrogen, the timescale for Jeans Escape is small enough such that you can't expect hydrogen gas to stay on planets like Earth for very long - which is why Earth doesn't have them. For gases such as oxygen, the timescale for Jeans Escape on Earth is extremely large - but still low for low mass objects like the moon.

Here's a good powerpoint describing Jeans escape of hydrogen: http://www.geosc.psu.edu/~jfk4/Abiol_574/Lectures/Lecture%209_Hydrogen%20escape.ppt

And an excellent pdf: http://faculty.washington.edu/dcatling/Catling2009_SciAm.pdf

• I can see how Jeans Escape applies to planetary processes. Would it also apply to deep space gases? I suppose it would, but over a much larger time frame. Jul 6, 2011 at 0:53
• It can basically apply to any gases that are gravitationally bound. So it might be relevant to the extended outer atmospheres of stars - although the stars that it would be most relevant on (like red supergiants) are also the types of stars that have the shortest lifespans Jul 6, 2011 at 0:59
• I'm not totally sure about deep space gases though - hopefully something else can address this or we can ask on Quora/Physics Forums/Reddit Jul 6, 2011 at 1:00