Did Maxwell discover the distribution of velocities in a gas during an exam set by Stokes?

Question

In some lecture notes I found online it was claimed that Maxwell discovered the Maxwell-Boltzmann distribution while in the course of solving a (then unsolved) examination problem given by Stokes. The reference is to Experiment, Theory, Practice by Kapitsa. I was able to find a Russian copy. Indeed the anecdote is there:

Приведу вам еще четвертый пример. Происходило это в Кембридже, во второй половине прошлого века. Теоретическую физику тогда преподавал Стоке. К нему пришел сдавать аспирантский экзамен один молодой человек. Аспирантский экзамен в те времена был довольно трудный, потому что аспирантур тогда было очень мало — всего две-три, и состязание за право попасть в аспирантуру былр очень трудным. Стоке давал задачу, причем система была такая: давался десяток задач, и студент сам выбирал те, которые он хотел решить. Ему давалось определенное число часов, и Стоке, не стесняясь, ставил часто неразрешимые задачи, чтобы посмотреть, знает ли студент, что эта задача неразрешима. Он ставил, например, такую задачу (то были домаксвелловские времена): найти распределение скоростей в газе. Тогда это распределение скоростей не было известно. Бернулли и все остальные считали, что скорости примерно равны. Молодой человек, к удивлению Стокса, решил эту задачу, и решил правильно. Вы догадываетесь, что этот молодой человек был не кто иной, как Максвелл. Таким образом, открытие закона распределения скоростей молекул в газе было сделано Максвеллом на экзамене.

Unfortunately Kapitsa doesn't cite a reference for this anecdote, nor does he appear to give enough detail for independent verification. The exam he refers to may be the Smith's Prize exam, on which (in a more well-known anecdote) Stokes also set the problem of proving the theorem that bears his name and Maxwell tied for first with Routh. This anecdote is also mentioned in the book, so I don't think it's likely that Kapitsa confused the two. (The problem about the velocity distribution of gases does not appear on the 1854 exam that contained Stokes's theorem.)

Answer 1

No, he did not. Stokes could not have offered a problem of the distribution of velocities of gas particles for the simple reason that he never thought of fluids as consisting of particles. He was a typical 19-th century continualist, as a look at his works shows. After Bernoulli's derivation of the ideal gas law in 1738 kinetic theory was moribund until late 1850-s, it could not have been put on an entrance exam in 1854, especially by Stokes.

Excerpts from the early works on statistical physics, and commentary, are collected in Uffink's Compendium of the foundations of classical statistical physics, where we read:

"Despite this initial success, no further results were obtained in kinetic gas theory during the next century. By contrast, the view that modeled a gas as a continuum proved much more fertile, since it allowed the use of powerful tools of calculus... Nevertheless, the kinetic view was revived in the 1850s, in works by Krönig and Clausius. The main stimulus for this revival was the Joule-Mayer principle of the equivalence of heat and work, which led to the First Law of thermodynamics, and made it seem more plausible that heat itself was just a form of motion of gas particles. (A point well-captured in the title of Clausius’ 1857 paper: “The kind of motion we call heat”, subsequently adopted by Stephen Brush (1976) for his work on the history of this period.)

[...] It was Maxwell’s paper of 1860 that really marks the re-birth of kinetic theory. Maxwell realized that if a gas consists of a great number N of moving particles, their velocities will suffer incessant change due to mutual collisions, and that a gas in a stationary state should therefore consist of a mixture of slower and faster particles. More importantly, for Maxwell this was not just an annoying complication to be replaced by simplifying assumptions, but the very feature that deserved further study."

Krönig's paper came out in 1856, Brush's book is available online. On the mathematical side the origin of Maxwell's insight is somewhat speculative, but it seems that he mimicked Herschel's derivation of the error curve. According to Brush,

"While Maxwell was indirectly influenced by earlier writers on statistics such as Laplace, Poisson, Cournot, and Quetelet, the most immediate stimulus for Maxwell's first derivation of his velocity-distribution law was a review of Quetelet's works on probability by Sir John Herschel in the Edinburgh Review of July 1850... Aside from the striking similarity to Maxwell's own derivation (see below) there is only a letter from Maxwell to Lewis Campbell, which is undated but thought to be written around June 1850 by Campbell and Garnett, who published it in their Life of Maxwell. According to C. W. F. Everitt, who discusses this question in his forthcoming biography of Maxwell, the letter was probably written by Maxwell just after he read Herschel's review, since it repeats some of Herschel's remarks on probability theory in only slightly different words."

Answer 2

Who am I to cast doubt on what Kapitsa wrote? But I thought Maxwell's first work on kinetic theory, including his first argument for the Maxwellian distribution, was presented in 1859 and had been stimulated by his reading of Clausius. He'd graduated from Cambridge in 1854.

Answer 3

Here is a letter that Maxwell wrote to Stokes on May 30, 1859, in which he states that he read a paper by Clausius in the Philosophical Magazine of 1859. Of course, this may not have been the first time he raised this issue, for there was another paper by Clausius in 1857.