Reading through Einstein's Brownian motion paper "On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat", it seems the final argument is that he can calculate the Avogadro's constant by using data on the diffusion rates, particle size and fluid viscosity. But its hard to see the connection that ultimately atoms must exist. Can someone lay out a flowchart of step-by-step reasoning leading to that conclusion? I see all the math steps, but need a more philosophical type of breakdown. Also can't the random movement and diffusion law movement of the particles be just as well explained by fluctuations of pressure and density of continuous matter?

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    $\begingroup$ Does the article really tries to prove that atoms exist? I think it takes this for granted (as the title suggests, since it refers to the Molecular-Kinetic theory) - that atoms exist was well-known before Einstein. $\endgroup$
    – Roger V.
    Oct 2, 2023 at 9:33
  • $\begingroup$ Could you look at the derived results and take appropriate limits? For example, the limit of continuous matter should be arrived at by taking, say, $N\to\infty$ and radius $P\to\infty$ and so forth, keeping density constant, etc. Some of the derived results would be ridiculous, and that could be taken as evidence in favour of atoms, since atoms would have specific fixed finite values for those quantities as opposed to the nonsensical limits. It would be the first few times after Planck's Law that continuum limits were nonsensical. $\endgroup$ Oct 2, 2023 at 9:43
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    $\begingroup$ @RogerVadim, in 1905, the existence of atoms was still an open question. The majority view was that they were real, but there was a minority (backed by some respectable arguments) that viewed atoms as merely a useful construct for figuring out which chemical reactions were possible. You can see a similar thing today with quarks: a majority view that sees them as real, and a minority view that they're merely a useful mathematical construct. $\endgroup$
    – Mark
    Oct 2, 2023 at 21:55
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    $\begingroup$ @Mark but the paper in question explicitly assumed that atoms exist. See also the answer by ThomasFritsch. $\endgroup$
    – Roger V.
    Oct 3, 2023 at 5:01
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    $\begingroup$ There are not absolute proofs in science at all. What we can do, though, is make things have to be "ever more convoluted" to not have a certain thing. Atoms and molecules are one of those things that it'd take an awful lot of "convolution" to "not have" given existing evidence, so we presume that that means they must (in some form) exist. FWIW, another good example from Einstein is the photoelectric effect and the photon. The photoelectric effect doesn't require a photon (see a perhaps-famous paper by Lamb and Scully), but it "makes things more convoluted" to explain without one. $\endgroup$ Oct 3, 2023 at 21:05

3 Answers 3


Strictly speaking, Einstein's paper does not prove the existence of molecules.
But: Assuming the existence of molecules, it correctly predicts the existence of Brownian motion and the diffusion law, as it is observed experimentally.

Einstein writes on the first page of the paper:

It is possible that the motions to be discussed here are identical with the so-called "Brownian molecular motion".

Furthermore, taking this molecular model of liquids to be true, it, for the first time, allowed to determine Avogadro's number ($6\cdot 10^{23}$) and the size of the molecules ($\approx 10^{-10}$ m). These are more predictions, which could later be verified with entirely different experimental methods (X-ray cristallography).

So it is not a proof with absolute mathematical rigor, but more like a proof by strong circumstantial evidence.

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    $\begingroup$ Proofs with absolute mathematical rigor are impossible in physics since we hove no axioms. $\endgroup$
    – John Doty
    Oct 2, 2023 at 17:43
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    $\begingroup$ @ChrisBecke We cannot unequivocally prove that jumping off a cliff this time will somehow not result in your broken body dashed upon the rocks below. All our theories are only modeled on our observations so far and it's entirely possible that there's something we've missed and some hitherto undetected force might see fit to counteract gravity for you as you jump. We think that's highly unlikely, but we're not going to say it's impossible. $\endgroup$
    – Shadur
    Oct 3, 2023 at 7:41
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    $\begingroup$ @Tom Regarding the theory of evolution by natural selection, it is not a single unified theory but a broad spectrum of findings and understandings continuously updated. I am not a biologists but I know that the way how Darwin saw it was not current any more at the times of Ronald Fischer, new insights were brought by Dawkins and his contemporaries and so on. If you show that some views of Darwin were incorrect, you falsify that specific view but not the whole principle of natural selection. It is a very broad framework. $\endgroup$ Oct 3, 2023 at 12:17
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    $\begingroup$ @Tom No. Not at all. There is simply no single "theory of evolution by natural selection" but there absolutely are many well-defined theories in various disciplines. There are various descriptions of natural selection and its connection to evolution in various papers that must be considered and potentially falsified independently. However, we are going astray and I suggest opening a question at philosophy.stackexchange.com $\endgroup$ Oct 3, 2023 at 12:20
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    $\begingroup$ @Blueriver Considering the Brownian motion of a body in a liquid, you can only prove that the liqid is made up of small particles, but not whether these are molecules or atoms. For proving that liquid is made up of molecules, and molecules are made up of atoms, you need other evidence (e.g. from chemistry). $\endgroup$ Oct 3, 2023 at 18:41

The scientific method doesn't "prove" that things are true in nature. A scientific theory is a model that can be used to make predictions about observations (e.g. the results of experiments, or natural occurrences).

When we make those observations, if they're consistent with the theory's predictions, we say it confirms the theory; if not, it refutes the theory. If there are enough observations that confirm the theory, no believable observations that refute it, and there are no other theories that are also confirmed, we treat the theory as true. This is typically when we elevate a theory to a "law", although often we stick with the old terminology out of habit (e.g. relativity is still called a "theory", although it's arguably as strongly confirmed as Newton's "laws" ever were).

So the basic logic of Einstein is:

  • If molecules exist, they will cause phenomenon X.
  • We observe phenomenon X.
  • Therefore, the hypothesis that molecules exist is confirmed.

"Confirmed" doesn't mean it's proved true, just that it's given more credence.

You'll observe that the above is not a normal logical progression. Scientific "proofs" are not like mathematical deductions, they're more like Bayesian induction -- each confirmation increases the likelihood that the hypothesis is true.


Thinking a little, I think my difficulty is due to a misunderstanding of what Einstein was trying to show. I assumed that he was trying to prove that water is made of molecules. Actually I don't think that was the objective at all now! In fact the molecular reality of water is somewhat irrelevant in Einstein's argument. Mathematically its treated as a continuum, and its role is to absorb momentum from the pollen. I think he was actually trying to show that kinetic theory is a mathematically self consistent theory, specifically it would apply equally well to macro particles such as pollen, as it would to molecules, such as sugar. This sentence I believe is key:

"According to this theory a dissolved molecule is differentiated from a suspended body solely by its dimensions, and it is not apparent why a number of suspended particles should not produce the same osmotic pressure as the same number of molecules"

So before 1905, this is what we know. Pollen is a particle. But a dissolvable substance such as sugar, is it a particle or continuous matter? I think this is what he is trying to prove, that sugar is a particle... not the water!

So how do we go about proving sugar is a molecule?

Step 1) Show that some property of sugar and pollen are similar.

Well, we know that there is this property of sugar called osmosis, and it's governed by Van't Hoff's law. So if we can show that Van't Hoff's law also applies to pollen, then the logic is pollen and sugar are similar in nature.

Step 2) Because we know pollen is a particle -> sugar must be a particle also

So then why not just measure the osmotic pressure of pollen and then the proof is done? Well I think the problem is that the osmotic pressure is going to be so tiny its undetectable. So back to step 1, he had to find a roundabout way to calculate osmotic pressure, thus kicking off the whole diffusion, random walk statistical analysis which ultimately culminates in Avogadro's constant. Since the same constant governs pollen and sugar, step 2 follows.

Btw I know this is not mathematical proof. Step 2 is not a hard corollary of step 1. To reiterate my key point, Einstein's analysis was not to say anything about the molecular nature of the liquid. It was merely showing that random motion of large particles and molecules from a solute are governed by the same equations.


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