# What does this observation of instantaneous velocity in Brownian particles mean?

“We’ve now observed the instantaneous velocity of a Brownian particle,” says Raizen. “In some sense, we’re closing a door on this problem in physics. But we are actually opening a much larger door for future tests of the equipartition theorem at the quantum level.”

• What did Einstein propose and what did they actually prove?
• What door does this experiment open?
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Einstein stated a technological limit ("we can't measure this", but not "this is in principle impossible to measure") which has only taken circa 100 years to beat. I tell, 'ya. It's sad, that guy just didn't know what he was talking about... – dmckee Mar 25 '11 at 23:04
What might half-amateurs write to get some attention? Maybe its the fault of some writer of that "Texas Science" paper? – Georg Mar 26 '11 at 11:17

The article makes no sense. Einstein realized that matter was composed out of atoms, so the number of collisions of a Brownian particle with the surrounding molecule is finite in a finite period of time.

However, for times $t$ much longer than the typical scale between the collisions, the particle moves by a distance scaling like $\sqrt{t}$. It follows that the velocity measured in time interval $t$ goes like $$v=s/t\sim\frac{\sqrt{t}}{t} =\frac{1}{\sqrt{t}}$$ and it diverges in the $t\to 0$ limit, that of the instantaneous velocity. This fact is described by the statements that the instantaneous velocity "cannot exist".

However, the scaling law above can't really be extrapolated for time scales shorter than the time between two collisions and what the people you mentioned may have succeeded in is to increase the time resolution so that the individual collisions may be distinguished - an advance that was impossible when Einstein wrote his paper 100+ years ago.

At any rate, it is not a big deal and it doesn't contradict anything Einstein believed - even though he may have failed to write down the explanation above explicitly.

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What you say is similar to my comment. This is an end to the scaling principle. The divergence indicates that one is on small enough time intervals that collisions or scattering of the particle by atoms is no longer a statistical or Langevin process. – Lawrence B. Crowell Mar 26 '11 at 13:06
Right, I agree, except that I don't see this point in your answer. ;-) – Luboš Motl Mar 28 '11 at 19:18

This is a bit strange. The Langevin equation $$\frac{dv}{dt}~+~\beta v~=~\frac{F}{m}$$ for the motion of a free particle under a stochastic force $F$ evaluates the velocity as an average or in an interval. The stochastic force has a Gaussian probability distribution $\langle F(t)F(t’)\rangle~=$ $2\beta kT\delta(t~-~t’)$, which is also a Markov process. The Langevin equation has an associated Fokker-Planck equation. This is a diffusion type of equation, $$\frac{\partial P}{\partial t}~=~\frac{\partial}{\partial v}{\beta v P}~+~\frac{D}{m^2}\frac{\partial^2P}{\partial v^2},~ D~=~m\beta kT$$ This looks a bit like a heat equation for the evolution of a probability function $P$ for a particle to be found in some region of space. Under a Fourier transform this is converted into a first order differential equation in $x$.

This experiment is measuring the “cut off” in the one dimensional fractal path of the particle. On a small enough of a scale and with time slowed down the motion is due to individual atoms recoiling off of the small particle. This is a domain where the thermal-classical world transitions into the quantum world. So in this transition place the above Fokker-Planck equation gives way to a Schrodinger equation. The above stochastic force in this scale becomes replaced by small change in momentum $\delta p$ which defines an operator ${\hat p}~=~\langle p\rangle~+~\delta p$, where $\delta p$ will have the Markovian properties above.

This is an interesting domain to think about, for it appears to be in the netherworld between thermal-classical physics and quantum mechanics.

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