I have heard many people say that the existence of atoms is proven by Brownian motion. Now, I understand how an atomic theory would suggest the existence of Brownian motion. However, who is to say that there is not another theory for what our world is composed of that can also predict Brownian motion (as well as the other phenomena predicted by atoms)? Of course, I am not sure what that theory would be, but I am wondering how one could say that Brownian motion proves the existence of atoms.
[Physics] How does Brownian motion prove the existence of atoms
atomic-physicsatomsbrownian motion
Related Solutions
So, whenever I want to find a nice introduction to a concept in physics, I check the American Journal of Physics, as it is full of articles with clever descriptions of phenomenon appropriate for presentation in university courses. In this case, this yields many results. In particular, I found the following three articles very helpful:
- The mathematics of Brownian motion and Johnson Noise - Daniel Gillepsie [doi][pdf]
- Two Models of Brownian Motion - David Mermin [doi][pdf]
- Fluctuation and Dissipation in Brownian Motion - Daniel Gillepsie [doi][pdf]
For completeness, I'll give my version of a modern approach to describing brownian motion, where I will borrow heavily from the above.
If you want to think in terms of Newton's laws, we will take an approach that in spirit is the same that Langevin gave three years after Einstein's paper (a translation of which also appeared in the American Journal of Physics [doi] [pdf]), that gives the same result.
If we imagine a pollen particle suspended in a liquid, we can assume that the forces on the pollen particle are given by a dissipative friction force, and some random jostling by impacts from the water molecules, we'll write
$$ m \dot v = -\gamma v + f \Gamma(t) $$ where $\gamma$ is the drag coefficient, $f$ is some constant we will have to determine, and $\Gamma(t)$ represents a random gaussian process. That is, we will assume the effect of all of the jostling by the water molecules amounts to drawing a random variable at every instant of time. Then, the next step of the argument is to make this whole deal consistent with statistical mechanics, namely the equipartition theorem, so in particular, if we look at long times, we should have $$ \frac 12 m \left\langle v^2(\infty) \right \rangle = \frac 12 k T $$ or in words, the average kinetic energy at long times should be a half $kT$ if we are going to be consistent with statistical mechanics.
So, we need only compute the average fluctuations in our velocity for long times. You can follow the papers to see a detailed mathematical account, but for just a taste, we can get the answer from dimensional analysis.
We are interested in determining $ \langle v^2(\infty) \rangle $, and this answer should only depend on the parameters in our equation for the forces the particle feels, namely $m$, $\gamma$ and $f$. The dimensions of $m$ and $\gamma$ are easy to read off of the equation
$$ [m] = [M] \qquad [\gamma] = [M T^{-1}] $$
but what about that $f$? Well, it depends on the dimensions of our random gaussian noise term, which is a bit tricky. But, the way we tried to describe it, the noise was supposed to be completely uncorrelated in time, so though I didn't detail it, this means that
$$ \langle \Gamma(t) \Gamma(t') \rangle = \delta(t-t') $$ in detail. And since we know that $\int dt\, \delta(t) = 1$, we have the dimensions $$ [ \delta(t) ] = [ T^{-1} ] \qquad [\Gamma(t) ] = [T^{-1/2}] $$ which tells us that $$ [f^2] = [ M L T^{-3} ] $$ which seems funny, but it enables us to determine that $$ \langle v^2 \rangle \propto \frac{ f^2 }{ \gamma m } $$ and in particular, we will assume that the proportionality constant is 1, which using equipartition, gives us $$ m \langle v^2 \rangle = f^2/\gamma = k T $$ or $$ f = \sqrt{ \gamma k T } $$ if we had done all of the math properly, the real answer turns out to be $$ \boxed{ f = \sqrt{ 2 \gamma k T } } $$ which is pretty darn close.
The point of all of that, and of Einstein's original paper is that we've shown that the fluctuations ($f$) causes by the jostling of the unseen water molecules is directly related to the dissipation ($\gamma$) you can observe in ordinary fluid experiments. This is the major result of Einsteins and Langevin's papers. With a bit more work, we can relate this to the diffusion constant, which tells us how the root mean square position increases linearly with time:
$$ \langle x^2(\infty) \rangle = D t $$
doing our dimensional analysis again, we discover we need a relation of the form
$$ D \propto \frac{f^2}{\gamma^2} $$ or, getting rid of this silly $f$ thing, using our other result above $$ \boxed{ D = \frac{ kT }{\gamma } } $$ which turns out to be right even if you do the math right (the proportionality constant is 1).
This was the actual formula Einstein got famous for in his paper, relating the diffusion constant, something you can measure in experiment, to the drag coefficient, something you can also measure from a different set of experiments. Giving in the end a quantitative theory of brownian motion that worked to help solidify the atomic hypothesis and some of the early results of statistical mechanics.
Best Answer
Einstein's mathematical model of brownian motion furnished strong support of the atomic model but did not furnish airtight proof of its uniqueness (that is, the nonexistence of alternative models) at the time it was proposed.
It is worthwhile to note that it wasn't his objective to logically exclude the possibility of alternative models but rather to demonstrate that the atomic model furnished a quantitatively consistent explanation of the phenomenon.
His work was an important piece of a larger puzzle, contributed to by a variety of other researchers in the field, and once his piece of it was in place, the rest of the picture came into better focus- and it became much more difficult to successfully argue against the atomic hypothesis.