Something is going on with the primes…

The chart below illustrates the Erdős–Kac theorem. This relates to the number of distinct prime factors of large numbers (integer sequence A001221 in the On-Line Encyclopedia):

Number No. of prime factors No. of distinct prime factors
1 0 0
2 (prime) 1 1
3 (prime) 1 1
4 = 2×2 2 1
5 (prime) 1 1
6 = 2×3 2 2
7 (prime) 1 1
8 = 2×2×2 3 1
9 = 3×3 2 1
10 = 2×5 2 2
11 (prime) 1 1
12 = 2×2×3 3 2
13 (prime) 1 1
14 = 2×7 2 2
15 = 3×5 2 2
16 = 2×2×2×2 4 1

The Erdős–Kac theorem says that, for large numbers n, the number of distinct prime factors of numbers near n approaches a normal distribution with mean and variance log(log(n)), where the logarithms are to the base e. That seems to be saying that prime numbers are (in some sense) randomly distributed, which is very odd indeed.

In the chart, the observed mean of 3.32 is close to log(log(109)) = 3.03, although the observed variance of 1.36 is smaller. The sample in the chart includes 17 numbers with 8 distinct factors, including 1,000,081,530 = 2×3×3×5×7×19×29×43×67 (9 factors, 8 of which are distinct).

The Erdős–Kac theorem led to an episode where, following the death of Paul Erdős in 1996, Carl Pomerance spoke about the theorem at a conference session in honour of Erdős in 1997. Quoting Albert Einstein (“God does not play dice with the universe”), Pomerance went on to say that he would like to think that Erdős and [Mark] Kac replied “Maybe so, but something is going on with the primes.” The quote is now widely misattributed to Erdős himself.

Brownian motion and molecular reality

Brownian motion is the random motion of tiny particle buffeted by molecular collisions. Robert Brown noticed it in 1827 with fragments of pollen grains in water (see his 1828 paper: “A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies”). This YouTube video shows Brownian motion in action:

A theoretical explanation was provided in “On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by the Molecular Kinetic Theory of Heat,” one of Albert Einstein’s Annus Mirabilis papers of 1905. In response, Jean Baptiste Perrin carried out a detailed experimental study, which he reported in his lengthy article “Mouvement brownien et réalité moléculaire” (“Brownian movement and molecular reality”) of 1909. The image below is redrawn from that publication (by “MiraiWarren”), and shows the positions of a particle (of radius 0.53 µm) at 30-second intervals. Perrin won the 1926 Nobel Physics Prize for this work, which helped to finally prove that atoms really existed.

In his 1913 book Atoms, Perrin describes the motion this way: “The trajectories are confused and complicated so often and so rapidly that it is impossible to follow them; the trajectory actually measured is very much simpler and shorter than the real one. Similarly, the apparent mean speed of a grain during a given time varies in the wildest way in magnitude and direction, and does not tend to a limit as the time taken for an observation decreases, as may easily be shown by noting, in the camera lucida, the positions occupied by a grain from minute to minute, and then every five seconds, or, better still, by photographing them every twentieth of a second, as has been done by Victor Henri, Comandon, and de Broglie when kinematographing the movement.

Before Perrin, some sceptics had doubted the existence of atoms and molecules – which were, after all, invisible. “And who has ever seen a gas molecule or an atom?” asked Marcelin Berthelot in 1877.

The decisive evidence for “molecular reality” was the fact that Avogadro’s number NA could be calculated in several quite different ways, including from Brownian motion. From the diagram above, for example, I calculate that the mean squared horizontal “jump” every 30 seconds is 3 (in grid square units). The first three horizontal “jumps” starting from the top right are 2.1, 0.2, and 2.5 in grid square units, and squaring those numbers gives 4.4, 0.0, and 6.3, for a mean square of 3.6 (but the mean square drops as more “jumps” are considered). According to Perrin, each grid square is 3.125 µm across, so that the mean squared “jump” in metric units is 2.9×10−11. Assuming a temperature of 20°C (293 K), at which water has a viscosity of 0.001002, Einstein’s theoretical work gives:

NA = 8.3144621 × 293 × 30 / (3 π × 0.53×10−6 × 2.9×10−11 × 0.001002) = 5.0×1023

This is a little too low, since 6.022×1023 is the true value, but it’s still remarkably close, and a tribute to the quality of Perrin’s work (although his final value was actually too high: 7.05×1023). Every different experimental approach gave (at least approximately) the same value for NA. Studies of the blue of the sky due to Rayleigh scattering, for example, gave a value between 3×1023 and 15×1023. Studies of radioactivity by Ernest Rutherford gave between 6×1023 and 7×1023. Perrin was able to list several other studies as well. Atoms existed!

In 1908, Wilhelm Ostwald (once a sceptic) wrote in the fourth edition of his textbook on chemistry: “I have satisfied myself that we arrived a short time ago at the possession of experimental proof for the discrete or particulate nature of matter – proof which the atomic hypothesis has vainly sought for a hundred years, even a thousand years. The isolation and measurement of gases on the one hand, which the lengthy and excellent works of J. J. Thomson have crowned with complete success, and the agreement of Brownian movement with the demands of the kinetic hypothesis on the other hand, which have been proved through a series of researches and at last most completely by J. Perrin, entitle even the cautious scientist to speak of an experimental proof for the atomistic constitution of space-filled matter.” (as quoted by Mary Jo Nye)

Perrin concluded his 1913 book Atoms with an acknowledgement both of victory and of further challenges: “The atomic theory has triumphed. Its opponents, which until recently were numerous, have been convinced and have abandoned one after the other the sceptical position that was for a long time legitimate and no doubt useful. Equilibrium between the instincts towards caution and towards boldness is necessary to the slow progress of human science; the conflict between them will henceforth be waged in other realms of thought.

But in achieving this victory we see that all the definiteness and finality of the original theory has vanished. Atoms are no longer eternal indivisible entities, setting a limit to the possible by their irreducible simplicity; inconceivably minute though they be, we are beginning to see in them a vast host of new worlds. In the same way the astronomer is discovering, beyond the familiar skies, dark abysses that the light from, dim star clouds lost in space takes aeons to span. The feeble light from Milky Ways immeasurably distant tells of the fiery life of a million giant stars. Nature reveals the same wide grandeur in the atom and the nebula, and each new aid to knowledge shows her vaster and more diverse, more fruitful and more unexpected, and, above all, unfathomably immense.