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I am not very knowledgable with respect to physics, I come from the math SE. I was wondering about Brownian motion and how close is the model to the phenomenon that started it all: the movement of a particle of pollen in a glass of water.

What I'm struggling to get a feel of is to what extent is ("theoretical") Brownian motion real.

Mathematically it's been established that Brownian motion is (1) nowhere differentiable and (2) continuous (and (3) has independent increments). Obviously the movement of a small particle is also continuous because the particle can't teleport, but physically is it possible for it to satisfy condition (2), that for any positive length of time it has been hit an infinite number of times by the water molecules?

Because that's what the non differentiability is. Otherwise, if the particle was hit $n<\infty$ times in the interval $[t,t+\delta]$, the movement would be piece-wise differentiable (with, like, $n+1$ pieces), and in physical terms, it would be rectilinear between two consecutive hits (or a polynomial if we consider gravity and other forces... still differentiable). But then for the event {the particle is hit} to happen a countable but infinite number of times in an interval $[t,t+\delta]$, and since there is a finite number of molecules in the glass of water, there must be at least one molecule that hits the particle an infinite number of times in the (arbitrarily!) short interval $[t,t+\delta]$.

So how is it possible for one (or more) molecule-s to hit an object infinitely many times in a short period of time?

John Cataldo
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    I would say it is just an approximation of small enough steps by infinitesimal steps to apply integral calculus. Gas molecules have collision frequency typically 10^10 / sec. The frequency for the pollen particle in condensed phase is much higher. – Poutnik May 28 '19 at 07:23
  • As @Poutnik, in the mathematical construction of the Brownian motion, "infinite" has been put there instead of the physical "many", or "more than observable". It works well for milimeter-micrometer scale, it does not work when you go at molecular scale (you will see ¨single, isolated, differentiable pollen-water collisions) – patta May 28 '19 at 11:25
  • Neither classical physical principles work, if we go to molecular scale, but do work well in macroscale.Any approximation is good, if it provides good enough predictions of a system behaviour for its domain of validity. – Poutnik May 28 '19 at 16:34

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