Statistical mechanics re-visited

Quite a while ago – in June and July 2015, to be precise – I wrote a series of posts on statistical mechanics, which included digressions on thermodynamics, Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics (probability distributions used in quantum mechanics), and so forth. I actually thought I had sort of exhausted the topic. However, when going through the documentation on that Stern-Gerlach experiment that MIT undergrad students need to analyze as part of their courses, I realized I did actually not present some very basic formulas that you’ll definitely need in order to actually understand that experiment.

One of those basic formulas is the one for the distribution of velocities of particles in some volume (like an oven, for instance), or in a particle beam – like the beam of potassium atoms that is used to demonstrate the quantization of the magnetic moment in the Stern-Gerlach experiment. In fact, we’ve got聽two聽formulas here, which are subtly – as subtle as the difference between v聽(boldface, so it’s a vector) and v (lightface, so it’s a scalar) 馃檪 – but fundamentally different:

velocity-distribution

Both functions are referred to as the Maxwell-Boltzmann density distribution, but the first distribution gives us the density for some v in the聽velocity聽space, while the second gives us the distribution density of the聽absolute value聽(or modulus) of the velocity, so that is the distribution density of the聽speed, which is just a scalar – without any direction. As you can see, the second formula includes a 4蟺路v2聽factor.

The question is: how are these formulas related to Boltzmann’s聽f(E) = C路e鈭抏nergy/kT聽Law? The answer is: we can derive all of these formulas – for the distribution of velocities, or of momenta – by clever substitutions. However, as evidenced by the two formulas above, these substitutions are not always straightforward. So let me quickly show you a few things here.

First note the two formulas above already include the e鈭抏nergy/kT聽function if we equate the energy E with the聽kinetic聽energy: E = K.E. = m路v2/2. Of course, if you’ve read those June-July 2015 posts, you’ll note that we derived Boltzmann’s Law in the context of a force field, like gravity, or an electric potential. For example, we wrote the law for the density (n = N/V) of gas in a gravitational field (like the Earth’s atmosphere) as n = n0e鈭扨.E./kT. In this formula, we only see the potential energy: P.E. = m路g路h, i.e. the product of the mass (m), the gravitational constant (g), and the height (h). However, when we’re talking the distribution of velocities – or of momenta – then the聽kinetic聽energy comes into play.

So that’s a first thing to note: Boltzmann’s Law is actually a whole set聽of laws. For example,聽the frequency distribution of particles in a system over various possible states, also involves the same exponential function: F(state)聽鈭澛e鈭扙/kT. E is just the聽total聽energy of the state here (which varies from state to state, of course), so we don’t distinguish between potential and kinetic energy here.

So what energy concept should we use in that Stern-Gerlach experiment? Because these potassium atoms in that oven – or when they come out of it in a beam – have kinetic energy only, our聽E = m路v2/2 substitution does the trick: we can say that the potential energy is taken to be zero, so that all energy is in the form of kinetic energy. So now we understand the e鈭抦路v2/2kT聽function in those聽f(v) and f(v) formulas. Now we only need to explain those complicated coefficients. How do we get these?

We get them through clever substitutions using equations such as:

fv(v)路dv聽 = fp(p)路dp

What are we writing here? We’re basically combining two normalization conditions: if fv(v) and fp(p) are proper probability density functions, then they must give us 1 when integrating over their domain. The domain of these two functions is, obviously, the velocity (v) and momentum (p) space. The velocity and momentum space are the same mathematical聽space, but they are obviously聽not聽the same聽physical聽space. But the two physical spaces are closely related: p = m路v, and so it’s easy to do the required transformation聽of variables. For example, it’s easy to see that, if E = m路v2/2, then E is also equal to E = p2/2m.

However, when doing these substitutions, things get tricky. We already noted that p and v are vectors, unlike E, or p and v – which are scalars,or聽magnitudes. So we write: p = (px, py, pz) and |p| = p, and聽v = (vx, vy, v z) and |v| = v. Of course, you also know how we calculate those magnitudes:

magnitude

Note that this also implies the following: pp = p2聽= px2聽+ py2聽+pz2聽= p2. Trivial, right? Yes. But have a look now at the following differentials:

  • d3p
  • dp
  • dp = d(px, py, pz)
  • dpx路dpy路dpz

Are these the same or not? Now you need to think, right? That d3p and dp are different beasts is obvious: d3p聽is, obviously, some infinitesimal聽volume, as opposed to dp, which is, equally obviously, an (infinitesimal) interval. But what volume exactly? Is it the same as that dp = d(px, py, pz) volume, and is that the same as the dpx路dpy路dpz聽volume?

Fortunately, the volume聽differentials聽are, in fact, the same – so you can start breathing again. 馃檪 Let’s get going with that d3p聽notation for the time being, as you will find that’s the notation which is used in the Wikipedia article on the Maxwell-Boltzmann distribution聽– which I warmly recommend, because – for a change – it is a much easier read than other Wikipedia articles on stuff like this. Among other things, the mentioned article writes the following:

fE(E)路dE = fp(p)路d3p

What is this? Well… It’s just like that fv(v)路dv聽 = fp(p)路dp equation: it combines the normalization condition for both distributions. However, it’s much more interesting, because,聽on the left-hand side, we multiply a density with an (infinitesimal) interval聽(dE), while on the right-hand side we multiply with an (infinitesimal) volume (d3p). Now, the (infinitesimal) energy interval dE must, obviously, correspond with the (infinitesimal) momentum聽volume聽d3p. So how does that work?

Well… The mentioned Wikipedia article talks about the “spherical symmetry of the energy-momentum dispersion relation” (that dispersion relation is just E = |p|2/2m, of course), but that doesn’t make us all that wiser, so let’s try a more聽heuristic聽approach.聽You might remember the formula for the volume of a spherical聽shell, which is simply the difference between the volume of the outer sphere聽minus聽the volume of the inner sphere: V = (4蟺/3)路R3聽鈭 (4蟺/3)路r3聽= (4蟺/3)路(R3聽鈭 r3). Now, for a very thin shell of thickness 螖r, we can use the following first-order approximation:聽V = 4蟺路r2路螖r.聽In case you wonder, I hereby copy a nice explanation from the Physics Stack Exchange site:

approximation

Perfect. That’s all we need to know. We’ll use that first-order approximation to re-write d3p聽as:

d3p聽= dp = 4蟺路|p|2路d|p| = 4蟺路p2路dp

Note that we’ll have the same formula for d3v, of course: d3v聽= dv = 4蟺路|v|2路d|v| = 4蟺路v2路dv, and also note that we get that same 4蟺路v2聽factor which we mentioned when discussing the f(v) and f(v) formulas. That is not a coincidence, of course, but – as I’ll explain in a moment – it is聽not聽so easy to immediately relate the formulas. In any case, we’re now ready to relate dE and dp so we can re-write that d3p formula in terms of m, E and dE:

substitution-2

We are now – finally! – sufficiently armed to derive all of the formulas we want – or need. Let me just copy them from the mentioned Wikipedia article:

momenta

energy

velocity

As said, you’ll encounter these formulas regularly – and so it’s good that you know how you can derive them. Indeed, the derivation is very straightforward and is done in the same article: the tips I gave you should allow you to read it in a couple of minutes only. Only the density function for velocities might cause you a bit of trouble – but only for a very short moment: just use the p = m路v equation to write聽d3p as聽d3p = 4蟺路p2路dp聽= 4蟺路m2路v2路m路dv = 4蟺路m3路v2路dv = m3路d3v, and you’re all set. 馃檪

Of course, you will recognize the formula for the distribution of velocities: it’s the聽f(v) we mentioned in the introduction. However, you’re more likely to need the f(v) formula (i.e. the probability density function for the speed) than the f(v) function.聽So how can we derive get the聽f(v) – i.e. that formula for the distribution of speeds,聽with the 4蟺路v2聽factor – from the f(v) formula?

Well… I wish I could give you an easy answer. In fact, the same Wikipedia article suggests it’s easy – but it’s not. It involves a transformation from Cartesian to polar coordinates: the volume element dvx路dvy路dvz聽is to be written as v2路sin胃路dv路d胃路d蠁. And then… Well… Have a look at this link. 馃檪 It involves a so-called聽Jacobian transformation matrix. If you want to know more about it, then I recommend you read some article on how to transform distribution functions: here’s a link to one of those, but you can easily google others. Frankly, as for now, I’d suggest you just accept the formula for f(v) as for now. 馃檪 Let me copy it from the same article in a slightly different form:density-formulaNow, the final thing to note is that you’ll often want to use so-called聽normalized velocities, i.e. velocities that are defined as a v/v0聽ratio, with v0聽the聽most probable聽speed, which is equal to聽鈭(2kT/m). You get that value by calculating the df(v)/dv derivative, and then finding the value v = v0聽for which df(v)/dv = 0. You should now be able to verify the formula that is used in the mentioned MIT version of the Stern-Gerlach experiment:mit-formulaIndeed, when you write it all out – note that 蟺/蟺3/2聽= 1/鈭毾 馃檪 – you’ll see the two formulas are effectively equivalent.聽Of course, by now you are completely formula-ed out, and so you probably don’t even wonder what that f(v)路dv product actually stands for. What does it聽mean, really? Now you’ll sigh: why would I even聽want聽to know that? Well… I want you to understand that MIT experiment. 馃檪 And you won’t if you don’t know what f(v)路dv actually represents.聽So think about it. […]

[…] OK. Let me help you once more. Remember the normalization condition once again: the integral of the whole thing – over the whole range of possible velocities – needs to add up to 1, so聽f(v)路dv is really the聽fraction聽of (potassium) atoms (inside the oven) with a velocity in the (infinitesimally small) dv interval. It’s going to be a聽tiny聽fraction, of course: just a tiny bit larger than zero. Surely聽not聽larger than 1, obviously. 馃檪 Think of integrating the function between two values – say v1聽and v2聽– that are pretty close to each other.

So… Well… We’re done as for now. So where are we now in terms of understanding the calculations in that description of that MIT experiment? Well… We’ve got the meat. But we need a lot of other ingredients now. We’ll want formulas for the聽intensity聽of the beam at some point along the axis measuring its聽deflection聽from its main direction. That axis is the聽z-axis. So we’ll want a formula for some I(z) function.

Deflection? Yes. There are a lot of steps to go through now. Here’s the set-up:set-upFirst, we’ll need some formula measuring the聽flux聽of (potassium) atoms coming out of the oven. And then… Well… Just have a look and try to make your way through the whole thing now – which is just what I want to do in the coming days, so I’ll give you some more feedback soon. 馃檪 Here I only wanted to introduce those formulas for the distribution of velocities and momenta, because you’ll need them in other contexts too.

So I hope you found this useful. Stuff like this all聽makes it somewhat more real, doesn’t it? 馃檪 Frankly, I think the math is at least聽as fascinating as the physics. We could have a closer look at those distributions, for example, by noting the following:

1. The probability density function for the momenta is the product of three normal distributions. Which ones? Well…聽聽The distribution of px, py聽and pz聽respectively: three normal distributions whose variance is equal to mkT. 馃檪

2. The fE(E) function is a chi-squared (蠂2) distribution with 3 degrees of freedom. Now, we have the equipartition theorem (which you should know – if you don’t, see my post on it), which tells us that this energy is evenly distributed among all three degrees of freedom. It is then relatively easy to show – if you know something about 蠂2聽distributions at least 馃檪 – that the energy per degree of freedom (which we’ll write as 蔚 below) will also be distributed as a chi-squared distribution with one degree of freedom:chi-square-2This holds true for any number of degrees of freedom. For example, a diatomic molecule will have extra degrees of freedom, which are related to its rotational and vibrational motion (I explained that in my June-July 2015 posts too, so please go there if you’d want to know more). So we can really use this stuff in, for example, the theory of the specific heat of gases. 馃檪

3. The function for the distribution of the velocities is also a product of three independent normally distributed variables – just like the density function for momenta. In this case, we have the vx, vy聽and vz聽variables that are normally distributed, with variance kT/m.

So… Well… I’m done – for the time being, that is. 馃檪聽Isn’t it a privilege to be alive and to be able to savor all these little wonderful intellectual excursions? I wish you a very nice day and hope you enjoy stuff like this as much as I do. 馃檪