1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

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1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

By the study of Boltzmann I have been unable to understand him. Hecould not understand me on account of my shortness, and his length wasand is an equal stumbling block to me.And Boltzmann (1868/1909, p. 49) indeed owned that it was difficult tounderstand Maxwell’s Deduktion (1867) because of its extreme brevity.2) Clausius. He (1857/1867, pp. 238 and 248) asserted that moleculesmoved with essentially differing velocities. Even Boscovich (1758, § 481)stated something similar but perhaps presumed that the differences betweenthese velocities were not large: The points [atoms] of a particle [of light, asin § 477, or of any body, as in § 478] move together with practically thesame velocity, and the entire particle will move as a whole with the singlemotion that is induced by the sum [the mean] of the inequalities pertainingto all its points. Clausius used a single mean velocity such as to make theentire kinetic energy of a gas equal to its actual value. Later he (1862/1867,p. 320) maintained that the velocities of molecules randomly differed onefrom another.And he (1858/1867, p. 268) studied the length of the free path of amolecule. Denote the probability of a unit free path by a, thenW = a x = (e –x ) α , α > 0will be the probability of its being equal to x; here, α is derived from themolecular constants of the substance. Similar considerations are in otherworks of Clausius (1862/1867, § 29; 1889 – 1891, pp. 70 – 71 and 119). He(1889 – 1891, pp. 70 – 71) also calculated the mean free path of a molecule.Actually, without writing it out, he considered free paths of random length ξand calculated the expected free path as an integral over all of its possiblevalues from 0 to ∞.Suppose now thatξ = ξ 1 + ξ 2 + … + ξ mwhere m is an arbitrary natural number. Then, according to Clausius’assumptions, ξ k , k = 1, 2, …, m, will not depend on (ξ 1 + ξ 2 + … + ξ k–1 ) andthe characteristic function for ξ k will be equal to the product of thesefunctions for the previous ξ’s. In this instance, all these functions areidentical, and F(s), the integral distribution function of ξ, is thereforeinfinitely divisible. Clausius’ achievements were interesting, but he did notattempt to construct the kinetic theory of gases on a stochastic basis.3) Maxwell (1860) established his celebrated distribution of the velocitiesof monatomic moleculesφ(x) =1 exp( 2 / α 2− x ).α πHe tacitly assumed that the components of the velocity were independent;later this restriction was weakened (Kac 1939; Linnik 1952). He thenmaintained that the average number of particles with velocities within theinterval [v; v + dv] was proportional to108

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