Accuracy of the Boltzmann equation

Question

I have had this question for some time now. Hopefully someone can answer it.

I know that the Boltzmann equation is widely regarded as a cornerstone of statistical mechanics and many applications have been explored with a linearized version.

I also know that it's extremely hard to obtain exact solutions, which has started a considerable amount of investigation looking for an equally good (or acceptable) formalism to analyse systems that otherwise would be impossible or would take a great deal of computational resources to obtain a solution using the Boltzmann equation.

In spite of this, I never heard a precise description about the degree of accuracy (in comparison with experiments) that can be drawn from the Boltzmann equation. Obviously, I expect that accuracy depends on the system at hand, however, it would be great to hear about some specific examples.

Recommended readings would also be appreciated.

Thanks in advance.

Answer 1

The Boltzmann equation is an approximation. It is obtained by closing the so-called BBGKY hierarchy that links N-particle distribution functions to N+1 particle distribution functions. The closing is obtained by introducing an assumption, the 'molecular chaos' assumption. In short, molecular chaos asumes the absence of correlation in the incoming state for particles that oollide. Detailed molecular dynamics simulations of many-particle model systems have revealed that the molecular chaos assumption indeed fails to describe quantitatively the transport coefficients of dense many-particle systems due to the presence of 'long-time tails'. In other words: correlations between particles don't die out exponentially. Interestingly, the lower the number of dimensions, the stronger this effect. The Boltzmann equation is not a good approximation to 2D liquids.

Answer 2

the discrepancy between the Boltzmann equation and the experiment may always be blamed upon the experiments. If one does things right, the Boltzmann equation is theoretically completely exact (issues will be discussed below). However, it's pretty hard to measure the distribution functions "directly", especially if you want the full dependence both on momenta and positions so one may say that the "weak link" is always on the experimental side (after errors are fixed).

The Boltzmann equation is mathematically accurate and a consequence of other laws of physics that have been tested pretty thoroughly. But there are a few subtleties here:

1. The Boltzmann equation in its general form contains a collision term which is a completely general function of all the variables - just like the distribution function itself. So unless one specifies what the collision term is, the Boltzmann equation is tautologically true for some collision term and may be viewed as a definition of the collision term.

2. The collision term which is a function of all variables is often approximated by the molecular chaos Ansatz - which depends on the distribution functions of one particle in a bilinear way; and on a function describing the rate of collisions at a given place and given momenta. Of course, one has to carefully calculate - or indirectly measure - the latter function. Even when one does it, the Boltzmann equation with the molecular chaos condition included may start to fail when the molecules are not in real chaos. The closer one is to an "ideal gas", the more accurately the molecular chaos assumption holds, but of course, the whole point of the equation is that it also works for non-ideal gases and liquids.

(It is a purely technical assumption to make further calculations more specific; obviously, the fact that Boltzmann or others made the molecular chaos assumption, to make their calculations specific, doesn't weaken the derivations of the H-theorem i.e. the second law of thermodynamics. It holds even without the molecular chaos assumption - the assumption or its equivalent is just needed to calculate the increase of entropy and other things quantitatively.)

1. Boltzmann equation is a classical beast that neglects quantum mechanics; however, one may still interpret the distribution functions as the appropriate expectation values - and these expectation values in quantum mechanics still have to satisfy an equation of the Boltzmann form. So the Boltzmann equation is valid even for the calculation of processes in the heavily quantum mechanical context. For example, it's been used to calculate the abundance of elements after nucleosynthesis - the period up to 3 minutes after the Big Bang when mostly light elements were produced in the hot "nuclear liquid" that was filling the Universe at the time. The abundances - mostly Hydrogen, some percentage of Helium and Lithium etc. - numerically agree with the observed percentages which is a nontrivial evidence for The Big Bang Theory (that will be aired on CBS at least until the 7th season, it was just decided).

Best wishes Lubos

Answer 3

I provided a response to question in the same context here:

Experiments that measure the time a gas takes to reach equilibrium

However, specifically to your question, this is a good quick overview: http://homepage.univie.ac.at/franz.vesely/sp_english/sp/node7.html

Where you find out that with certain simplifying assumptions, the Boltzmann distribution is a solution to the Boltzmann equation.

Again, it is interesting to understand how boltzmann distribution are related to the Arrhenius equation: http://en.wikipedia.org/wiki/Arrhenius_equation

The fact that these theories underpin most of 20th century technology I think is testament enough to accuracy within relevant regimes.

Answer 4

@Robert Smith Well you're asking a question about the accuracy of an equation. The equation is created under certain assumptions about small objects sometimes called "particles" or "atoms". So if the equation accurately describes the system in question, then the behavior of the sytem should be described by solutions to the governing equations. So questions as to accuracy of the equations, is really revealed by whether the systems behavior follows the solutions to the equations.

Since the boltzmann's distribution is a solution to boltzmann's transport equation under certain conditions, it is the question of whether the system deviates from a boltzmann distribution under situations where it should hold that would indicate a situation where the boltzmann equation may no longer be valid.

So if you are still confused as how my answer relates to the question, it might be that you are not asking the right questions.

http://arxiv.org/PS_cache/astro-ph/pdf/9807/9807078v1.pdf