Müller I. A history of thermodynamics. The doctrine of energy and entropy (Müller I. A history of thermodynamics. The doctrine of energy and entropy.pdf), страница 64

549–560.All of Einstein’s early papers on the Brownian motion were later edited by R. Fürth:“Untersuchungen über die Theorie der Brownschen Bewegungen.” [Investigations on thetheory of the Brownian movement] Akademische Verlagsgesellschaft, Leipzig (1922).This collection has been translated into English by A.D. Cowper and is available as aDover booklet.7 S.G. Brush: “The kind of motion we call heat.” loc. cit. p.

673.8 P.A. Schilpp (ed.): “Albert Einstein Philosopher-Scientist”. New York. “Library of LivingPhilosophers” (1949).The Schilpp collection contains an autobiographical note by A. Einstein from which theabove quotation is taken.9 Schilpp collection. Autobiographical notes. loc.cit p. 17/18.10 A. Einstein: “Kinetische Theorie des Wärmegleichgewichtes und des zweiten Hauptsatzesder Thermodynamik.” [Kinetic theory of heat equilibrium and of the second law ofthermodynamics] Annalen der Physik (4) 9 (1902 )pp.

417–433.2769 Fluctuationsto deal with such a process.11 We shall consider a one-dimensional andsimplified version of his argument:Let the x-axis be subdivided into equal intervals of length ǻ and let aBrownian particle jump – right or left with equal probability, i.e. probability1/2 – to neighbouring intervals after each time interval IJ. The jumps occurbecause the particle is hit by solvent molecules but no explicit account isgiven of the mechanics of the collisions.From what has been said, the probability w(x,t) of finding the particle atposition x at time t must satisfy the difference equationw( x, t )12w( x ', t W ) 12 w( x ', t W ) .If ǻ and IJ are small, one may expand the right hand side into a Taylorseries breaking off at the leading non-zero terms in ǻ and IJ.

Thus oneobtains the differential equationwwwV22' w w.2W wZ 2Einstein says: This is the well-known diffusion equation and we recognizethat D = ǻ2/2IJ is the coefficient of diffusion.Many solutions of this equation are known – primarily through Fourier’swork, cf. Chap. 8. In particular, if at time t = 0 the particle was in theinterval at X, its probability to be at position x at time t is given byw( x, t )È ( x X )2 Ø1exp É 4 Dt ÙÚ4πDtÊand the root mean square distance Ȝ from X comes out asλ(x X )2ȇØ2(xX)w(x,t)dxÉÔÙÊ ‡Ú122 Dt ,so that it is determined by the diffusion coefficient. Thus by repeatedcareful observations of Brownian motion and averaging over the results onecould determine D.Einstein, however, favoured another application of the formula for Ȝ.

Hehad determined a relation between the unknown diffusion coefficient D – ofa Brownian particle of radius r in a solvent – and the known viscosity Ș ofthe solvent, viz, cf. Insert 9.111A. Einstein: “Investigations ... ’’ loc.cit. § 4.Brownian Motion as a Stochastic ProcessDkT6πηrso that he could write λkT3πηr277t.Thus measurements of Ȝ for known values of Ș and r could determine thevalue of the Boltzmann constant k.

Therefore Einstein concludes his paperwith the words: It is to be hoped that some enquirer may succeed shortly insolving this problem [the experimental determination of k]… which is soimportant in connection with the theory of heat.This remark is obviously a reflection of the then still ongoing – albeit obsolete –discussion between Mach and Boltzmann in Vienna, where the former maintainedthat atoms were a fiction of imagination, since their properties could not bedetermined; [Mach ignored Loschmidt’s rough and ready calculation of 1865, cf.Chap. 4.] The rest of the world watched this out-dated debate in amazement12 butEinstein seems to have taken it seriously.13I cannot help feeling that the importance and feasibility of measuring k inthis manner is somewhat exaggerated here by Einstein.

After all, this recipewould involve a cumbersome observation of the mean motion of aBrownian particle. No doubt that it can be done, but why should it be done?A good value of the Boltzmann constant was already known from theRayleigh-Jeans formula, cf. Chap. 7, which was perfectly convincing andindubitably correct for low-frequency radiation.Relation between diffusion coefficient D and viscosity ȘWhen Brownian particles of mass µ, radius r, and with particle density n(x,t) aresuspended – macroscopically at rest – in a solvent of temperature T, they are denserat the bottom than on top, because they must satisfy the stationary momentumbalance12Robert Andrews Millikan (1868-1953) – the man who determined the elementary charge e– writes in his autobiography: The amazing thing is that this question could be debated atall at that time [1904] … and that even the brilliant philosopher Ernst Mach could at thatepoch oppose atomic theories.R.

A. Millikan: “The autobiography of Robert A. Millikan.” Arno Press, New York(1980).D. Lindley, the author of “Boltzmann’s atom” loc.cit. writes: To an audience of youngNew World scientists, this debate must have seemed an intrusion into their fresh universefrom the Old World’s attic.13 Einstein writes: If the movement discussed here can actually be observed … an exactdetermination of actual atomic dimensions is then possible. On the other hand, had theprediction of this movement proved to be incorrect, a weighty argument would beprovided against the molecular kinetic conception of heat.2789 Fluctuations˜p ( n , T )˜x nP g or with pnkT, according to van't Hoff's law fordilute solutions, cf. Chap. 5 :˜n˜xPgkTn(barometric formula).We may think of the particles as being macroscopically at rest, because two flowvelocities compensate each other:a downward flow withXPg‘6S K raccording to Stokes´s law for a sphericalparticle under gravity, cf.

Chap. 8 andan upward flowX−D1 ˜nn ˜xaccording to Fick´s law, cf. Chap. 8.Hence follows with the barometric formulaPg6SKrDPgkTorDkT.6SKrThis is Einstein’s relation between D and Ș.Insert 9.1Einstein’s paper carries the mark of genius in a positive and negativesense: The positive aspect is that the paper introduces stochastic argumentsinto Brownian motion and this made such arguments acceptable to thermodynamicists. But then the paper is also carelessly written, it shows a benignneglect of detail and direction that might – and did – throw people off thetrack. Thus Brush14 complains about the muddled presentation. He says thatEinstein did not emphasize very strongly the significance of his result thatȜ is proportional to the square root of time, and in fact it is quite probablethat most early readers of the paper gave up in bewilderment before theygot to the result.Indeed, it makes no sense that the initial growth rate of Ȝ is infinite as isimplied by the result.

And surely this prediction should have warranted aremark. It may in fact be understood as a shortcoming of the stochasticmodel by which the Brownian particle, – in executing its random jumps – is14S.G. Brush: “The kind of motion we call heat.” loc.cit. p.

681.Mean Regression of Fluctuations279not ascribed an inertia. The physicist Paul Langevin (1872–1946)15 lookedinto the argument and he came up with an improved equation of the formλ2ËÛµ6πηr2 D Ìt 1 exp µ t ÜÍ 6πηrÝby taking inertia into account. To be sure, for typical values of Ș, µ, and rthe second term in the square brackets is usually negligible, so thatEinstein’s results holds approximately. But this is not so for small times.Mean Regression of FluctuationsIn the Brownian motion we see a nearly macroscopic body – the Brownianparticle – kicked around by the atoms or molecules in the manner envisagedby Poincaré, see above. The force F(t) of impact by the molecules on theparticle fluctuates, and it stands to reason that, averaged over a long time, oraveraged – at one time – over many Brownian particles, the force is zero.Since the particle moves in a viscous fluid, its equation of motion reads6πηrXXµ1µF (t )This equation is known as the Langevin equation.

On the basis of thatequation Langevin was able to correct Einstein’s result for the root meansquare distance Ȝ, see above.If the mass µ of the particle is very big, its equation of motion isunaffected by the fluctuating force F(t) and the velocity decaysexponentially as a function of timeXX(t )r(t0 ) exp 6πηµ (t t0 ) .I shall refer to this solution as the macroscopic law of decay. For theBrownian motion the decay is exponential, but this need not be so in othercases of fluctuating quantities; indeed, the decay may be a damped oscillation on other occasions.On the other hand, when the particle has a small mass, the fluctuatingforce makes its velocity fluctuate as well about an average velocity zero asillustrated in the upper part of Fig. 9.1.

The graph of this velocityfluctuation seems totally irregular, and certainly in no way related to themacroscopic law of decay. And yet, some regularity is hidden in thefluctuations; and that regularity is brought forth, if we construct the meanregression of a fluctuation.15P. Langevin: Comptes Rendues Paris 146 (1908) p. 530.2809 FluctuationsFig. 9.1.

Top: Velocity fluctuations of a Brownian particle. Bottom: Mean regression of afluctuationWhen we consider very many, say N , velocity fluctuations of a particularfixed size Xȕ which occur at the times tĮȕ (Į = 1,2,…N), we may ask for thesizes of the fluctuation at a later time tĮ + IJ. They are all different, of course,but upon averaging we obtainXX(τ , vβ )1βN (t τ ) .Çα1 αNThis function of IJ is the mean regression of the fluctuation XǪ.