The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval. Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 1021 collisions per second. Thus Einstein was led to consider the collective motion of Brownian particles. He showed that if ρ(x, t) is the density of Brownian particles at point x at time t, then ρ satisfies the diffusion equation:
where D is the mass diffusivity.
Assuming that N particles start from the origin at the initial time t=0, the diffusion equation has the solution
This one is going to be fun. What did Einstein tell us? He identified some approximations in the relative movement of two, mixed chemical solutions. These approximations let us use Isaac's Rules of grammar.
Can we define how the actual molecules do it without Isaac's rules? Can we eliminate the Greek symbols and dump the t thing? We sure can. I am working on it, but likely I will only get us part way through as I am a hack. This is a work in progress, I am going to dump the e thing and make finite log, replace the t thing with the fine structure spectrum, and let the pi thing fall out as a result of local action by the bubbles of the universe.
I think we will find molecules at the markets, and they keep marching further into the market where trades expand mostly as the square of the market 'quant' and they rarely have to wait in line. I am going to do it with the Frank Lucas rules of grammar.
It was not until 1918 that a proof (using hyperelliptic functions) was found for this remarkable fact, which has relevance to the bosonic string theory in 26 dimensions. Elementary proofs have been more recently published.
Uh Oh, he only does 2D boson's. How are we going to fix that?Hopefully Andrey Markov had some ideas. Do geodesic integers exist? Bubbles make spheres easily enough but they need to connect them. Or we can make unit ellipsoids using Mr. Markov's triples. Where are the brilliant mathematicians who work this out? Let me go look for them and save myself time and effort.