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

sheynin.de
  • No tags were found...

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

In a manuscript of the same year (1873) Maxwell (Campbell & Garnett,p. 360), remarked thatThe form and dimensions of the orbits of the planets […] are notdetermined by any law of nature, but depend upon a particular collocationof matter. The same is the case with respect to the size of the earth.This was an example illustrating Poincaré’s statement concerningrandomness and necessity (§ 1.1), but it was not sufficiently specific; theeccentricities of planetary orbits depend on the velocities of the planets, cf.end of § 7.3.And here is Maxwell’s position (1875/1890, p. 436) concerningrandomness in the atomic world:The peculiarity of the motion of heat is that it is perfectly irregular; […]the direction and magnitude of the velocity of a molecule at a given timecannot be expressed as depending on the present position of the moleculeand the time.At the very end of his life Maxwell (1879/1890, pp. 715 and 721)introduced a definition for the probability of a certain state of a system ofmaterial particles:I have found it convenient, instead of considering one system of […]particles, to consider a large number of systems similar to each other […].In the statistical investigation of the motion, we confine our attention to thenumber of these systems which at a given time are in a phase such that thevariables which define it lie within given limits.Boltzmann (1868, § 3) defines the probability of the system being in aphase […] as the ratio of the aggregate time during which it is in that phaseto the whole time of the motion.4) If the classical definition of probability is included here, we can saythat Boltzmann used three formulations. Maxwell (item 2 above) mentionedone of them, and another reference can be added: Boltzmann (1895 – 1899,1895, Bd. 1, p. 50). Yet another one was that applied by Maxwell (see samesubsection) although sometimes Boltzmann (1878/1909, p. 252) did notindicate which one he was employing. He (1872/1909, p. 317) apparentlythought that these posterior probabilities were equivalent.In other words, with respect to separate molecules Boltzmann introducedthe time average probability, – and maintained that it was equivalent to theusual phase average probability. When studying polyatomic gases,Boltzmann (1871) defined the probability of its state as a product such asfdω where f was some function, varying in time, of the coordinates andvelocities of the separate molecules and dω, the product of the differentialsof those parameters. For stochastic processes, such functions determine thedistribution of a system of random variables at the appropriate moment.Zermelo (1900, p. 318) and then Langevin (1913/1914, p. 3) independentlystressed the demand to provide a definition correcte et claire de laprobabilité (Langevin). Like Maxwell, Boltzmann (1887/1909, p. 264; 1895110

More magazines by this user
Similar magazines