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.
