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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 velocities**of** 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