home about github

Guessing Physical Laws

On mobile devices, read the page in horizontal/"landscape" mode. Otherwise math may not fit the screen.

Theoretical physics is a precise business. You choose your principles carefully, put them in to mathematical form and use rigorous logical deduction to infer consequences. Or perhaps you take a hithertho unexplained phenomenon and start from a known physical theory, using judicious approximations and clever calculation techniques to produce a brilliant explanation for its features. You might even write a computer program to apply numerical methods to theories and get your results that way. Either way, it’s clearly a job for the logical goober! That’s how you should do physics.

Well, except if you just want to play a guessing game. I mean…

  • Bohr deduced his famous atomic model by guessing physical principles that were totally wrong, he managed to get the right result anyway;

  • Maxwell used gears and other mechanical devices to explain his theory of electromagnetism, and those turned out to be totally useless;

  • Modified Newtonian dynamics was originally based on just guessing a generic form for Newton’s law of gravity;

  • Dirac derived his equation using mathematical methods and simply guessed the positron solutions were actually real particles

and on and on it goes.

So, hell, it’s pretty useful to know how to just wing it. Back-of-the-envelope calculations and leaps of logic abound in physics departments everywhere, and great discoveries are often made just as much by inspired guesswork as they are by logic. Still, there’s rules even to leaps; no matter how magical Michael Jordan’s jumps look, he can’t actually fly. Well, I don’t think so anyway. To be fair, a few of the videos I’ve seen seem to defy the laws of physics – for the sake of argument, suppose he doesn’t know how to fly to make my point make sense. Magical guesswork isn’t actually magical!

In this text, I’ll go over a few methods to just guess, using simple physical intuition, one particular example of a physical law: the Maxwell-Boltzmann speed distribution.

What the Hell Do You Know, Anyway?

Our problem is to find the distribution of the speed of gas molecules in a room full of molecules. What is it that we know about this problem?

Well, any schoolboy knows that rooms feel different at different temperatures. Even a 5 year old pipsqueak can tell you that temperature differences are caused by the speed of the molecules whizzing about in the room. You can guess that the kind of the molecules matters, too – different speeds for different types. And you probably already knew in your mother’s womb that distributions are normalized; that is, if we have a speed distribution, then

\[\int f(v) dv = 1\]

with $f(v)$ our mysterious distribution. We can perhaps make a further guess: presumably there’s nothing special about any of the directions – the distribution looks the same in $x$, $y$ and $z$ directions, so we can deal with them separately.

Well, you say, without further information this doesn’t amount to much. Unless I can use the principles I learned in my statistical physics class, there’s no way to make progress!

Or that’s what you would say if you were a pathetic little weakling. Are you a pathetic weakling? No? Then let me introduce you to dimensional analysis.

There’s Rules To Combining Quantities

You see, we just identified a number of things a speed distribution probably depends on:

  • The actual speed of the particles (obviously, since it’s a speed distribution)

  • The kind of molecule

  • The temperature of the room

The first one is evident, but the other two aren’t. Let’s talk about them.

First of all, we’re not really interested in the kind of molecule in the room, just some property of it. Unless you’re feeling confident and think you can solve this problem by starting from the quantum mechanics of molecules (if so, good luck, see you in 5 years). No, what we really want is just the inertia of the molecule, how hard it is to accelerate. In principle, if there were enough molecules in the room, or if they were big enough, then we might have to care about intermolecular interactions, but let’s just suppose they’re far enough apart to not really interact in a meaningful way. So we want the mass of the molecules.

As for the temperature, I already told you that even a schoolboy knows temperature differences are measured by the speed – or more accurately, the kinetic energies – of the particles. So let’s simply call the quantity of interest here "energy", for that is in fact what it is.

What now? We have three quantities, $v, m$ and $E$. We also know the normalization of the distribution. But this normalization gives us our first clue.

Notice how the end result of $\int f(v) dv$ is a pure number – no dimension? Let’s call the units of mass $[m] = M$, units of time $T$ and units of length $L$ (these could be anything – meters, inches, a fruit fly’s average wingspans – as long as you keep your units consistent). Then we know that $[f(v)] = [1/v] = T/L$, because $[dv]=L/T$. See how that works? The units of $f$ have to cancel out the units of integration measure $dv$, since the end result 1 doesn’t have units.

So we’re looking for a distribution that has units of $T/L$. How could we combine our quantities $m$, $E$ and $v$ to get units like that? Notice that $[E] = ML^2/T^2$ Think about it for a while, come up with options. Here are the easiest options I came up with:

\[\begin{align}f(v) &\propto 1/v g\bigg(\frac{v^2m}{E}\bigg)\\ f(v) &\propto \sqrt{\frac{m}{E}}g\bigg(\frac{v^2m}{E}\bigg) \\ f(v) &\propto \frac{vm}{E}g\bigg(\frac{v^2m}{E}\bigg)\end{align}\]

The function $g$ is unknown, but must be a function of dimensionless quantity, since $1/v$ and $\sqrt{\frac{m}{E}}$ and $\frac{vm}{E}$ already have the right dimensions (check for yourself!). The simplest dimensionless quantity you can construct from $v$, $m$ and $E$ is $v^2m/E$.

Alright, is there any way to eliminate any of these possibilities using our intuition? How do you think the distribution of the speeds should behave?

Well, I don’t know about you, but I don’t think that the distribution blowing up as $v\rightarrow 0$ seems very reasonable. I mean, maybe the slower speeds are more likely, but is that prefactor really supposed to be divergent? How would you even find a reasonable $g$ that would still integrate to a finite number over the infinite interval $v\in [0, \infty)$?

As for the third one, it suggests that the distribution is proportional directly to the speed and mass of the particles. Does it seem likely that $f(0) = 0$? Really, just a straight up zero at the origin? Maybe it’s possible – we can keep it in mind.

However, the middle one is the least offensive choice for at least my sensibilities (I also happen to know this guess produces the right answer, so my "intuition" is greatly aided by foreknowledge!) Since the dimensionful part doesn’t contain $v$, there’s no immediate pathologies that jump out.

Let’s go with the middle guess there. Our distribution is of the form (with $\alpha$ and $\gamma$) some yet to be determined constants)

\[f(v) = \alpha \sqrt{\frac{m}{E}} g\bigg(\gamma \frac{v^2m}{E}\bigg)\]

We now need to find an appropriate $g$. It has to be some function that is integrable from 0 to $\infty$. Presumably, the speeds closer to 0 are way more likely than the tail end – we don’t have too many infinitely fast particles flying about, that would just be straight up painful. Try to find some functions that satisfy that property. Go on, I’ll wait.

You can probably find a bunch. You can plot them to see what they would look like, just pick some easy values for $m$ and $E$. However, we now remember the Central Limit Theorem, which says that everything is always the damn exponential distribution (well, it doesn’t quite say that, but close enough), so that

\[g\bigg(\frac{v^2m}{E}\bigg) = e^{-\frac{\gamma v^2m}{E}}\]

Plot this – it certainly seems reasonable, right?

Supposing we’re right, is there some way to go even further? Can we guess $\alpha$ and $\gamma$?

Well, notice what we have in the exponential function: $mv^2$. Does that look like any form of energy you know of?

Right, it’s the kinetic energy of a particle if we take $\gamma = \frac{1}{2}$. So let’s plug that in! And, lucky for us, we now remember the normalization of the distribution:

\[\begin{align}\int _0^\infty f(v)dv &=\int _0^\infty\alpha \sqrt{\frac{m}{E}} g\bigg( \frac{v^2m}{2E}\bigg)dv = 1 \\ &\implies \alpha \frac{1}{2}\sqrt{2\pi } = 1 \\ &\implies \alpha = \sqrt{\frac{2}{\pi }}\end{align}\]

So, our answer is

\[f(v) = \sqrt{\frac{2}{\pi }} \sqrt{\frac{m}{E}}e^{-\frac{\gamma v^2m}{E}}\]

which, wouldn’t you know it, is exactly right.

You Cheated!

"Ah yes", you say, "of course you’re able to do this because you cheated! You already knew the end result, and used that as a guide! You can’t use this for anything real!"

First of all, who the hell do you think you are, assaulting me with the truth??

It’s not quite that simple, though. As I said, real physicists making real discoveries have used various forms of guesswork all throughout history (see Anthony Zee’s book "Fly by Night Physics" for plenty of examples of exactly this sort of historical reasoning). Order of magnitude estimates, dimensional analysis, intuitively inspired approximations – all of these are tricks you can find in the literature that have really been used for research, not just when you know the end result.

It might also help you if you’re a student. Suppose you only very vaguely remember some formula or end result of a calculation, and it’s asked on the exam. Well, if you mostly remember it – like was the case for me and the Maxwell-Boltzmann distribution when I did this calculation – you can guess your way to the right solution using this kind of reasoning. Mr. Zero Points turns in to Mr. Partial Credit.

What are the principles we used here?

  1. Identify the relevant quantities at play; in our case, mass, energy, speed.

  2. Make the necessary approximations. We implicitly approximated that the particles are non-relativistic, and we don’t care about intermolecular forces or quantum mechanical effects; basically, it’s a normal room full of something like air.

  3. Perform dimensional analysis on the quantity you’re interested in calculating. Find any combinations of the quantities that you identified that give the right units and satisfy any other constraints you identified. Eliminate the combinations that don’t fit the constraints; if more than one combination remains, pick the one that seems right to you based on your intuition.

  4. Try to fix any undetermined constants by some bullshit argument if possible; if not, just leave them there.

It is clear that this sort of reasoning can’t give you the same certainty as a logical deduction from well-justified principles. It’s equally clear that it’s far easier to use this method than to come up with the correct fundamental principles for a problem you’re not all that familiar with. If you got it wrong – eh, it happens, whatever.