Brownian motion and continuity

[DesertFox]

It is said often that in 1905 Einstein “mathematically proved” the existence of atoms. More precisely, he worked out a mathematical atomic model to explain the random motion of granules in water (Brownian motion). According to that mathematical model, if the atoms were infinitely small and infinitely numerous, the effect of the collisions would balance at each instant and the granules would not move; so, the finite size of the atoms and the fact that these are present in finite (rather than infinite) number, cause there to be fluctuations.

However, in mathematics there is the so-called continuous model which is a limiting form. The limiting form of Brownian motion is The Wiener Process.

So, here it is my question. Can The Wiener Process be considered (in purely mathematical sense) as “mathematical proof” that there are no atoms?

I know this is layman’s question; my level is layman. That’s why I will try to re-phrase the question (in order to articulate it precisely as far as I can). Is The Wiener Process a mathematical model which consistently explains the Brownian motion without postulating atoms?

 

[TeethWhitener]

I think it’s probably better to say that the Wiener process is the continuous time limit of a discrete random walk, rather than Brownian motion (which can mean a number of different things, depending on the context).

From a practical standpoint, the “continuous time” aspect of the Wiener process simply means that the time between atomic collisions with a Brownian particle can be arbitrarily short, not a process where atoms don’t exist. Technically, the process only involves a random fluctuation force and is agnostic as to the actual cause of that force, but if atoms are imparting a (pseudo)random force, it will look a lot like a Wiener process.

 

[DesertFox]

I think it’s probably better to say that the Wiener process is the continuous time limit of a discrete random walk, rather than Brownian motion (which can mean a number of different things, depending on the context).

From a practical standpoint, the “continuous time” aspect of the Wiener process simply means that the time between atomic collisions with a Brownian particle can be arbitrarily short, not a process where atoms don’t exist. Technically, the process only involves a random fluctuation force and is agnostic as to the actual cause of that force, but if atoms are imparting a (pseudo)random force, it will look a lot like a Wiener process.

So, there is not any known consistent mathematical non-atomic model of the Brownian motion?

 

[TeethWhitener]

So, there is not any known consistent mathematical non-atomic model of the Brownian motion?

Again, Brownian motion just postulates a randomly fluctuating force (with a few technical parameters). It doesn’t say anything about what the cause of that force is. It could be atoms, or it could be tiny gremlins. But if atoms are causing a randomly fluctuating force (subject to the aforementioned parameters), then the observable behavior of a particle acted on by those atoms will be well-described by a Wiener process.

 

[DesertFox]

Again, Brownian motion just postulates a randomly fluctuating force (with a few technical parameters). It doesn’t say anything about what the cause of that force is. It could be atoms, or it could be tiny gremlins. But if atoms are causing a randomly fluctuating force (subject to the aforementioned parameters), then the observable behavior of a particle acted on by those atoms will be well-described by a Wiener process.

If we assume that atoms are not the cause, is the behavior "wel-described"?

 

[TeethWhitener]

If we assume that atoms are not the cause, is the behavior "wel-described"?

What does well-described mean?

 

[DesertFox]

What does well-described mean?

It is your phrase...You tell me

 

[TeethWhitener]

Like I said, there’s nothing in the Wiener process that explicitly mentions atoms. That’s why you can apply the same math to certain aspects of the stock market. Stock options are not made of atoms, and yet these processes still underlie certain price fluctuations.

 

[DesertFox]

Like I said, there’s nothing in the Wiener process that explicitly mentions atoms. That’s why you can apply the same math to certain aspects of the stock market. Stock options are not made of atoms, and yet these processes still underlie certain price fluctuations.

But my last question was: is there any known consistent mathematical non-atomic model of the Brownian motion?

 

[TeethWhitener]

It is your phrase...You tell me

Ah, I see. I mean that observations will match closely with predictions from the model. See post 8 for the answer to your question. There are no atoms in the model. But the predictions from the model seem to match experiment pretty well. So it’s a pretty safe bet that there is a random fluctuating force. If you want to say it’s gremlins instead of atoms, that’s your prerogative. Though atomic theory has plenty of support from elsewhere in science.

 

[TeethWhitener]

But my last question was: is there any known consistent mathematical non-atomic model of the Brownian motion?

See post 8. Spend more than 6 minutes digesting my answer.

 

[TeethWhitener]

Also, just a heads up: none of this has anything to do with QM, so I’m requesting the mentors move the thread.

 

[DesertFox]

Though atomic theory has plenty of support from elsewhere in science.

Oh, I see: perhaps that plenty of support is even more conclusive than the mathematical atomic model of the Brownian motion.

See post 8. Spend more than 6 minutes digesting my answer.

Or, better you read the question in post #5 and spend more than 10 minutes before you digest and answer it?

Also, just a heads up: none of this has anything to do with QM, so I’m requesting the mentors move the thread.

I really appreciate your care for the thread.

 

[TeethWhitener]

Or, better you read the question in post #5 and spend more than 10 minutes before you digest and answer it?

I guess beggars can be choosers.

 

[BWV]

FWIW Jean Baptiste Perrin received the 1926 Nobel Prize for experimental confirmation of Einstein’s 1905 theoretical paper

http://scihi.org/jean-baptiste-perrin-and-the-brownian-motion/

 

[PeroK]

Oh, I see: perhaps that plenty of support is even more conclusive than the mathematical atomic model of the Brownian motion.

The atomic model no longer rests on anything done in 1905 or before. That may have been part of the historic development/confirmation of the theory, but there have been 117 years of scientific development since then.

 

[vanhees71]

Again, Brownian motion just postulates a randomly fluctuating force (with a few technical parameters). It doesn’t say anything about what the cause of that force is. It could be atoms, or it could be tiny gremlins. But if atoms are causing a randomly fluctuating force (subject to the aforementioned parameters), then the observable behavior of a particle acted on by those atoms will be well-described by a Wiener process.

Well, in the Langevin equation you need the correct dissipation-fluctation relation between the drag and diffusion coefficients to be thermodynamically consistent. It's not clear to me, how else than with (classical or quantum) kinetic theory you can get to this result.

 

[DesertFox]

Well, in the Langevin equation you need the correct dissipation-fluctation relation between the drag and diffusion coefficients to be thermodynamically consistent. It's not clear to me, how else than with (classical or quantum) kinetic theory you can get to this result.

Is the Langevin equation employed in the Wiener process?

 

[vanhees71]

https://encyclopediaofmath.org/wiki/Langevin_equation#:~:text=dv(t)=−,1930 [a2] (cf.

 

[BWV]

The continuity of the Weiner process is in regards to time, so I fail to see what difference that makes vs a discrete time model of Brownian motion.

 

[DesertFox]

The continuity of the Weiner process is in regards to time, so I fail to see what difference that makes vs a discrete time model of Brownian motion.

In the Weiner process, random vectors with infinitely divisible distributions are at work.

 

[TeethWhitener]

Well, in the Langevin equation you need the correct dissipation-fluctation relation between the drag and diffusion coefficients to be thermodynamically consistent. It's not clear to me, how else than with (classical or quantum) kinetic theory you can get to this result.

This is a really good point. If I can play devil’s advocate for a minute, though, all you really need from kinetic theory for the fluctuation-dissipation theorem is the correct energy distribution (Boltzmann for classical and BE or FD for QM), so I suppose it’s conceivable that the fact that kinetic theory and FDT rely on the same distribution is a mere coincidence. But yes, the fact that the FDT relies on Boltzmann and Boltzmann leans very heavily on atomic theory is very compelling.

 

[vanhees71]

The continuity of the Weiner process is in regards to time, so I fail to see what difference that makes vs a discrete time model of Brownian motion.

The Langevin equation is a stochastic differential equation, where time is continuous. The fuctuating force is described by a Wiener process.

 

[vanhees71]

This is a really good point. If I can play devil’s advocate for a minute, though, all you really need from kinetic theory for the fluctuation-dissipation theorem is the correct energy distribution (Boltzmann for classical and BE or FD for QM), so I suppose it’s conceivable that the fact that kinetic theory and FDT rely on the same distribution is a mere coincidence. But yes, the fact that the FDT relies on Boltzmann and Boltzmann leans very heavily on atomic theory is very compelling.

It's not a coincidence but derived from it. One should also be aware that the Langevin equation (or equivalently the corresponding Fokker-Planck equation) is an approximation to the Boltzmann equation.

The Boltzmann equation tells us, via the famous H-theorem, what the equilibrium phase-space distributions must be, namely the said MB, BE, or FD distributions. The H-theorem relies on the principle of detailed balance, which itself is based on the very fundamental property that the S-matrix of scattering theory is unitary (note that you don't need to assume P and/or T invariance of the interactions!).

Now the Langevin equation describes a fluid in equilibrium within which some "mesoscopic" particles are immersed. Since these particles are much heavier than the molecules making up the fluid in each collision the momentum transfer to the particle is very small, and thus the Boltzmann collision term can be approximated by an expansion up to 2nd order in the momentum transfer, which leads to the Fokker-Planck equation, which is equivalent to the Langevin equation in the sense of a stochastic process.

Now the Boltzmann equation tells you that the particles after a long time should equilibrate with the fluid at the given temperature, and this leads to the fluctuation-dissipation relation a la Einstein.