The concept of velocity in the history of Brownian motion -- From physics to mathematics and vice versa
TThe concept of velocity in the history of
Brownian motion
From physics to mathematics and vice versa
Arthur Genthon
Gulliver, ESPCI Paris, PSL University, CNRS, 75005 Paris, France * [email protected] Brownian motion is a complex object shared by different communities: first ob-served by the botanist Robert Brown in 1827, then theorised by physicists in the1900s, and eventually modelled by mathematicians from the 1920s. Consequently,it is now ambiguously referring to the natural phenomenon but also to the theo-ries accounting for it. There is no published work telling its entire history from itsdiscovery until today, but rather partial histories either from 1827 to Perrin’s exper-iments in the late 1900s, from a physicist’s point of view; or from the 1920s froma mathematician’s point of view. In this article, we tackle a period straddling thetwo ‘half-histories’ just mentioned, in order to highlight its continuity, to questionthe relationship between physics and mathematics, and to remove the ambiguitiesmentioned above. We study the works of Einstein, Smoluchowski, Langevin, Wiener,Ornstein and Uhlenbeck from 1905 to 1934 as well as experimental results, throughthe concept of Brownian velocity. We show how Brownian motion became a researchtopic for the mathematician Wiener in the 1920s, why his model was an idealizationof physical reality, what Ornstein and Uhlenbeck added to Einstein’s results andhow Wiener, Ornstein and Uhlenbeck developed in parallel contradictory theoriesconcerning Brownian velocity.
1. Introduction
Brownian motion is in the first place a natural phenomenon, observed by the Scottish botanistRobert Brown in 1827. It consists of the tiny but endless and random motion of small particles,from pollen grains, at the surface of a liquid. It was naturally an interest for botanists untilBrown and some physicists brought it into the field of physics. The latter built the first quan-titative theories to account for this motion, that culminated with Albert Einstein, Marian vonSmoluchowski and Paul Langevin in the 1900s. In the 1920s, Brownian motion knew a seconddomain shift to mathematics with Norbert Wiener’s early works, while continuing to be studiedby physicists. From that moment on, Brownian motion was an object of interest shared by differ-ent communities, referring to both the natural phenomenon and the physical and mathematicalmodels associated with it, themselves significantly different. This polysemy is responsible forsome confusions between model and reality, and between models themselves, for which we wishto make some clarifications in this article. Brownian motion is a highly remarkable object dueto its transdisciplinary, which allows us to address the relationship between scientific disciplines1 a r X i v : . [ phy s i c s . h i s t - ph ] J un nd more particularly between physics and mathematics; and also because of its long history ,which had an important role in the atomic hypothesis debate and which gave birth to the fieldof stochastic processes, at the boundary between theoretical physics and mathematics.Although Brownian motion is well documented in the literature, its history is often split intoparts that prevent us to appreciate its continuity and especially the dialogue between disciplines.We can easily find some excellent reviews of Brownian motion from a physical point of view , thatstart in 1827 and usually end around 1910, when physicists succeeded in building satisfactorytheories, which were in addition confirmed by Jean Perrin’s experiments.After the 1910s, the ‘second-half’ of the Brownian motion history started with ground-breakingprogress made by Norbert Wiener from the 1920s and with the numerous enhancements to theexisting physical theories made by Leonard Salomon Ornstein from the late 1910s onward, andlater joined by George Eugene Uhlenbeck. This second history is hardly-ever told, or is writtenin difficult mathematical language . In any case, the continuity between the two half-histories isalmost never dealt with. As a result of this assessment, the goals of this article are the following.We wish to study the exchanges at the boundary between physics and mathematics throughthe example of Brownian motion. We try to understand how an object of interest for physicistscould become a research topic for a young visionary mathematician, what the mathematicaldevelopments could offer to physics and how mathematicians and physicists dialogued whenworking on the same subject at the same time. Many aspects of Brownian motion can bestudied, so we chose in this article to focus on one property of Brownian motion that is relevantboth in physics and in mathematics, as well as in their connection: velocity. Indeed, the velocityof Brownian particles was one of the most difficult concept to agree on for experimenters andtheorists in the 1900s, and its understanding was one of Wiener’s motivations when he readPerrin’s account on irregular trajectories. Velocity is also at the heart of Ornstein’s work, itappears then to be a perfect common theme to our study. This work is also the occasion toexplain some simple results from Wiener’s theory to physicists, which are difficult to read inoriginal form though useful to understand the birth of the field of stochastic processes.We start by giving a short review of Brownian motion history from 1827 to 1905, to setimportant landmarks and describe the context in which Einstein published his first article.Secondly, we sum up the history from 1905 to 1910, which includes the theories proposed byEinstein, Smoluchowski and Langevin, the experiments carried out by Theodor Svedberg, MaxSeddig, Victor Henri and Perrin and the debates between the two communities. These elementsclearly set, we have all the required background knowledge to study in detail in a third partWiener’s work from 1921 to 1930, in a fourth part Ornstein’s and Uhlenbeck’s works from 1917to 1930 (with a glimpse at the 1945 article), and to finally compare these theories. We restrictourselves to the study of texts published, translated or commented in English or French.
2. Historical background
Since Brown’s article in 1827, physicists got interested in Brownian motion, and some sparseprogress was made between 1827 and 1905, when the first of Einstein’s articles on the subjectwas published. We aim to give a quick review of this time period to offer the reader usefulclues for the understanding of later works. Even if the history of Brownian motion is often started with Brown’s discoveries in 1827, it can even be tracedto the first century as mentioned in a poem by Lucretius, discussed by Louis Georges Gouy (Gouy, 1915). See (Nye, 1972; Brush, 1976; Maiocchi, 1990) and (Duplantier, 2007) which is a reviewed and extended versionof the original paper (Duplantier, 2006) See (Kahane, 1998). For in-depth studies on this period, one can read (Brush, 1976; Maiocchi, 1990; Duplantier, 2007). . Doing this, he put an end to the vitalist theories relying on thehypothetical vital force animating living particles, and thus explaining the motion. From thatmoment on, he aimed at eliminating some physical explanations to this movement, such as theevaporation-induced fluid flows, or the interaction between suspended particles, with success.Between Brown and Einstein, few discoveries were made, mainly because experiments wereincomplete and too qualitative, thus leading to diverging interpretations. The authors neitheragreed on the origin of the motion nor on the experiments themselves. Concerning the origin ofthe motion, Christian Wiener, Louis Georges Gouy, P`ere Julien Thirion, Ignace Carbonnelle andothers invoked the kinetic theory of gas, introduced by Maxwell and Boltzmann, which we callthe atomic hypothesis; while other claimed that the motion was caused by various phenomenalike lightning or electricity. Those last fanciful theories were refuted, at least qualitatively, duringthe twentieth century. Concerning the facts, for most physicists the motion was truly random,whereas it was a deterministic oscillatory movement for Carbonnelle and Svedberg for example.The influences of different parameters were also called into question, since for Gouy and theExner family, temperature increased the motion but for Thirion and Carbonnelle the oppositewas true .In spite of this lack of agreement, we sum up the main results obtained during this period, toease the future reading of Einstein’s article. During the 30 years that followed Brown’s articlethere was no sign of explanation, it was not until the 1860s that we see emerge the first attempts,including the atomic hypothesis.In 1863, Christian Wiener published his results on Brownian motion. He was one of the firstto propose a version of the atomic hypothesis (Nelson, 1967). It was however a primitive versionof atomism, formulated in terms of an ether and prior to Maxwell’s version (Brush, 1976).In 1867, Siegmund Exner published his observations in which he indicated that the intensity ofmovement seemed to increase with the liquid’s temperature and also when the liquid’s viscositydecreased. (Pohl, 2006).In 1879, Karl Wilhelm von N¨ageli proposed a counter argument to the atomic hypothesis. Hewas a botanist who had the advantage of being familiar with the kinetic theory of gases andknowing the orders of magnitude of masses and speeds. He developed a theory of displacementsof small particles of dust in the air and calculated that given the mass ratio between a gasmolecule and a dust particle, the speed communicated by the collision between the two wouldbe much too weak to explain the velocities observed experimentally.In 1888, Gouy published an article in which he recognized that uncoordinated collisions wouldnot be enough to account for the motion of suspended particles, and thus a correlation would beneeded on a space of about one micron. He was not the first one to make this observation, butbeing a physicist and speaking their language, he had more impact and hence was often wronglypresented as the discoverer of the origin of Brownian motion. Gouy’s major contribution wasto point out a theoretical difficulty with the atomic hypothesis: it seemed to violate the secondprinciple of thermodynamics, as the thermal energy from molecular agitation was converted intomechanical work that moved the suspended particles.In 1900, Felix Exner, Siegmund Exner’s son, was the first one to carry quantitative andrepeated measurements to study the influence of the particle’s size and of the temperature onthe velocity of suspended particles. He measured that the intensity of movement increased as thesize of the suspended particles decreased. Whereas the kinetic theory predicted a proportional In fact, Brown himself cited a 1819 work on this point, by Bywater from Liverpool, as related in (Duplantier,2007), but denied the construction of his experiment. On this point, Perrin later showed that the influence of the temperature had never been truly measured sinceviscosity also depends on temperature, therefore experimenters rather measured the influence of viscosity. T = − ◦ C. He himself had no opinion on his result, he didnot see it as an argument in favour of a theory or another (Maiocchi, 1990).In 1902, Richard Adolf Zsigmondy invented the ultramicroscope, which allowed much moreprecise measurements of Brownian movement in colloids. He won the Nobel Prize in Chemistryin 1925 for his invention and the work that followed.Several factors were invoked to explain this lack of strong result during the nineteenth century,including the lack of interest of physicists for this phenomenon. Yet the revival of the kinetictheory of gases and James Clerk Maxwell’s and Ludwig Boltzmann’s ground-breaking works inthe years 1860-1890 boosted the researches on the links between microscopic theory and heat.The lack of suitable mathematical tools was also highlighted for the first observations, sincethey occurred before or shortly after the 1860s, in which statistical methods from the kinetictheory of gases became available . That said, according to Roberto Maiocchi, it is important notto take the lack of mathematical tools as solely responsible for the difficulties. As highlightedbefore, the set of experimental data was quite fuzzy so physicists were far from the ideal casewhere a strong set of cross-confirmed experimental data was only waiting to be accounted forby a theory. As we shall see, Einstein’s theory on Brownian motion did not emerge from theknowledge of the experiments conducted in the nineteenth century, and more importantly, itconcerns a quantity (displacement) that had never been measured during this period.
3. Brownian motion as a physical concept
After nearly 80 years without a satisfactory theory for Brownian motion, Einstein, Smoluchowskiand Langevin published their works over a period of only four years, between 1905 and 1908.All three theories rely on the atomic hypothesis, that is to say they explained the displacementsof suspended particles by the collisions with the particles composing the liquid. At that time,the atomic hypothesis was not yet accepted by the whole community and those theories alongwith Perrin’s experiments played a major role in its acceptance.Einstein published a series of five articles on Brownian motion between 1905 and 1908, gath-ered in (Einstein, 1926). We decided to study the first one (Einstein, 1905), which contains allthe ingredients of his theory and which is one of the reference article on the subject; and thefourth one in which he tackled the issue of the experimental measurement of the velocity ofBrownian particles. The other three are not particularly relevant for our study.Smoluchowski started to work on Brownian movement before Einstein but he did not publishhis results before 1906 (Smoluchowski, 1906), as he was waiting for more experimental evidenceand was finally pushed by Einstein’s publication. He continued to work and publish on Brownianmovement until his death in 1917.Langevin wrote only one article on Brownian motion, in 1908 in the
Comptes rendus del’acad´emie des sciences (Langevin, 1908).Their articles have been much discussed in the literature so we do not attempt to give afull rendition of their ideas but rather to highlight the main reasonings and the main results,because they are the starting point of Wiener’s, Ornstein’s and Uhlenbeck’s works, as we shallsee in the following sections. Despite the link between the kinetic theory of gases and Brownian motion, it is striking to note that neitherMaxwell nor Rudolf Clausius published on Brownian motion. Boltzmann was aware of some of the Brownianmotion experiments, which he mentioned in a letter to Ernst Zermelo in 1896 (Darrigol, 2018), but he nevertackled this issue, whereas it could have been a great test for his theory. see Brush, 1976; Duplantier, 2007; Maiocchi, 1990; Nelson, 1967; Nye, 1972; Piasecki, 2007 for detailed analysis.
4e propose to analyse each theory through three questions: (i) what are their physical in-gredients? (ii) how do they introduce stochasticity into the equations? and (iii) what are theirhypotheses concerning velocity?We then look at the reception of these theories in the experimenters’ world, through theworks by Svedberg, Seddig, Henri and Perrin and the corresponding answers from Einstein,Smoluchowski and Langevin; because it offers some insights on the thorny understanding ofBrownian velocity.
In his 1905 article, Einstein obtained two major results: the relation between the diffusioncoefficient and the properties of the medium; and the correspondence between Brownian motionand diffusion. Interestingly enough, it was not directly an article about Brownian motion sincehe declared that he did not know if the phenomenon he studied was what experimenters calledBrownian motion, but it could be. His aim was clearly not to account for experimental resultsbut rather to propose a test for the validity of the kinetic theory of gases. If the kinetic theoryof gases was true then microscopic bodies in suspension in a liquid should be in movement,and this motion should be observable with a microscope. On the other hand, such behaviourwas forbidden by classic thermodynamics which predicted an equilibrium, thus putting the twotheories in conflict. He then defined a measurable quantity which can weigh in favour of a theoryor the other: the mean of the squares of displacements.Einstein’s invention of the physical theory of Brownian motion was discussed in detail in(Renn, 2005). In particular, J¨urgen Renn analysed how Einstein combined his ideas comingfrom his 1901-1902 work on solution theory and his 1902-1904 work on the statistical interpre-tation of heat radiations, to come up with the idea that the atomic hypothesis could be testedby observing fluctuations from particles in solution.Einstein’s reasoning was built in three steps. In the first step, he related the diffusion coeffi-cient to the properties of the medium, in a second step he derived the diffusion equation from aseries of hypotheses on the particle’s motion, and lastly he combined the two results.Let us analyze the physical ingredients used in the first step, without going into details.Einstein used two physical ingredients from different validity domains, which was one of hismaster ideas. The first one is Stokes’ law, which describes the force (cid:126)F that undergoes a sphericalbody of radius a when in movement at constant velocity (cid:126)v in a fluid of viscosity µ : (cid:126)F = − πµa(cid:126)v .The second one is van ’t Hoff law, similar to ideal gas law, which relates the pressure increaseΠ, called osmotic pressure and due to the addition of dilute particles in a solution; and theconcentration n of those dilute particles: Π = nRT /N A , where N A is Avogadro constant, T the temperature and R the gas constant. Even if Einstein’s theory was based on the atomichypothesis and thus on the collisions between particles, it was not directly an ingredient he tookinto account in his calculations.First, Einstein considered the equilibrium between two force densities: the gradient of osmoticpressure, and an external force density (in this case the viscous force described to Stokes’ law): nF − ∂ Π /∂x = 0. Second, at equilibrium two processes act in opposite directions: a movement ofthe suspended particles under the influence of the force F , and a diffusion process considered asa result of the thermal agitation. This can be written by canceling the number of particles thatcross a unit area per unit time due to both processes: nF/ πµa − D∂n/∂x = 0. By combiningthe two equilibrium relations with the definition of F and Π, Einstein obtained the first and An Australian physicist named William Sutherland, derived a very similar equation in 1904, before Einstein.His equation was D = RTN A πµa µ/βa µ/βa , where β came from a generalized Stokes’ law, and should be taken D = RTN A πµa , (3.1)of which we can read the full derivation in (Duplantier, 2007) (extended version of the originalarticle (Duplantier, 2006)).In a second phase, he examined ‘the irregular movement of particles suspended in a liquidand the relation of this to diffusion’. To introduce the irregularity in his equations, he usedprobability distributions, in the fashion of the kinetic theory of gases. Particles are localised bytheir positions x in one dimension. They undergo displacements ∆ over a time τ . Displacementsare random and distributed according to a probability law φ τ , normalised as (cid:82) + ∞−∞ φ τ (∆)d∆ = 1.Einstein called f ( x, t ) the number of particles having a position between x and x + d x at time t . f is normalised at any moment t as (cid:82) + ∞−∞ f ( x, t )d x = N , where N is the total number ofparticles. Einstein next made a series of hypotheses:(i) Displacements of each particles are independent of that of others,(ii) We work at a timescale τ smaller than the observation time, but large enough for thedisplacements of a particle to be independent on two consecutive intervals of length τ ,(iii) The function φ τ is non-null only for small values of ∆, in other words only smalldisplacements are allowed over a time τ ,(iv) The space is isotropic, thus there is no privileged direction, and the probability dis-tribution for displacements is even: φ τ (∆) = φ τ ( − ∆).As we will see in a moment, future theories did not necessarily accept these theories. Einsteinnevertheless judged them natural and used them to write the relation between the distribution f at time t + τ and that at time t as follows f ( x, t + τ ) = (cid:90) + ∞−∞ f ( x + ∆ , t ) φ τ (∆)d∆ . (3.2)Using hypotheses (ii) and (iii), he expanded the left-hand side at first order in τ and the right-hand side at second order in ∆, thus obtaining ∂f∂t = ∂ f∂x · τ (cid:90) + ∞−∞ ∆ φ τ (∆)d∆ , (3.3)in which he recognised a diffusion equation with the diffusion coefficient: D = 1 τ (cid:90) + ∞−∞ ∆ φ τ (∆)d∆ . (3.4)Lastly, the distribution f obeys a diffusion equation , which reads ∂f∂t = D ∂ f∂x . (3.5) infinite to compare to Einstein’s result. He presented his derivation in January 1904 in an Australian congressand published his result in the beginning of the year 1905 in the proceedings of the congress and then in March1905 in Philosophical Magazine , two months before Einstein’s article. He was however completely forgottenfor Einstein’s benefit. To explain this historical curiosity, several hypotheses have been emitted like a misprintin his first article of 1905, Sutherland’s weak influence in Europe or the chemistry-rooted style he used. Formore details, see (Duplantier, 2007; Home, 2005). Einstein was not the first to establish the link between a random process and the diffusion equation. In factLouis Bachelier, working under the direction of Henri Poincar´e, published a memoir in 1900 (Bachelier, 1900)in which he found the diffusion equation for options prices in market economy. His contribution to Brownianmotion is studied in Dimand, 1993. f ( x, t ) = N √ πDt exp (cid:18) − x Dt (cid:19) . (3.6)Einstein noticed that thanks to the independence described by his hypothesis (i), he could choosethe starting point of each particle as the origin of the associated coordinate system, rather thana common one. Thus f ( x, t ) becomes the number of particles having undergone a displacement x between time 0 and time t . The probability distribution for the displacements is naturally f /N .Einstein computed the second moment of this distribution, which is the mean of the squaresof displacements (because the distribution is even), as λ x = (cid:104) x (cid:105) = 2 Dt . (3.7)In the last phase, Einstein combined the results of the two first parts to obtain λ x = (cid:115) RTN A πµa √ t . (3.8)This is probably the most famous result on Brownian motion, and Einstein presented it as aphysically measurable quantity, that could be the test-quantity we talked about in the introduc-tion of the article. From this perspective, he computed the numerical value λ x = 0 . µ m, withstandard values for the parameters and for one second of time.Einstein later proposed other approaches and gave other proofs for this relation. In particular,in his second paper on the subject (Einstein, 1906), in which he was convinced at that timethat the phenomenon he described and Brownian motion were the same thing, he gave a moretheoretical and general approach. He proposed in this paper a theory that accounted not onlyfor translational Brownian motion but also for rotational Brownian motion, and highlighted thefundamental role of Boltzmann’s distribution for Brownian motion.Einstein developed a third derivation of this relation in a lecture given in Zurich in 1910, en-titled ‘On Boltzmann’s Principle and Some Immediate Consequences Thereof’. This lecture hasbeen translated in English and published in (Einstein, 2006). Einstein discussed the concept ofirreversibly in physics through the statistical interpretation of the second law of thermodynam-ics, and Boltzmann’s statistical entropy formula S = k B log W . He considered, as an examplefor his reasoning, the case of a particle in a fluid, subjected to gravitation, and computed itsheight distribution using Boltzmann’s entropy formula. By simply assuming the stationarity ofthis distribution he computed the mean square height and found quite elegantly eq. (3.8).We note that Einstein only worked in displacements space and did not refer at all to the ve-locity of particles. Neither did he speak of the true length of a particle trajectory, but rather ofthe displacement which is the difference between two positions at different times. This confirmswhat has been said in section 2: Einstein introduced the suitable quantity to describe Brownianmotion, which had not been not discussed by experimenters in the nineteenth century. This wasa point of conflict with later experimenters as we will discuss in a moment.Before closing this section, let us take a few lines to analyse the introduction of the timescale τ from a critical point of view. It is defined between the microscopic timescale τ corr for whichthere are correlations between displacements, and the macroscopic timescale τ macro which is thecharacteristic time of variation for observable quantities, such as f ( x, t ). Thus, τ cannot be7aken to be 0, however there are two steps in Einstein’s calculation which implicitly supposea τ → τ ; second, in the identificationof the diffusion coefficient with the integral term in eq. (3.4). Indeed, D should not dependon an arbitrary timescale (besides, Einstein did note write D τ ), whereas the right-hand sideexplicitly depends on τ . The only escape from this contradiction is that in the τ → τ . How to satisfy both conditions? This is discussed indetails in (Ryskin, 1997) and the answer is that the limit is not formally reached, but peoplestill write τ → τ (cid:28) τ macro . The supposition of the existence of τ and the lack ofmathematical rigour in the treatment of its limit are clearly the weak points of Einstein’s article.On this particular point Ornstein and Uhlenbeck sharpened Einstein’s theory as we shall see insection 5, and Wiener diverged from the physical reality, as we will examine in section 4.2.3. Smoluchowski published his article in 1906, pushed by the first two articles published by Einsteinin 1905 and 1906. Einstein’s and Smoluchowski’s articles are very different in style.Firstly, Smoluchowski knew in details all experimental works carried on Brownian motionbefore 1906, he gave a clear account of them at the beginning of his article, and he constructedhis theory in order to account for these observations; whereas Einstein was not sure that theproblem he dealt with really was Brownian motion. Secondly, Smoluchowski’s calculations weredirectly based on the collisions between particles, which was better in his opinion because itoffered an intuitive understanding of the microscopic mechanism, even though both theoriesgave the same results. Thirdly, he introduced stochasticity by the mean of average quantitiesfrom the beginning of his article, while Einstein’s computation dealing with distributions wasmore general. Lastly, he examined more cases than Einstein did, by considering the case wherethe particle dimension is small compared to the free mean path of the solution’s particles.Smoluchowski must also be credited for the counter-argument by which he debunked N¨ageli’scriticism of the atomic hypothesis, evoked in section 2, and which had remained unanswereduntil 1906. His idea was that, even if the velocity communicated to a suspended particle bya collision is tiny (around 2 × − mm s − ), as pointed out by N¨ageli, one must not deducethat collisions are unable to move suspended particles at the measured velocities, if they acttogether. Indeed, even though the average position is null, due to space isotropy, the meanof the deviation (a positive quantity) from the initial position is non-null, and evolves as thesquare root of the number n of collisions . Thus, if n is large enough, most collisions cancelbut √ n collisions contribute to a displacement in one direction. According to him, there are10 collisions per second in a liquid with makes 10 collisions contributing to the displacement.Taking N¨ageli’s value for the velocity communicated by one collision, then 10 collisions givethe particle a velocity 10 cm s − . This value is false as well, due to voluntary omissions. Indeed,according to Smoluchowski, the absolute value of the change in velocity depends on the absolutevalue of the velocity before the collision, and is therefore different for each collision; and theprobability of a collision that slows down the movement is greater than the probability of acollision that speeds it up. However, this was a victory against N¨ageli’s argument. This wasonly a qualitative answer for Smoluchowski who continued with a quantitative argument. Thetrue value of the velocity is given by the equipartition of energy (eq. (3.9)), and should thereforebe v = 0 . − , which is still not in agreement with the experimental values. Nevertheless,for Smoluchowski, this was the good value. Indeed, it is impossible to follow experimentally thetrue trajectory of a particle that undergoes 10 collisions per second. Observed trajectoriesare averaged trajectories for which the length of the path is greatly underestimated. His valueshould be the good one between two collisions, which is not a measurable timescale. See (Duplantier, 2007) for the complete derivation. (cid:104) v (cid:105)√ m = (cid:104) v (cid:48) (cid:105)√ m (cid:48) , (3.9)where v and m are as usual the velocity and the mass of the Brownian particle and v (cid:48) and m (cid:48) are the velocity and the mass of the medium particles. Since Smoluchowski worked along waywith average quantities, we stop writing brackets from now on. Moreover, he considered thatBrownian particles were weakly deviated at each collision, by a constant small angle ε = 3 v/ v (cid:48) .If we note λ the mean free path of medium particles, and a the Brownian particle radius, thereare two cases: (i) a < λ and (ii) a > λ . The second case is the most common one, experimentedin the lab and described by Einstein, where suspended particles are significantly bigger thansolution particles. Figure 1: Example of trajectory where P i are the collision pointsand ε is the deviation angle.From (Smoluchowski, 1906).Let us look at case (i) first. Smoluchowski consid-ered the simple case where the particle travels exactlythe distance l between each collision. Thus, the veloc-ity of the particle, the distance it travelled between twocollisions and the angle by which it is deviated when acollision occurs are constant; the only random param-eter is the direction of the particle after a collision, forwhich Smoluchowski takes a uniform probability on thecone of angle ε . By writing Λ n = OP n , the end-to-enddistance travelled between the starting point and then-th collision, Smoluchowski aims to express (cid:112) (cid:104) Λ n (cid:105) asa function of the problem parameters. After some cal-culations, which can be found in (Duplantier, 2007),he demonstrates that (cid:112) (cid:104) Λ n (cid:105) = 83 v (cid:48) (cid:114) tn ∗ , (3.10)where n ∗ iss the number of collisions per second, de-fined by n = t · n ∗ . The case (ii) is more difficult andgives a very similar result, so we choose not to analyseit, but rather to look at the comparison between theabove result and Einstein’s formula eq. (3.8).Smoluchowski sought a relation between the friction coefficient S and the parameters m and n ∗ , in order to compare his result to Einstein’s one. According to him, usual methods gave S = 2 m (cid:48) n ∗ /
3, which allowed him to replace n ∗ in eq. (3.10). He also used Stokes’ law S = 6 πµa to substitute S and finally obtained (cid:112) (cid:104) Λ n (cid:105) = 89 (cid:115) m (cid:48) v (cid:48) πµa √ t . (3.11)Although Smoluchowski did not do that, we can use the equipartition of energy to replace m (cid:48) v (cid:48) by k B T , for the one-dimensional case, in the above equation, giving (cid:112) (cid:104) x (cid:105) = (cid:114) (cid:115) k B T πµa √ t , (3.12) In fact, there is a mistake in his calculations. Instead of the numerical factor 8 /
3, he found 4 √ /
3, but for thesake of clarity we choose to give the correct result. (cid:112) /
27. This small differenceis not surprising given all the approximations Smoluchowski made.Unlike Einstein, Smoluchowski wanted to compare his result with already existing data. Hetook for comparison Felix Exner’s values, which he quoted in the introduction of his article, v = 3 . × − cm s − . According to eq. (3.11) with similar parameters to that of Exner, hegot (cid:112) (cid:104) x (cid:105) /t = 1 . × − cm s − . To compensate for this difference he introduced a rathermysterious coefficient π √ / π/ √
10 factor is much more unjustified. By dividingExner’s value by this coefficient, he obtained v = 1 . × − cm s − , which was in agreementwith his value.At the end of his article, Smoluchowski obtained a relation between the diffusion coefficient D and the parameters of the problem, by combining a qualitative reasoning on the mean freepath and a result from his previous article on mean free path, which read D = 16243 m (cid:48) v (cid:48) µπa . (3.13)This result is directly comparable to Einstein’s in eq. (3.1), though still differing by a factor64 / n . In addition to his numerous personal contributions to physics, Paul Langevin is known to haveread, understood and diffused Einstein’s ideas on relativity and Brownian motion in France.He published his only article on the subject in 1908 in the
Comptes rendus hebdomadaires del’acad´emie des sciences in full knowledge of Einstein’s and Smoluchowski’s articles.Langevin’s derivation is so short and powerful, and radically different in the way he introducedrandomness in the equations, that even if it has already been discussed in the literature,it isworth being given in details.Langevin started with the announcement of the exact correspondence between Einstein’sand Smoluchowski’s results, which differed until now by a factor (cid:112) /
27, if one applies somecorrections to Smoluchowski’s derivation, even though he did not say which corrections.Langevin introduced randomness in a new way. Einstein (and also Smoluchowski in laterarticles we did not discuss) worked on probability distributions to establish partial differentialequations. His equations were deterministic, in the sense that they admit an exact solution, butthe unknowns of his equations were the distributions, which are of probabilistic nature. On thecontrary, Langevin used a probabilistic equation, which includes a stochastic noise and thereforecannot be solved directly, but governs a deterministic variable: the velocity . This equation,now known as Langevin equation, is m d x d t = − πµa d x d t + X . (3.14) The additional factor was mentioned by Smoluchowski himself in his article, but with the value (cid:112) /
27 due tothe error of a factor √ Both ways of introducing the stochastic aspect of a problem in the equations are the pillars of the stochasticprocesses, as a branch of mathematics and theoretical physics. We still talk about Langevin equation (orstochastic equation) to refer to the case where a random variable appears in a partial differential equation,and about Fokker-Planck equation when the unknowns of the deterministic partial differential equation areprobability distributions. Fokker-Planck equation is of the same family as the one first used by Einstein andSmoluchowski, but was named after Adrian Fokker who worked with Max Planck on his thesis in 1913. Hisformula contains a convection term which makes it more general than the one applied to Brownian motionwhich only contains diffusion. X .Langevin justified this additional force by explaining that the viscous friction force only de-scribes the average effect of the resistance of the medium, which is in reality fluctuating becauseof the irregularity of the collisions with the surrounding molecules. The stochastic force X thenaccounts for the fluctuations around this average value. The two forces are therefore due to thesame phenomenon: the collisions with medium particles, but one is averaged, deterministic andis in the opposite direction of the drift velocity, while the other is fluctuating, stochastic andhas no privileged direction. Moreover, the value of X is such that it maintains the particle’smovement, which would stop otherwise because of the dissipative force.Because of X , this equation cannot be solved exactly, so Langevin multiplied it by x to obtain m x d t − m (cid:18) d x d t (cid:19) = − πµa d x d t + xX . (3.15)He took the mean of the above equation over a large number of particles, making the term (cid:104) xX (cid:105) vanish because of the irregularity of the collisions described by X . He also replaced the term m (cid:104) (d x/ d t ) (cid:105) by RT /N A using the equipartition of energy. Thus, he obtained a deterministicequation governing the newly-defined variable z = d (cid:104) x (cid:105) / d t , written as m z d t + 3 πµaz = RTN A . (3.16)This equation is similar to Einstein’s type of equation discussed earlier since it is a deterministicequation governing a random variable, with is in this case not a probability distribution but thetime derivative of one of its moments.The solution is given by z ( t ) = RTN A πµa + C exp (cid:18) − πµam t (cid:19) , (3.17)where C is a constant of integration. The second term in the right-hand side decreases expo-nentially with a characteristic time θ = m πµa , (3.18)and then becomes negligible when time t is around θ , whose value is approximately given by10 − s. This value is much smaller than measurable intervals so experimenters are always is thecase where the first term in the right-hand side prevails.Finally, he replaced z by its definition and integrated once to obtain the exact same result asEinstein’s eq. (3.8). Even though his final result was a relation governing (cid:104) x (cid:105) and not includingvelocity, the latter was thought to exist and to be well defined since it was used in the very firstequation, which was a fundamental difference with Einstein’s and Smoluchowski’s theories. Since Svedberg’s and Seddig’s articles have not been translated into English, the following mainlycomes from (Kerker, 1976) for Svedberg and from (Maiocchi, 1990) for Seddig.Perrin’s work has been deeply documented, and is often the logical following step in Brow-nian motion histories, after Langevin’s theory. Therefore there is not much we can add to theexisting literature, but we can instead describe preceding works by Svedberg, Seddig and Henri, Langevin wrote ‘The mean value of Xx is obviously null due to the irregularity of complementary actions X ’.This physical intuition has been discussed and criticised in (Naqvi, 2005). v = 0 .
03 cm s − . even when varying the viscosityand the particle size.Svedberg later discovered Einstein’s 1905 article but did not understand it and tried to connecthis misconceptions on Brownian motion to Einstein’s results in a second paper (Svedberg, 1906b).He replaced, in Einstein’s eq. (3.8), (cid:112) (cid:104) λ x (cid:105) by four times the amplitude of his supposed sinusoid,whereas the first quantity is stochastic and the second one is deterministic; and replaced time t by the period of the oscillations, whereas the first one is non-specified and the second one isa property of the motion. These two confusions show the misunderstanding of Einstein’s workin the experimental world in the first years. Svedberg then checked his data against the newformula and the results differed by a factor 6 or 7, which he judged tolerable.It is striking that in spite of this accumulation of mistakes, which Einstein, Langevin andPerrin soon remarked on, Svedberg continued to trust his theory all his life and was evenawarded the chemistry Nobel prize in 1926, the same year Perrin received the physics Nobelprize, both for their contributions to Einstein’s theory. Let us look at how physicists reacted toSvedberg’s experiments.Perrin was the most severe regarding Svedberg’s theory and several signs of his criticism canbe found in work. Here is a sample: Until 1908, there had not been published any verification or attempt that gave a clue aboutEinstein’s and Smoluchowski’s remarks. [Then in footnote] Svedberg’s first work on Brownianmotion is no exception [Svedberg, 1906a; Svedberg, 1906b]. Indeed:1. The lengths given as displacements are 6 to 7 times too high, which, supposing they arecorrectly defined, would be no progress, especially on the discussion due to Smoluchowski;2. Much more gravely, Svedberg thought that Brownian motion became oscillatory for ultra-microscopic particles. It is the wavelength (?) of this motion which he measured and usedas Einstein’s displacement. It is obviously impossible to test a theory taking as a startingpoint a phenomenon which, supposed exact, would be in contradiction with this theory . Iadd that, at no scale Brownian motion shows an oscillatory behaviour.(Perrin, 1913, p. 178-179)
The most mysterious reaction was surely that of Smoluchowski. When Smoluchowski’s 1906article arrived in Uppsala, where Svedberg worked, the latter had already published his second1906 article, in which he compared his results to Einstein’s formula. However, due to the12umerical error in Smoluchowski’s article, discussed in section 3.1.2, Svedberg’s result were closerto Smoluchowski’s predictions than to that of Einstein. He thus wrote a letter to Smoluchowskito show him his results, to which the latter replied enthusiastically. From that moment on,the two physicists started a correspondence, as documented in (´Sredniawa, 1992). Thanks tothe help of Smoluchowski who suggested some small modifications, Svedberg published anotherarticle in 1907, in which his results were only 3 to 4 times too large compared to theoreticalvalues, against 6 to 7 times in his 1906 article. Their scientific collaboration spanned the periodfrom 1907 to 1914, including works on Brownian motion as well as on density fluctuations.Up until 1916, Smoluchowski cited Svedberg’s articles on Brownian motion, even after Perrin’sresults published in 1908.In his only article, Langevin briefly criticised Svedberg’s results for two reasons. First, hisvalues differed by a factor 1 / (cid:104) x (cid:105) .Einstein’s answer is surely the most interesting for us because it offers some insights into whythe experimental measure of Brownian velocity is in fact not possible. He published in 1907 hisfourth article on Brownian motion (Einstein, 1907), which opened with a reference to Svedberg’sworks, of which he wished to clarify some theoretical points for experimentalists. He startedfrom the equipartition of energy, written as m (cid:104) v (cid:105) = 3 RT /N A and used Svedberg’s values fortemperature and particles mass to compute the square root of the mean squared velocity as (cid:112) (cid:104) v (cid:105) = 8 . − . (3.19)He then questioned the possibility to observe such a gigantic velocity. He used a simple reasoningto show that it is in fact not possible. If one takes the simplified model in which the particleis only submitted to the frictional force, the equation governing the evolution of its velocity isthen m d v/ d t = − πµav and the velocity decreases exponentially. Einstein computed the time θ after which the velocity is only 10% of its initial value, as θ = m ln(10)6 πµa , (3.20)which for Svedberg’s parameters takes the value θ = 3 . × − s . (3.21)This timescale is clearly not accessible experimentally, therefore it is not possible to observethe value of velocity given by eq. (3.19). Moreover, Einstein considered a simplified case but inreality one has to take collisions into account, which makes the measure even more impossible.Indeed, for the mean velocity to be maintained at equilibrium according to the equipartitionof energy, the velocity decrease due to viscosity must be balanced by collisions which transferimpulses to the particles. Since collisions are extremely frequent, the particle movement isaltered even during the short timescale θ , which makes it impossible to define a velocity .At the end of his article, Einstein gave a more theoretical argument to explain that velocity isnot a suitable quantity to describe Brownian motion. Using his eq. (3.8), he defined a quantitywhich had the meaning of the average velocity of a particle during a time t , expressed as λ x t ∝ √ t , (3.22) Einstein did not do it, but we can picture the number of collisions in question with the help of Smoluchowski’svalue given in (Smoluchowski, 1906). According to the latter, there are 10 collisions per second in a liquid, soduring the time θ for which the particle loses 90% of its velocity, the particle undergoes 3 . × collisions. Itis therefore impossible to assign neither a value nor a direction to the velocity of the particle at this timescale. t , and thus does not reach any limiting value as t decreases , while remaining larger than θ . Thus, the value of this mean-velocity-like quantity has no meaning since it depends on theobservation time, and therefore all velocities that are measured experimentally, since they aremean velocities by nature because of the experimental incapacity to follow the true path, aredoomed to be dependent on the measurement time.Seddig’s case is more subtle since his misconceptions are less obvious. He knew Einstein’sworks when he published in 1907 and 1908 his articles, in which he seemed to check the rela-tion λ x ∝ (cid:112) T /µ , when varying T with t constant. Once again, Perrin later said that it wasdifficult to draw conclusions from these experiments, because the viscosity µ also depends onthe temperature. We must wait 1911 for Seddig to add details to his experiments from 1907and 1908. From these new details it appears that he misunderstood λ x for the actual lengthtravelled by particles during a time t and not the displacement. To measure the length of thepath, he tried to take long exposure pictures but this was too complex since the light requiredfor the picture brought energy to the liquid and then distorted the results. For the sake ofhis experiment, he was therefore forced to send only two very close flash lights and to measurethe distance travelled in straight line during these two flash, which was in fact the good read-ing of the quantity λ x although he was unaware of it. He tried to find a way to recover thetrue length of the path, which he thought to be the true meaning of λ x , from the displacement,but never succeeded, leading to the publication of his results which are in agreement with theory.Perrin’s name is associated with the experimental verification of Einstein’s results, but in facthe was interested in statistical physics questions, close to Brownian motion, even before readingEinstein’s articles. For example, as early as 1906, he gave a lecture in which he discussed the linkbetween the second law of thermodynamics and the atomic hypothesis, leading to the subject ofBrownian motion (Brush, 1976). He ended up proposing a protocol invented in order to violatethe second principle .In 1906, Perrin published an article unrelated to Brownian motion, in which he spoke forthe first time of the interest for physics of mathematicians’ functions without tangents (Brush,1976). These functions are useful as an analogy for Perrin to describe the discontinuity ofmatter. Indeed, even if matter seems smooth and continuous it is in fact heterogeneous anddiscontinuous when looked through a microscope. This mathematical concept was later usedagain by Perrin to describe Brownian trajectories, which was a starting point of Wiener’s work,as we shall se in section 4.1.On May 11, 1908 Perrin published his first results on Brownian motion in the Comptes rendusde l’acad´emie des sciences (Perrin, 1908a). At first sight, this article was very surprising becausePerrin announced that he verified Einstein’s theory, but there was no sign of Einstein’s workin this article. Perrin rather tested the altitude distribution of particles suspended in a liquid, Einstein already mentioned this idea at the end of his 1906 article on Brownian motion (Einstein, 1906), ina section named ‘On the limits of application of the formula for (cid:112) (cid:104) ∆ (cid:105) ’. He defined the same quantity λ x /t ∝ / √ t diverging as t →
0, which is physically impossible. Einstein explained it by the fact that anhypothesis he used when deriving his result is caught off guard when taking the limit t →
0: the independenceof collisions. Therefore, velocity values obtained by this calculation bear no meaning. His idea was to take advantage of osmotic pressure, which was also a fundamental ingredient of Einstein’stheory, even though the two were unrelated, to recover the mechanical work performed by suspended particleslocalised in one half of the solution, onto a semi-permeable membrane that split the solution in two halves.This idea of proposing thought experiment to call into question the second principle dates back to Maxwell’sdemon, invented in a private letter he send to Peter Guthrie Tait in 1867 (Maxwell, 1867). It is interestingto note that Smoluchowski also actively worked on Maxwell’s demon and its relation to Brownian motion(Smoluchowski, 1912). The common history of Maxwell’s demon and Brownian motion was very rich andfruitful, but we cannot study it further because it would take us away too far from our subject. , which links the mean square of displacementsto time and other parameters. He used a complex photographic set-up, working with two flash0.05 seconds away, to test the formula. Unfortunately, he found that his results were 4 timeslarger than the ones predicted by the theory. This was a new failure for the atomic hypothesis,considering that this time the experiment and its interpretation were faultless. On July 6, 1908he published another article (Henri, 1908b), which dealt a new blow to the theory. He found thatthe increase of the solution’s acidity slowed down Brownian motion, whereas Brownian motionshould only be impacted by one solution property: its viscosity, and viscosity was not changedby this small rise of acidity. No one ever detected errors or flaws in Henri’s experiments, andthus no one could explain why his results were diverging from the theory. Perrin declared The method was fully correct, and had the merit of being used for the first time. I do not knowthe cause that distorted the results. (Perrin, 1913, p.180, footnote 2)
Perrin later obtained good results using the same method.On July 13, 1908 Jacques Duclaux took Svedberg’s and Henri’s experiments as an argumentagainst the atomic hypothesis (Duclaux, 1908). He particularly criticized the use of Stokes’ lawoutside its domain of validity. Indeed, Stokes’ law is supposed to be used for larger particles,around the millimetre, and the solution is supposed to be continuous, while neither of thetwo hypotheses is satisfied. Perrin answered this criticism on September 7, 1908 by publishinghis conclusive test of Stokes’ law validity at the scale of Brownian particles (Perrin, 1908b).In reality, his reasoning was circular and was not a real proof, as demonstrated in detail in(Maiocchi, 1990). However, the mistake was not revealed soon and Perrin’s article scored apoint.Eventually, Joseph Ulysses Chaudesaigues, who was working in Perrin’s lab on Brownianmotion experiments at that time, published on November 30, 1908 the article that put a stopto the debate on the theory’s validity (Chaudesaigues, 1908). Perrin invented a protocol toprepare emulsions containing particles of the exact same size, which was a great advantage sinceit greatly reduced the uncertainty due to the particle size. Thanks to this particular method,they successfully tested eq. (3.8) and also checked that the influence on the mean square ofdisplacements of the particle size, the liquid viscosity, and the experiment duration were thosepredicted by Einstein’s formula. Perrin’s numerous experiments on Brownian motion from 1908to 1913 are gathered in his 1913 best-selling
Les Atomes .
4. Norbert Wiener’s theory of Brownian motion
By the end of the 1910s, the physicists’ Brownian movement reached a satisfactory stage oftheorization since the theories proposed by Einstein, Smoluchowski and Langevin were exper-imentally confirmed by Perrin. In addition, the atomic hypothesis gained a lot of ground andconvinced some energetists until then doubtful. Many physicists then turned to the applicationof these theories to the determination of fundamental quantities such as Avogadro number andthe elementary charge or molecular sizes. Although some theoretical physicists continued to Indeed, Seddig tested it in 1907, but as we saw he misunderstood the quantity λ x , whereas Henri did not. What were the reasons for Norbert Wiener, a young 25 years old mathematician, to publish hisfirst article on Brownian motion in 1921, when it was not yet a subject for mathematicians?Perrin’s description of Brownian trajectories by mathematicians’ functions without tangent isoften presented as the starting point of Wiener’s interest in the question.Indeed, Wiener spoke of Perrin’s description of Brownian trajectories in his autobiography
Iam a mathematician - the later life of a prodigy as follows
Here the literature was very scant, but it did include a telling comment by the French physicistPerrin in his book
Les Atomes , where he said in effect that the very irregular curves followedby the particles in the Brownian motion led one to think of the supposed continuous non-differentiable curves of the mathematicians. He called the motion continuous because theparticles never jump over a gap and non-differentiable because at no time do they seem to havea well-defined direction of movement. (Wiener, 1956, p.38-39)
Pesi Masani, Norbert Wiener’s biographer, author of
Norbert Wiener 1894 - 1964 , discussedWiener’s reading of Perrin’s
Les Atomes , and referred in particular to the following quote
Those who hear of curves without tangents or of functions without derivatives often think at firstthat Nature presents no such complications nor even suggests them. The contrary. however,is true and the logic of the mathematicians has kept them nearer to reality than the practicalrepresentations employed by physicist (Perrin, 1913, p.25-26, Masani, 1990, p.79) which was ‘music to Wiener’s ears’. In his 1923 article, Wiener quoted Perrin, as translated byFrederick Soddy in 1910
One realizes from such examples how near the mathematicians are to the truth in refusing, bya logical instinct, to admit the pretended geometrical demonstrations, which are regarded asexperimental evidence for the existence of a tangent at each point of a curve. (Perrin, 1909,p.81, Wiener, 1923, p.133)
It is clear then that the mathematical hypothesis expressed by Perrin played a role in the birthof Wiener’s interest in Brownian motion, but was this the only reason? To answer this question,16t is necessary to briefly study Wiener’s biography, his connection, his mathematical interestsand the publications preceding that of 1921.The following biographical elements are taken from Wiener’s biographies (Wiener, 1956;Masani, 1990).Norbert Wiener was a mathematician born in 1894 in the United States. He entered TuftsUniversity in Boston at only 12 to study mathematics and biology, obtained his A.B. degreein mathematics, and then entered Harvard Graduate School for Zoology in 1909. Unwillingto continue in this branch, he was transferred to Harvard Graduate School for Philosophy in1911, where he studied philosophy and mathematics. In 1913, just 18 years old, he obtained hisPhD in mathematical logic and went to study logic, philosophy and mathematics in Cambridge(England) with Bertrand Russell, thanks to a Harvard post-doctoral fellowship.During his stay in Cambridge in 1913-1914, Russell advised him to open up to disciplinesother than pure logic and mathematics foundations, and Russell mentioned the interface be-tween mathematics and physics. Wiener followed his advice and read Rutherford’s work onelectron theory, Niels Bohr’s atomic theory, Einstein’s and Smoluchowski’s works on Brownianmotion, and Perrin’s
Les Atomes . It is interesting to note that neither Wiener nor Masanimentions reading Langevin’s article, which may explain why all of Wiener’s work was basedon the Einstein-Smoluchowski approach; while Ornstein, Uhlenbeck and Doob took Langevin’sformalism with stochastic noise as a starting point, as we shall see in section 5.In Cambridge, Wiener was also attending mathematics lectures by Godfrey Harold Hardy, whowould be the most influential teacher for the young Wiener. He discovered with Hardy otheraspects of mathematics and especially Lebesgue integration, named after Henri-L´eon Lebesgue.Russell’s lessons also introduced him to Einstein’s theory of relativity. His interest in the interfacebetween mathematics and the physical sciences arose at this time from his reading and from theinfluence of his two professors Russel and Hardy. However, Wiener did not decide to work onmathematical physics before 1921. What happened between 1914 and 1921 that led Wiener tostudy Brownian motion?The period 1913-1919 was very scattered since he worked and studied successively in G¨ottin-gen, Columbia, MIT and Harvard and saw his activity disturbed by the World War I. Duringthese years, he focused on the foundations of mathematics and their structure, he then studiedalgebra, postulates systems and philosophy.In 1919 he obtained a professorship at MIT, where he met Henry Bayard Philips, who ‘morethan anyone else’ introduced him to the physical aspect of mathematics with Willard Gibbs’work on statistical mechanics, which was a key element of his understanding of the role ofstatistics in physics.Also in 1919, he inherited analytical mathematics books after the death of the mathematicianGabriel Marcus Green of Harvard, at that time his sister’s husband. He began to read thefundamental works on analysis, which he had until now left aside, with a particular interest forLebesgue’s and Maurice Fr´echet’s works, the latter whom he later met at the 1920 Strasbourgcongress.His interest in probabilities came from another meeting, with Isaac Albert Barnett in 1919.Wiener said he asked him a mathematical subject to study and Barnett suggested to him the fieldof probabilities where random events were not points but curves. Indeed, at this time probabilitytheory dealt only with discrete problems based on random variables, and there was no continuousprobability theory, based on measure theory from mathematical analysis. According to Wiener,‘The world of curves has a richer texture than the world of points. It has been left for thetwentieth century to penetrate into this full richness.’ (Wiener, 1956, p.36).Wiener thus spent a year trying to apply the Lebesgue integral to intervals whose points arethemselves curves but it was too difficult. However, Wiener knew mathematician Percy JohnDaniell’s work, who had formulated a new theory of integration in 1918. He then wrote an17rticle in 1920 (Wiener, 1920) to propose developments on Daniell’s theory in the direction ofthe integration on function spaces. This work was purely mathematical and had no direct linkwith Brownian movement but in 1920, Wiener read Geoffrey Taylor’s work on turbulence andsaw a perfect subject to apply the ideas developed in his 1920 article. In fact, turbulence theorywas based on average quantities depending on the whole movement. This attempt was a failurebecause the problem of turbulence was too tough to be solved so early, but Wiener knew anothersubject, distantly related to the problem of turbulence: Brownian motion.
Here I had a situation in which particles describe not only curves but statistical assemblagesof curves. It was an ideal proving ground for my ideas concerning Lebesgue integral in a spaceof curves, and it had the abundantly physical texture of the work of Gibbs. It was to this fieldthat I had decided to apply the work that I had already done along the lines of integrationtheory. (Wiener, 1956, p.38)
Wiener published an article in 1921 (Wiener, 1921a), in which he applied the ideas of his 1920article to Brownian movement, as he wanted to do for turbulence. This was his first proper articleon Brownian movement, written in 1921 when he did not yet know the incredible richness of itsmathematical structure.
In 1921, Wiener published his first article on Brownian movement, and since that date this topicremained a thread of his entire mathematical career. Norbert Wiener is now recognized as animmense twentieth century mathematician, for his contributions to the theorization of Brownianmotion, the invention of Wiener’s measure, his contributions to Gibbs’ statistical mechanics andquantum physics, but especially his invention of cybernetics. In all his works, the style of mathe-matical reasoning developed during the study of Brownian motion from the 1920s is recognizable.Wiener’s work on Brownian movement was centred on problems of different nature from thoseof the physicists mentioned earlier. Wiener said in his biography
The Brownian motion was nothing new as an object of study by physicists. There were fun-damental papers by Einstein and Smoluchowski that covered it, but whereas these papersconcerned what was happening to any given particle at a specific time, or the long-time statis-tics of many particles, they did not concern themselves with the mathematical properties of thecurve followed by a single particle. (Wiener, 1956, p.38)
The study of these trajectories and the functions of these trajectories was what guided Wiener’sstudy of Brownian motion.We focus here on his work on Brownian motion, published between 1920 and 1933, which wedivide into three periods, for each of which he developed a different model of Brownian motion.The first period extended mainly between 1920 and 1922, during which Wiener developed hisideas on functional (i.e. functions depending on other functions and not points) averages, fol-lowing Daniell’s work. He developed an axiomatic theory of integration, not based on measuretheory. We have already mentioned two articles written during this period in the section onWiener’s motivations (Wiener, 1920; Wiener, 1921a), but there were two other articles (Wiener,1921b; Wiener, 1922), published in 1921 and in 1922. The first one was extremely little quotedin the secondary literature (with the exception of (Doob, 1966), which did not however analysethe article in detail). This article, however, was in the logical continuity of the two previousones because it was based explicitly on their results and, unlike the other two, it was muchcloser to physics and rich in lessons on Wiener’s Brownian movement. The second one was theculmination of Wiener’s ideas on axiomatic integration where he developed a model that wouldlater be taken up and improved in his 1930 article (Wiener, 1930), which we analyse in the thirdsub-section. 18he second period began in 1923 with the publication of
Differential Space (Wiener, 1923),the article often cited as the one in which Wiener founded his theory of Brownian motion. Hedeveloped a different approach from that used until now, based on measure theory. Indeed, itwas in this article that Wiener defined what mathematicians today call the Wiener measure.It was also in this article that Wiener gave the first argument for the non-differentiability ofBrownian trajectories by defining a coefficient of non-differentiability.Lastly, Wiener returned to the question of Brownian motion in 1930, in his memoir (Wiener,1930) on harmonic analysis. Based on the approach developed in the article (Wiener, 1922),he constructed a third model of Brownian motion, based on the Lebesgue measure. In orderto do this, he established a mapping between the set of continuous functions and the interval[0 , α ∈ [0 , To fully understand what is at stake in Wiener’s articles, it is important to make a naive andvery quick point on the state of the art of integration in 1920.In the second half of the nineteenth century, the first rigorous theory of integration had beendeveloped by Bernhard Riemann. This theory was fundamental but had limits, which we do notexpose here but which pushed mathematicians to seek another approach to integration. Thus, in1902, Lebesgue proposed his version of integration, which made it possible to integrate functionson more complex spaces than the intervals of R N , as for example sets of discrete points. Wewrite his integral (cid:82) f d µ where µ is the Lebesgue measure that gives a weight to each subset ofthe integration space. There is no general expression for this measure, but it is the simplestone in the sense that on simple spaces it corresponds to the intuitive notion of measure. Forexample, the Lebesgue measure of a segment is its length, the Lebesgue measure of a surface isits area, and so on. These two versions do not directly allow integration on sets of functions.Moreover, both are measure-based theories, that is, based on the measure d x or d µ which weighseach element of integration.In order to generalize the notion of integration to infinite-dimensional spaces, Percy JohnDaniell proposed in 1918 an axiomatic theory of integration, not based on measure theory. Hedefined an abstract object I , which satisfied some axioms so that I ( f ) represented the integral ofthe function f , and coincided with the prior definitions of the integral under certain conditions.Like the previous constructions (Riemann, Lebesgue), Daniell first defined his integral on avery small set T of functions, then showed how to extend the definition to a much larger set T , in the same way Riemann first defined his integral on step-functions before defining it onthe set of piecewise continuous functions using his step-functions.In the series of articles we examine in the following section, Wiener took up the ideas ofDaniell’s integration theory, to explicitly compute functional averages over function spaces. Letus then expose the premises of Daniell’s theory, which is important to study Wiener’s way ofthinking and to understand his major results.Daniell defined two abstract objects ( I, T ). T is a space of simple functions on which theintegration is simply defined, and I is the integration operator on T . Thus Daniell’s integral is19oted in all generality I ( f ) , f ∈ T . Functions in T are required to have the following stabilityproperties ∀ ( f , f ) ∈ T , f + f ∈ T , ∀ f ∈ T , ∀ c ∈ R , cf ∈ T , ∀ f ∈ T , | f | ∈ T . (4.1)Similarly, to match the intuitive idea of integration, the I operator must satisfy the followingaxioms ∀ ( f , f ) ∈ T , I ( f + f ) = I ( f ) + I ( f ) , ∀ f ∈ T , ∀ c ∈ R , I ( cf ) = cI ( f ) ,f ≥ ⇒ I ( f ) ≥ ,f ≥ f ≥ ... ≥ f n → ⇒ lim n →∞ I ( f n ) = 0 . (4.2)Daniell’s major theorem is to extend the integrability of the functions of T to a much largerclass of functions T , defined by the functions of T . Theorem 1
Let there be an increasing sequence of functions f n belonging to T , such that thereexists a function g greater than all the functions f n , then the limit f of the sequence f n issummable in the sense of Daniell, with I ( f ) = lim n →∞ I ( f n ) . (4.3)The set T is then defined as the set of functions f described in the theorem. The 1920 article, as discussed in section 4.1, was unrelated to Brownian movement but exposedWiener’s progress on Daniell’s integration. However, his ideas were used on the one hand in thefollowing article (Wiener, 1921a) dealing with Brownian motion, and on the other hand in thearticle (Wiener, 1922).Wiener noted that Daniell had established a method to go from T to T but left open howto build I and T in the first place. Wiener proposed in this article to build these two objectsand to apply them to the case of functionals. For this, he used a simple notion of step functionsfor T . We present here his construction step by step.Let K be a set, we call I n a division of the set K depending on a parameter n , a divisionof K being defined as a finite set of subsets, also called intervals, which cover K at least once.The intervals of the division I n are denoted i ( I n ), ..., i m ( I n ). We no longer use the I notationfor the Daniell integral, so there is no confusion with the divisions. We can also assign a weight w I n (denoted w n if there is no ambiguity) to each interval of a division I n , so that the interval i j ( I n ) has the weight w n [ i j ( I n )]. The division I n is then said to be weighted by w . Finally, asequence { I n } n ∈ N of divisions weighted by w is called a partition P K of the set K if it satisfiesthe following properties(i) Each interval of I n +1 is included in an interval of I n and only one,(ii) The weight w n [ i j ( I n )] of an interval i j ( I n ) is the sum of the weights w n +1 [ i l ( I n +1 )]of the intervals i l ( I n +1 ) included in i j ( I n ).There is a third condition that does not contribute anything to understanding, which we do notgive here for the sake of synthesis. 20ith these definitions, Wiener could then define his step functions. A function f defined on K is called a step function on P K if there is a division I n belonging to the partition P K suchthat f is constant on each interval i j ( I n ). Then Wiener defined the average A P K ( f ) of a stepfunction f on P K intuitively as A P K ( f ) = (cid:80) mj =1 w n [ i j ( I n )] f ( x j ) (cid:80) mj =1 w n [ i j ( I n )] , (4.4)where x j ∈ i j ( I n ).Wiener then showed that his step functions satisfied the conditions of the set T , given ineq. (4.1) by Daniell; and that his definition of the mean (eq. (4.4)) satisfied the axioms of the I operator, given in eq. (4.2). Using Daniell’s theorem, Wiener proved that all bounded anduniformly continuous functions on P K are summable in the sense of eq. (4.4). He thus had aconstruction of the mean of a function defined on any set K , potentially of infinite dimension.To conclude his article, Wiener took some examples. By defining the I n divisions in a simpleway and taking a segment for K , his definition of the average gave back Lebesgue’s one. Moreinterestingly, when he took for K the set of continuous functions defined on the interval [0 ,
1] andnull in 0, which we note C [0 ,
1] from now on, which are in addition bounded and Lipschitzian,then the functions defined on K were functionals by definition, and the application of the theoremgave that all continuous and bounded functional was summable in the sense of Wiener.The foundations of his axiomatic theory were laid in this article and were developed in thefollowing articles in which he gave more explicit definitions for the mean of a functional and inwhich he applied his theory to the study of Brownian motion.The following article (Wiener, 1921b) was the first to explicitly deal with Brownian motion.Wiener noted at the bottom of the first page that the theory of functional average had alreadybeen addressed by Ren´e Gateaux but that his own version was more adapted to the case ofBrownian movement than that of Gateaux. He began his article with a reference to Einstein’swork, and recalled this result: if a particle is free to move on the x axis and is subjected toBrownian motion, and if we assume that the probability that it moves a certain value over acertain time interval is independent of(i) its starting point,(ii) its starting absolute time,(iii) its directionthen Einstein showed that the probability, that after a time t the particle reached the position x , written f ( t ) by Wiener, between x and x , was under certain assumptions P ( x ≤ f ( t ) ≤ x ) = 1 √ πt (cid:90) x x exp (cid:18) − x t (cid:19) d x , (4.5)where Wiener voluntarily omitted to note the physical parameter 4 D , present in eq. (3.6), bysetting it equal to 1. The assumptions in question were not explained by Wiener but it is clearthat the one of the existence of a time scale τ on which the displacements are independentof previous displacements, was essential to the establishment of the Gaussian probability forEinstein. Wiener did not deal with the question of this time scale and used eq. (4.5) for all times,which made his object a simplified model of Brownian motion. In fact, he freed himself from thephysical difficulties that appeared when the mean free path was approached, and constructed amathematical model that made it possible to study Brownian motion by extending the range ofvalidity of the Gaussian distribution. It is this model that he continued to use in his subsequentarticles and which he described lucidly as follows21 n the physical Brownian motion, it is of course true that the particle is not subject to anabsolutely perpetual influence resulting from the collision of the molecules but that there areshort intervals of time between one collision and the next. These, however, are far too short tobe observed by any ordinary methods. It therefore becomes natural to idealize the Brownianmotion as if the molecules were infinitesimal in size and the collisions continuously described.It was this idealized Brownian motion that I studied, and which I found to be an excellentsurrogate for the cruder properties of the true Brownian motion. (Wiener, 1956, p.39) Let us go back to the article of 1921. To meet the conditions of his previous article, Wienerrestricted the parameter t ∈ [0 , f ( t ), describing Brownian trajectories fellwithin the framework of the second example given in the previous article, and he could computethe average of the continuous and bounded functions defined on this set K of functions f ( t ).Wiener gave in this article a more explicit formula for the average of these functionals.Let us recall that functionals are defined as functions of functions, i.e. functions which do notdepend on a finite number of variables but of an entire function, which can be seen as an infinityof variables (i.e. f ≡ { f ( t ) } t ∈ [0 , ). Physically, these functionals of trajectories can be anyquantity that depends on the complete trajectory, such as the maximum value of the function,which represents the maximum distance the particle has moved away from its origin; or thelength of the trajectory.Wiener started with a simple case where the functional F [ f ], which we note with bracketsto differentiate it from a simple function, depended on f only for a finite number of values f ( t ), ..., f ( t n ) in polynomial form: F [ f ] = f ( t ) m ...f ( t n ) m n . In this case F was rigorously afunction and not a functional, and the average of F was conventionally defined as the averageof a function A ( F ) = 1 (cid:112) π n t ( t − t ) ... ( t n − t n − ) + ∞ (cid:90) −∞ d x ... + ∞ (cid:90) −∞ d x n x m ...x m n n exp (cid:34) − n (cid:88) k =1 ( x k − x k − ) t k − t k − (cid:35) . (4.6)These integrals are analytically computable. Let us turn now to the most general case of truefunctionals. Wiener defined a general functional F [ f ] by the expression F [ f ] = a + (cid:90) f ( t )d ψ ( t ) + ... + (cid:90) ... (cid:90) f ( t ) ...f ( t n )d ψ n ( t , ..., t n ) + ... . (4.7)It was natural to require from the average operation to be stable by permutation with the sumof a series, by permutation with an integral, and by multiplication by a constant, which ledWiener to define the mean on this functional class as A { F } = a + (cid:90) A ( f ( t ))d ψ ( t ) + ... + (cid:90) ... (cid:90) A ( f ( t ) ...f ( t n ))d ψ n ( t , ..., t n ) + ... , (4.8)where we take care to note A {·} the average of a functional, defined by this formula, and A ( · )the classical average of a function. The right-hand side is computable using eq. (4.6) for theterms A ( f ( t ) ...f ( t n )). As soon as this series converges, we have a definition for the mean of afunctional and a method relying on means of functions to compute it.From here one can read Wiener’s work following two paths: the article (Wiener, 1922) fol-lowed the logic of the two articles that we just presented to refine the theory of the calculation The date of this article is uncertain, because its heading specified ‘Received February 27th, 1922.—Read March9th, 1922.’ and Wiener cited it in his 1930 article (Wiener, 1930) with the date 1922. However, this articleis mentioned with the date 1924 in the literature, as for example in Masani, 1990. Furthermore, in thisprecise article Wiener cited his 1923 article
Differential Space , whereas in
Differential Space he mentioned a‘forthcoming paper in Proc. Lond. Math. Soc.’ which is the 1922/24 article in question. It is then likely thatboth articles were written almost at the same time, during the year 1922, but were published in two differentjournals one year apart.
22f functional average, and that of (Wiener, 1921b) deviated from this problem to work on aconcrete case coming from Brownian movement. We choose to break the chronological orderhere to present the article (Wiener, 1922) first, because of its thematic filiation with the previousarticles. The article (Wiener, 1921b) deserves a sub-section (section 4.2.3) for itself because itwas the only article closely related to the physicists’ Brownian movement and to experiments,and moreover it has hardly been discussed in the literature.The purpose of this article was to specify the partition P K used in the case of Brownianmotion, which had not been done in the article (Wiener, 1921a), and to use this partition toexpress the average of a functional having for argument a function representing the trajectoryof a Brownian particle.Wiener chose once again the set K = C [0 , n time values between 0 and1: 0 ≤ t ≤ ... ≤ t n ≤ x i and x i defining [ x i , x i ] windows through whichthe function f had to pass at times t i . Constraints on f were then expressed as x ≤ f ( t ) ≤ x ,x ≤ f ( t ) ≤ x ,... ,x n ≤ f ( t n ) ≤ x n . (4.9)The probability of observing a trajectory satisfying these constraints therefore was1 (cid:112) π n t ( t − t ) ... ( t n − t n − ) x (cid:90) x d ξ ... x n (cid:90) x n d ξ n exp (cid:34) − n (cid:88) k =1 ( ξ k − ξ k − ) t k − t k − (cid:35) . (4.10)Wiener called the set of trajectories that satisfied the constraints of eq. (4.9) an interval, anddefined the weight of this interval as the probability given by eq. (4.10). We directly see the linkwith the construction of the article (Wiener, 1920), though Wiener let the reader make the linkwith the concepts defined in the 1920 article. We can recognize the following ones. The intervaldefined by eq. (4.9), is what was previously noted i , the probability of this interval is the weight w ( i ), and n which represents the number of time points, that is to say the fineness of the [0 , n on which the I n divisions previously defined depend.To pursue the calculations, Wiener set explicit values for t i , x i and x i , as follows ≤ h ≤ n ,t h = h/ n ,x h = tan( k h π/ n ) ,x h = tan(( k h + 1) π/ n ) , (4.11)where k h was an integer between − n − and 2 n − − n , we regularly split the time interval [0 , n time values t h . The first time value is t = 1 / n because the value of the function isset to 0 for t = 0. There are 2 n possible values for each k h , so k h π/ n is ranging from − π/ π/ x h and x h scan all ] − ∞ , + ∞ [. A specific choice of the value of k h for each23 corresponds to an interval i ( I n ). Since there are 2 n values of h and 2 n choices for k h , thereare therefore (2 n ) n intervals in I n . An example of trajectory is presented on fig. 2 on page 24for the division I . The curved line represents the Brownian trajectory, that is, a continuousfunction on [0 ,
1] starting at 0 for t = 0. The horizontal dotted lines represent the possiblevalues of the x h and the vertical dotted lines represent the 8 possible values of t h . The blackvertical segments highlight the windows through which the function must pass at each time t h .The choice of these windows is therefore an interval i ( I ) among the 8 possible. NO RB E R T W I E N E R o f t x a nd t b e t w ee n a nd , i n c l u s i v e , [ M a r c h , w h e r e h i s a g i v e n po s iti v e i n t e g e r . T h e s e f un c ti on s a r e m a n i f e s tl y w h a t A s c o li * ca ll s e qu a ll y c on ti nuou s , a nd a r e bound e d a s a s e t . T h e y t h e r e f o r e f o r m w h a t F r ec h e tt ca ll s a c o m p ac t s e t , a nd t h e r e i s no d i ff i c u lt y i n * A s c o li , " L e c u r v e li m iti d i un a v a r i e t a d a t a d i c u r v e " , L i n e d , V o l . , pp . - G . t M . F r ec h e f c ,. " S u r qu e l qu e s po i n t s du ca l c u l f on c ti onn e l " , R e nd . C i v . M a t I ' a L , V o l . , pp . G , . Figure 2: Example of a path going through the windows of a particular interval i ( I ). The imagehas been rotated by 90 ° compared to the one in the original article.It was relatively easy for Wiener to verify that this definition of the divisions I n , as well as thatof the weight function w defined by the eq. (4.10) satisfied the properties (i) and (ii) establishedin his 1920 article (Wiener, 1920).Wiener thus established an explicit partition P K on K = C [0 , i ( I n ) of a certain division I n . The average of a step functionalwas then naturally given by A { F } = (cid:88) k F [ f ] w [ i k ( I n )] , (4.12)where f ∈ i k ( I n ).Wiener was finally able to apply Daniell’s theorem and he deduced that all bounded andcontinuous functionals, defined on Brownian trajectories, were summable. Moreover the averagevalue was given by the formula A { F } = lim max( t i +1 − t i ) → π − n/ n (cid:89) k =1 ( t k − t k − ) − / ∞ (cid:90) −∞ d x ... + ∞ (cid:90) −∞ d x n F [ { ψ ( t , ..., t n )( x , ..., x n ) } ( t )] exp (cid:34) − n (cid:88) k =1 ( x k − x k − ) t k − t k − (cid:35) . (4.13)where Wiener set F [ { ψ ( t , ..., t n )( x , ..., x n ) } ( t )] = F t ,...,t n [ f ] the step functional only dependingof f ( t ) , ..., f ( t n ). The article on which we focus now was quite particular in Wiener’s bibliography. As we al-ready noted, it was hardly ever discussed in the literature, maybe because it was the only onethat did not offer novelties from a mathematical point of view, but it is even more interesting tonote that Wiener took a different position compared to his other articles by assuming that thevelocities of Brownian particles existed. Indeed, the major results of the mathematical modelof Brownian motion came from the properties of its trajectories which are neither bounded nordifferentiable. These results were the culmination of several years of work, starting with the firstarticles already discussed and concluding in 1933 as we shall see in section 4.2.5, thus confirmingPerrin’s hypothesis. This article was a parenthesis in this journey, inside which he used a differ-ent model where the velocity was well defined, which is in fact close to the Ornstein-Uhlenbeckmodel, which will be discussed in section 5 .For the first time, Wiener gave a review of the physicists’ Brownian movement, which wasquite detailed, quoting Einstein and Perrin, explaining the physical origin of the phenomenon interms of collisions and noting the difference between theoretical and measured velocities due tothe ‘extreme sinuosity of the trajectories’. According to him, Einstein made two assumptions:(i) the validity of Stokes’ law at the scale of Brownian particles,(ii) the independence of increments over time intervals τ .His objective was to use the results obtained in his previous article to demonstrate that Ein-stein’s formula on the mean square of displacements (eq. (3.8)) did not in fact require the secondhypothesis to be true.Wiener began by defining a function f whose meaning is different from that of the previousarticle. Here f ( ct ) was the total momentum of the Brownian particle acquired by the collisionswith the other molecules only, up to the time t . This was a different definition from the previousarticle where f ( t ) was the position of the particle at time t . The constant c depended on thephysical parameters, and must now be noted explicitly because it had an impact on numericalcalculations. Wiener could have redefined the normal distribution of eq. (4.5) to include c butchose to use c as an argument of the f function directly, to preserve the form of f ( t ) and thus25ake the results obtained in the previous article usable. According to him the distribution of f was then given by P ( a ≤ f ( ct ) ≤ b ) = 1 √ πct (cid:90) ba exp (cid:18) − x ct (cid:19) d x . (4.14)One must note that a , b and x have the dimension of a momentum, that is to say kg m s − ,which makes c a quantity expressed in kg m / s .Wiener’s starting point was the following equation, similar to Newton’s second law m [ v ( t + d t ) − v ( t )] = f ( ct + c d t ) − f ( ct ) − πµav ( t + δ · d t )d t (0 ≤ δ ≤ . (4.15)In this equation the role of collisions was taken into account by f ( ct + c d t ) − f ( ct ), which thenappeared as an analogue of Langevin’s force X in eq. (3.14). The last term accounted for theviscosity and δ could vary between 0 and 1, probably to allow the choice of any point of theinterval [ t ; t + d t ] for the value of the velocity. However this factor did not matter as it woulddisappear in the next calculation step. Wiener did not make any assumptions about the function f , in particular he did not suppose it differentiable, which explained this writing in infinitesimalform. Since he could not derive this equality, he integrated it to obtain m [ v ( t ) − v ] = f ( ct ) − πµa (cid:90) t v ( t (cid:48) )d t (cid:48) , (4.16)where v was the initial velocity of the particle. He solved this equation and wrote mv ( t ) = mv e − βt + f ( ct ) − βe − βt (cid:90) t f ( ct (cid:48) ) e βt (cid:48) d t (cid:48) , (4.17)where we set, in order to simplify the writings, β = 6 πµam . (4.18)We can note that β is the inverse of the characteristic time θ defined by Langevin in eq. (3.18).Since he obtained the velocity, Wiener had to integrate to get the position, then square theexpression and take the mean to shape the result in a similar way to that of Einstein. The firststep reads x ( t ) = (cid:90) t v ( t (cid:48) )d t (cid:48) = v β (cid:16) − e − βt (cid:17) + e − βt mc (cid:90) τ e βτ (cid:48) /c f ( τ (cid:48) )d τ (cid:48) , (4.19)where he made the change of variable τ = ct in the integral. This τ was not the same as Einstein’stime scale τ , which Wiener did not use in his calculations. Let us recall that his goal was todemonstrate that there is no need for the hypothesis on the independence of displacements, andtherefore no need for the definition of a physical time scale from which independence is verified.Since x ( t ) is a functional of f , the average of x ( t ) in the sense of Wiener is therefore writ-ten A (cid:8) x ( t ) (cid:9) (analogue of λ x in Einstein’s notation) and it is an average over the differentrealizations of the function f weighted by their probabilities according to eq. (4.14). A (cid:8) x ( t ) (cid:9) = A (cid:26) v β (cid:16) − e − βt (cid:17) + e − βt m c (cid:90) τ (cid:90) τ e β ( τ + τ ) /c f ( τ ) f ( τ )d τ d τ + 2 v e − βt mβc (cid:16) − e − βt (cid:17) (cid:90) τ e βτ (cid:48) /c f ( τ (cid:48) )d τ (cid:48) (cid:27) . (4.20)26t was at this moment that Wiener used eqs. (4.6) and (4.8), which allowed him to suppress theterm in { f ( τ ) } , and to explicitly compute the term in { f ( τ ) f ( τ ) } as A (cid:26)(cid:90) τ (cid:90) τ f ( τ ) f ( τ ) H ( τ , τ )d τ d τ (cid:27) = (cid:90) τ (cid:90) τ τ H ( τ , τ )d τ d τ , (4.21)in the case where H ( τ , τ ) = H ( τ , τ ), which is our case. After replacement, the formula read A (cid:8) x ( t ) (cid:9) = v β (cid:16) − e − βt (cid:17) + e − βt m c (cid:90) τ (cid:20)(cid:90) τ τ e β ( τ + τ ) /c d τ (cid:21) d τ . (4.22)Wiener then computed the double integral of the right-hand side and wrote the final result asfollows A (cid:8) x ( t ) (cid:9) = v β (cid:16) − e − βt (cid:17) + c m β (cid:104) βt − e − βt − e − βt (cid:105) . (4.23)It appears that the linear term in time is given by the contribution ct/ (2 m β ) in the right-handside. If one wishes to obtain a result similar to Einstein’s one, one must make sure that this termis dominant, so that there is a linearity between the average of the squares of displacements andthe time. Hence, Wiener chose to express the absolute value of the difference between A (cid:8) x ( t ) (cid:9) and this value (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A (cid:8) x ( t ) (cid:9) t − c m β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 t (cid:12)(cid:12)(cid:12)(cid:12) v β (cid:16) − e − βt (cid:17) + c m β (3 − e − βt )(1 − e − βt ) (cid:12)(cid:12)(cid:12)(cid:12) . (4.24)From this equality, Wiener deduced without justification the following inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A (cid:8) x ( t ) (cid:9) t − c m β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ v β + 3 c m β . (4.25)It appears that he replaced the exponentials by 0 to obtain an upper bound of the right handside, giving that all the terms are positive, but that he forgot to put back the term 1 /t , thusgiving a result that was not homogeneous from a physical point of view. For the sake of clarity,we from now on write this missing factor explicitly in the equations.The last step of Wiener’s reasoning consisted in defining the relative deviation as follows (cid:12)(cid:12) A (cid:8) x ( t ) (cid:9) /t − c/ (2 m β ) (cid:12)(cid:12) c/ (2 m β ) ≤ t (cid:20) m v c + 32 β (cid:21) . (4.26)In order to estimate the right-hand side, Wiener finally referred to Perrin’s values on gamboge,without precisely mentioning which ones, and obtained the value 10 − for the absolute relativeerror.Let us investigate how Wiener obtained the value 10 − . First, we replace the mathematicalnotations by their physical meanings. In Wiener’s calculation, the nominal value of λ x was c/ m β and in Einstein’s theory it was k B T / πmua , hence by equalizing the two quantities,we obtain c = 2 m β k B T πµa . (4.27)We can then replace β by its value, given in eq. (4.18) to get c = 24 πµak B T . (4.28)27y replacing all of these quantities by their physical meanings in eq. (4.26) we get | λ x /t − k B T / (3 πµa ) | k B T / (3 πµa ) ≤ t (cid:20) m v πµak B T + m πµa (cid:21) . (4.29)We can suppose the equipartition of energy for the initial velocity, as done by Ornstein insection 5, and thus replace mv by k B T (in the one-dimensional case) to obtain | λ x /t − k B T / (3 πµa ) | k B T / (3 πµa ) ≤ t m πµa . (4.30)Wiener did not mention the particular set of Perrin’s values on gamboge he used but we lookedat the ones Perrin gave in Les Atomes . On page 182 appears a table of results for severalexperiments with gamboge, let us choose the line 6 for which the experiment seems to haveworked the best, since it was the one that gave the best value for Avogadro’s number. Thevalues are µ = 1 .
25 kg m − s − , a = 0 . µ m and m = 0 . × − kg. Replacing these valuesin eq. (4.30) ans choosing t = 1 s, we get 6 . × − , which is in agreement with the order ofmagnitude calculated by Wiener.We can also note that the right-hand side of eq. (4.30) is indeed 2 θ/t where θ is the character-istic time scale defined by Langevin (eq. (3.18)), whose value is 10 − s according to him. Onceagain the result is compatible with Wiener’s one.Let us analyse how Wiener’s mistake impacted the assumption he was trying to prove. Hewanted to show that the hypothesis made by Einstein, on the independence of the displace-ments on disjoint successive time intervals, was in fact not necessary since, even without thisassumption, the discrepancy between both calculations was much too small to be observed ex-perimentally. Therefore, the ‘slightly different value of λ x /t for small values of the time thanfor larger values’ were to be traced to a violation of Stokes’ law at this scale. However, time t did not appear in the final formula that bounded the absolute relative mistake, so maybeWiener thought that his value 10 − held for any time t , and thus even for very small values of t , so precisely in the case where discrepancies had been observed. In reality, the upper boundon the relative error is inversely proportional to time t , so for very small values of t the upperlimit becomes large, making Wiener’s conclusion possibly false. This bound reads 2 /βt , so ifthe ‘small values of time’ that invalidated Einstein’s result are small but still large enough toensure t (cid:29) β − , then the relative error remains small and Wiener’s reasoning still holds.Be that as it may, Wiener’s calculation had the merit to offer an explicit formula for the upperbound and undoubtedly gave him a legitimacy to later study the idealized Brownian motion, inwhich he considered that the Gaussian distribution of displacements was valid for all times, assmall as desired. This seemed legitimate to him since the physical reality, too complex to model,actually deviated little from the Gaussian result. However, he did not mention this article whenhe later used his ideal model of Brownian motion. In 1923, Wiener published
Differential Space , which would later be cited as the article in whichWiener laid the groundwork for his model of Brownian motion. He used a different approachfrom the one used so far since he did not base it on Daniell’s integral but rather privileged anapproach like that of Paul L´evy. Wiener explicitly quoted the exchanges he had with L´evy onthe correspondence between his work and L´evy’s work, as a starting point for the article. Indeed, since this factor t did not appear in Wiener’s equation at this stage of the calculation, it seemed logicalto us to compute this quantity numerically with the value t = 1 s. Of course, in reality the error increases as t decreases so the value 10 − is not a physical limit. x ( t ) definedon the interval [0 , n , whichhave constant values on each interval [0 , /n ], ..., [( n − /n, x , ..., x n . A simple function of order n is thus represented by a point in a n -dimensional space. We now consider a particular volume V in the space of functions definedon [0 , R , defined by (cid:90) x ( t )d t = R . (4.31)Simple functions of order n that belong to this domain form a another domain, whose volume V n is the volume inside the sphere defined by n (cid:88) i =1 x i = nR . (4.32)Finally, a functional F defined and continuous on the volume V , is also continuous and definedon the volumes V n , on which it has a mean value µ n , calculated as a mean of a classical function.L´evy therefore defined the average of F on V as the limit of the averages on V n A { F } = lim n → + ∞ µ n . (4.33)Wiener’s first task was to build his ‘differential space’. He assumed that it was not thesuccessive positions of a Brownian particle that were independent of each other but rather theincrements over regular and disjoint time intervals. He then defined the n increments x n fromthe division of the time axis [0 ,
1] into n equal segment, as x = f ( n ) − f (0) = f ( n ) ,x = f ( n ) − f ( n ) ,... ,x n = f (1) − f ( n − n ) . (4.34)These n quantities were independent and had the same statistical weight in the sense that theycontributed to the displacement of the particle over equal periods of time, so it seemed naturalfor Wiener to use L´evy’s formulation and to consider the sphere defined by n (cid:88) i =1 x i = R n . (4.35)To test the relevance of this definition, Wiener raised the question of measuring the inner regionof the sphere where the position f ( a ) was between y and y , i.e. µ ( y ≤ f ( a ) = (cid:80) nak =1 x k ≤ y ),where the notation µ is used for the measure. In the case where R n = R was a constant, Wienerproved that this measure took the value µ ( y ≤ f ( a ) ≤ y ) = 1 √ πaR (cid:90) y y exp (cid:16) − u aR (cid:17) d u . (4.36)This measure was of the same form as the probability given by eq. (4.5), which showed that theL´evy sphere seemed to be an appropriate tool to study Brownian trajectories. Wiener then called29he set of functions f ( t ), along with the measure µ defined as the limit of the measures given ineq. (4.36), the differential space, to reflect the fact that it is the differences that are independent.Wiener returned to his major concern: the definition of the average of a functional. He beganonce again with the simple case of a functional that depended on a function only by a finitenumber of its values. He took n points 0 ≤ t ...t n ≤ ,
1] into ν parts,all of size 1 /ν . ν must be large enough for only one t i value to be included in each interval. Hethen defined the step function f ν ( t ) which took the values of f for each value of t i : ∀ i, f ν ( t i ) = f ( t i ). He could then define the differences on the function f ν : x k = f ν ( k/ν ) − f ν (( k − /ν ).Finally, the values f ( t ), ... f ( t n ) on which depended the functional F could be expressed interms of these differences: f ( t i ) = f ν ( T i /ν ) = x + x + ... + x T i where he defined T i = νt i if νt i was an integer, or the first higher integer otherwise.He was thus in the position of applying the formalism he established before, placing himselfin the volume defined by the interior of the sphere (cid:80) ni =1 x i = R . The average of the functionalwithin this sphere was properly defined, and if it reached a value when taking the limit, he couldspeak of the average of this functional on the differential space. The average of F in the spheredefined above was A { F } = (2 πR ) − n/ n (cid:89) k =1 ( t k − t k − ) − / × + ∞ (cid:90) −∞ d y ... + ∞ (cid:90) −∞ d y n F [ y , ..., y n ] exp (cid:34) − n (cid:88) k =1 y k R ( t k − t k − ) (cid:35) . (4.37)Moreover, Wiener defined the measure of a domain of this sphere by choosing for the functional F the value 1 if the argument function laid in the considered domain and 0 otherwise. To givean example, he defined the following domain y ≤ y ≤ y ,y ≤ y ≤ y ,... ,y n ≤ y n ≤ y n , (4.38)which was none other than the domain previously defined in eq. (4.9). Wiener measure of thedomain defined by eq. (4.38) was(2 πR ) − n/ n (cid:89) k =1 ( t k − t k − ) − / y (cid:90) y d y ... y n (cid:90) y n d y n exp (cid:34) − n (cid:88) k =1 y k R ( t k − t k − ) (cid:35) . (4.39)This long article was full of other rich ideas but we chose to focus on the one that was a firststep towards the non-differentiability of Brownian trajectories.Wiener reversed the previous perspective, in which he was interested in the distribution of f ( a ) = (cid:80) nak =1 x k with the constraint (cid:80) ni =1 x i = R . Inversely, he raised the question of thedistribution of (cid:80) ni =1 x i = (cid:80) ni =1 [ f ( i/n ) − f (( i − /n )] assuming that the increments f ( t ) − f ( t ) were independent and had a Gaussian distribution. He demonstrated that the quantity (cid:80) ni =1 x i only deviated from D with a small probability, where D was the physical diffusioncoefficient. ∀ ε > , ∀ δ > , P (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim n → + ∞ n (cid:88) (cid:20) f (cid:18) kn (cid:19) − f (cid:18) k − n (cid:19)(cid:21) − D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > δ (cid:33) < ε , (4.40)30ext, he considered the particular case of a continuous functions f , of limited total variation. n (cid:88) k =1 (cid:20) f (cid:18) kn (cid:19) − f (cid:18) k − n (cid:19)(cid:21) ≤ max k (cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) kn (cid:19) − f (cid:18) k − n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) k =1 (cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) kn (cid:19) − f (cid:18) k − n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ T ( f ) · max k (cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) kn (cid:19) − f (cid:18) k − n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (4.41)where T ( f ) was the total variation of the function f over [0 , f is a possible measure of its arc length and can be defined as T ( f ) =sup P (cid:80) nk =1 | f ( k/n ) − f (( k − /n ) | where P denotes the set of partitions (in the modern sense,not that of P K ) of [0 , n tend to infinity in eq. (4.41), the upper bound in the right-hand side went to 0.Since Wiener showed that in the case of Brownian trajectories the quantity (cid:80) ni =1 [ f ( i/n ) − f (( i − /n )] did not tend to 0 but was close to D , he therefore deduced that it was infinitely unlikely forBrownian trajectories to be of limited total variation and therefore infinitely unlikely that theyhave a bounded derivative.Wiener called the coefficient lim n →∞ (cid:80) ni =1 [ f ( i/n ) − f (( i − /n )] ‘a kind of coefficient ofnon-differentiability of the function f ’ because it measured a degree of non-differentiabilityequal to its deviation from 0. The next advances in Brownian motion modelling came later, in the 1930s, with a thesis onharmonic analysis published in 1930 by Wiener (Wiener, 1930) and with the collaboration withmathematicians Paley and Zygmund in 1932-1933, which culminated to a fundamental articlein 1933 (Paley et al., 1933). In his 1930 article, Wiener gave a new formulation of Brownianmotion, in terms of Lebesgue measure, and it was this formulation that allowed Brownian mo-tion to be considered as a stochastic process in the 1933 article. This led to the proof of thenon-differentiability of Brownian trajectories, which ends our history of mathematical Brownianmotion. From this time on, Wiener’s work became harder to follow for non-mathematicians, sowe try to make the most of the arguments without going into technical details.From 1924, Wiener had been interested in harmonic analysis, which was the study of the prop-erties of Fourier series expansions. He wrote a series of articles on harmonic analysis, includingan important memoir in 1930 (Wiener, 1930). In this thesis, Wiener studied general questionsabout the probability distributions of the frequency or energy spectra of time-dependent func-tions. It was therefore a science at the border between statistics and harmonic analysis. Chapter13 of this thesis named ‘Spectra and Integration in Functional Space’ opened with a discussionof the coexistence of these two sciences in physics. According to Wiener, the question had beenmainly raised in optics, when it was necessary to obtain statistical behaviours of the superposi-tion of electromagnetic waves, as for their amplitude or their intensity. Rayleigh had contributedthe most to this study using a theory based on statistical harmonic analysis. Wiener’s goal wasto show that a better and more rigorous approach than Rayleigh’s one could be thought out.The following general question arose: if a resonator received a chaotic sequence of input pulses,it would reemit the input spectrum but amplify certain contributions and reduce others; howthen to study the statistics of this output? To approach the problem it was necessary to have aclear vision of what ‘chaotic’ meant and the simplest chaotic system for Wiener was Brownian It should be noted that in the article it was the quantity lim n →∞ (cid:80) ni =1 (cid:2) f ( i/n ) − f (( i − /n ) (cid:3) that Wienercalled the coefficient of non-differentiability but it was a printing error, as there were several in the article. , R . Indeed, a function defined on R could be expanded in Fourier integral and allowed theuse of all the results developed by Wiener in the previous chapters of his memoir; while thefunctions on [0 ,
1] could be studied as Fourier series, which was a slightly different framework.Wiener recalled that he had dealt with the problem of integration on functions space in thepast, with Daniell’s theory of integration, but that his new approach was different: his goalwas to achieve a mapping between the functions of C [0 ,
1] and the segment [0 , f ( t ) ∈ C [0 ,
1] would be uniquely represented by apoint of the segment [0 ,
1] and vice versa. Eventually, this correspondence made it possible tointegrate on the [0 ,
1] segment with Lebesgue measure rather than on the set C [0 ,
1] with Wienermeasure, which was more convenient for the rest of the memoir. We do not discuss how thisnew representation contributed to the rest the memoir but we rather outline the constructionof the mapping, which would be taken for granted in the 1933 article (Paley et al., 1933).The idea of the construction was quite simple and relied entirely on the one made in the article(Wiener, 1922), discussed in the section 4.2.2. For the sake of clarity, we use the definitions ofthe divisions I n , intervals i ( I n ), and weights w [ i ( I n )], given by eqs. (4.9) and (4.10). To avoidconfusion and preserve the coherence of the notations we keep the name interval only for theintervals i ( I n ) of a division, and refer to the interval [0 ,
1] as segment. His idea was to firstestablish the mapping between a division I n and the segment [0 ,
1] by placing the intervals ofthis division on sub-parts of the segment whose lengths were equal to the weights of the intervals.To illustrate this, let us take the easiest case: n = 1. According to the definition of thedivisions, there are two time values t = 1 / t = 1. For these values, the functions f mustpass through the windows [ x , x ] and [ x , x ]. The possible values for k h are 0 and 1, sothere are 4 intervals in this division, defined by the choices of ] − ∞ ,
0] or [0 , + ∞ [ independentlyfor [ x , x ] and for [ x , x ]. The 4 intervals are obviously of equal weight, worth 1/4. Thuswe cut the segment [0 ,
1] into four sub-segments of length 1/4 and we assign to each intervalone of the four sub-segments of [0 , n = 2 and the 4 = 256 intervals of I . If, for a set of valuesof the parameters k h , which defines an interval i j ( I ), the interval i j ( I ) is found to be includedin an interval of the type i ( I ) (and the uniqueness of the latter is ensured by the hypothesis(i) of the partitions, discussed in section 4.2.2), then we place the sub-segment correspondingto i j ( I ), the length of which is defined by the weight of this interval, inside the sub-segmentcorresponding to i ( I ).Wiener repeated the operation with n → + ∞ . The intervals of the successive divisions thusfitted in infinitely until reaching sub-segments of zero length. Thus each point of the segmentwas determined by an infinite sequence of intervals whose weights tend to 0, defining a uniquetrajectory of zero probability. Wiener therefore completed the construction of the mapping be-tween the points of the [0 ,
1] segment and the functions of C [0 , one of the components of the displacement of the moving particle in the Wiener theory by χ ( α, t ), where t is the time and α the parameter on which Lebesgue integration is performedfor the purpose of averaging over all functions and determining probabilities. Wiener’s χ ( α, t )is then, as he shows, a continuous function of t for almost all values of α , defined for 0 ≤ α ≤ ≤ t ≤ t = 0. This definition may seem surprising at first glance, but it was nevertheless the way to definethe stochastic processes later (for example in (Doob, 1966)) and it still is today. α is a numberbetween 0 and 1, which according to the mapping established by Wiener in the previous articlerepresents a function of C [0 , α . We observe a change of perspective,rather than denoting α ( t ) a component of the trajectory of a Brownian particle, we use a χ function with two variables, taking as argument both the time t and the trajectory α . Anenlightening interpretation of this quantity was given in (Doob, 1966). χ ( α, t ) must be understood as the value of the α trajectory at time t , i.e. χ ( α, t ) = α ( t ).Then χ ( α, · ), seen as a function of t , is a trajectory, or a realization, of the stochastic process.With this formalism, Joseph L´eo Doob summarized the properties Wiener had established onBrownian motion as(i) Its increments are Gaussian: χ ( · , t ) is a random variable on C [0 , χ ( · , t ) − χ ( · , t )) ∼ N (0 , σ | t − t | ),(ii) its increments on disjoint time intervals are independent: ∀ ≤ t ≤ ... ≤ t n ≤ χ ( · , t ) − χ ( · , t ), ..., χ ( · , t n ) − χ ( · , t n − ) are independent.Let us go back to the 1933 article. The authors obtained, once more, the formula of theaverage of a functional when this one depended only on a finite number of values given at times t = 0 ≤ t ≤ ... ≤ t n ≤
1, but with the χ formalism A { F } = (cid:90) F [ χ ( α, t ) , χ ( α, t ) − χ ( α, t ) , ..., χ ( α, t n ) − χ ( α, t n − )] d α = π − n/ n (cid:89) k =1 ( t k − t k − ) − / ∞ (cid:90) −∞ d x ... + ∞ (cid:90) −∞ d x n F [ x , ..., x n ] exp (cid:34) − n (cid:88) k =1 x k t k − t k − (cid:35) . (4.42)At the end of the article, the authors proved for the first time the property of non-differentiabilityof mathematical Brownian trajectories in a rigorous way. The result can be written as follows Theorem 2
The values of α for which there exists a t such that lim ε → χ ( α, t + ε ) − χ ( α, t ) ε λ < ∞ ( λ > / , form a set of zero measure In other words, the probability of choosing a Brownian trajectory from the set C [0 , λ = 1 would have been enough, since this is the definition of differentiability,but the theorem actually expressed a stronger property.33t was also in this article that the authors introduced the first stochastic integral, formallynoted as (cid:90) f ( x, t ) c ( t )d t χ ( α, t ) , (4.43)where one had to make sense of ‘d t χ ( α, t )’, since they showed that the function χ did not have aderivative. This problem is part of what is called stochastic calculus, which was strongly devel-oped by Doob in the years that followed and which allowed to connect Langevin’s and Einstein’sapproaches by proving a formal equality between Langevin’s white noise and this Brownian mo-tion’s ‘derivative’. We do not discuss this point nor Doob’s stochastic calculation because it is avast subject going far beyond our scope, but it is historically important enough to be mentioned.Let us close this section with Wiener’s own words To my surprise and delight I found that the Brownian motion as thus conceived had a formaltheory of high degree of perfection and elegance. Under this theory I was able to confirm theconjecture of Perrin and to show that, except for a set of cases of probability 0, all the Brownianmotions were continuous non-differentiable curves. (Wiener, 1956, p.39)
5. Physicists’ Brownian motion after 1908
After Perrin’s experiments, as we discussed in section 4, the attention of physicists largely turnedto the experimental application of the physical theory of Brownian motion. The formulas derivedby Einstein, Smoluchowski and Langevin were used to measure fundamental quantities such asAvogadro’s number, appearing explicitly in eq. (3.8), or the elementary charge of the electron.For example, a series of articles on the measurement of the elementary charge using Brownianmotion, written by Jean Perrin, Louis de Broglie, or Felix Ehrenhaft, was published in the
Comptes rendus de l’acad´emie des sciences de Paris from 1908.However, other physicists followed the path opened by the first theories of Brownian motion,in order to enhance them and give them more generality. This movement came mainly fromthe Dutch school, with the two authors we are going to study: Leonard Ornstein and GeorgeUhlenbeck, but also with other figures who played a role, such as Gertruida de Haas-Lorentzor Willem Rhijnvis van Wijk. The period we discuss in detail began with Ornstein’s articlewritten in 1917 but published only in 1919 (Ornstein, 1919), and ends in 1934 with an articleby Ornstein and van Wijk (Ornstein and Wijk, 1934). This story ran parallel to Wiener’s one,although Wiener and the physicists did not communicate and adopted different approaches topursue different goals.Wiener was interested in the study of the properties of the curves representing idealizedBrownian trajectories, that is to say whose probability distribution of increments remainedGaussian for any duration, however small, which was not the physical reality. On the contrary,the Dutch movement focused on the transformation of results at short times, wanting to give ageneral theory of Brownian motion valid for all times, thus respecting the physical reality.Wiener took Einstein’s results as a starting point for his construction, without mentioningLangevin’s approach, while the Dutch worked both from Langevin’s equation, to obtain the dif-ferent moments of physical quantities, and from Einstein’s approach on probability distributions,called the Fokker-Planck method , to obtain the partial differential equation of the problem. As we mentioned in section 3.1.3, although Einstein wrote a Fokker-Planck equation in 1905, which consistedof starting from an integral relation on a probability distribution to obtain a partial differential equation onthe same distribution, it was named after the physicists Adrian Fokker and Max Planck who obtained it in1913, in a more general framework than that of Einstein. τ . This article, unique in Wiener’s bibliography, can be compared verydirectly to some of Ornstein’s results, as we see in the following sections.In this section, we discuss Ornstein’s and Uhlenbeck’s works from 1917 to 1934, focusing onthe progresses made on the understanding of Brownian motion at short times, as well as on thedefinition of the velocity of Brownian particles. We then discuss the relation between Ornstein-Uhlenbeck’s theory and the mathematical model of Brownian motion, by presenting two latertexts: Joseph L´eo Doob’s article of 1942 (Doob, 1942), in which he proposed the first rigorousmathematical theory of Ornstein-Uhlenbeck process, and Uhlenbeck and Ming Chen Wang’s1945 review article (Uhlenbeck and Wang, 1945), in which they mentioned Wiener’s and Doob’sworks. Leonard Ornstein was a Dutch physicist who contributed a lot to statistical physics and stochas-tic processes, and gave his name to the Ornstein-Uhlenbeck process, which we discuss later.He studied physics and obtained his thesis at Leiden University under the direction of HendrickLorentz in 1908. He then became professor of mathematical physics at Groningen Universityin 1909 and professor of theoretical physics at Utrecht University, succeeding Peter Debye, in1914. He remained in Utrecht, but his interest turned to experimental physics when he tookover the post of lab head of experimental physics in Utrecht in 1925. His laboratory gained agreat renown under his direction thanks to the measurements of luminous intensity which heobtained by revolutionary methods of photometry (Mason, 1945).The first article on Brownian movement communicated by Ornstein in 1917, but publishedonly in 1919, cited Gertruida de Haas-Lorentz’s work as the starting point. De Haas-Lorentzwas Hendrick Lorentz’s daughter, under whose direction she obtained her thesis in physics fromLeiden University in 1912. She was, in her thesis, the first to use calculations of fluctuationson electrons seen as Brownian particles. Her thesis was cited in numerous articles on Brownianmotion by Dutch physicists, for the theory that was exposed but also for her historical reviewof the work on Brownian motion up to 1912. Unfortunately it has not yet been translated intoEnglish. In his article, Ornstein gave a generalization of Einstein’s eq. (3.8), valid for all times,as well as a similar relation for speed increments. He also developed a Fokker-Planck relationon probability distributions, similar to that on position given by Einstein (eq. (3.5)), but for thedistribution of velocities, and valid for all times.Ornstein wrote a second article on Brownian movement in 1918, also published in 1919, withthe physicist Frederik Zernike (Ornstein and Zernike, 1919). We do not analyse this articlebecause its subject is out of the scope of our study.The following biographical elements about George Uhlenbeck’s can be found and supplementedby reading (Cohen, 1990; Dresden, 1989). Uhlenbeck was also a Dutch physicist, twenty yearsyounger than Ornstein. He entered Leiden University in 1919, and studied physics with PaulEhrenfest, under whose direction he obtained his thesis in 1927. Indeed, in 1912 Lorentz decidedto end his teaching activities at the university to devote more time to research, and proposed35o Einstein to take his place. The latter refused because he had just accepted a job at ETHZurich, and Ehrenfest took over from Lorentz in Leiden.After his thesis, Uhlenbeck held the position of professor of physics at Michigan University inAnn Arbor from 1927 to 1935, then became professor of theoretical physics at Utrecht Universityin 1935, replacing Hans Kramers, who himself inherited the chair of theoretical physics in Leiden,left empty after Ehrenfest’s suicide. In 1939, Uhlenbeck returned to Michigan University andremained there until 1960. Between 1935 and 1939, both Ornstein and Uhlenbeck were atUtrecht University.Uhlenbeck marked the history of physics thanks to several major works including the intro-duction of the concept of spin in 1925 with Samuel Goudsmit, also Ehrenfest’s PhD student,and his work on stochastic processes including the introduction of Ornstein- Uhlenbeck processand the first appearance of the master equation in an article on cosmic rays in 1940.Uhlenbeck’s interest in Brownian movement came from his reading of Ornstein’s works, how-ever his first article on the subject was not co-written with him. Indeed, Uhlenbeck was workingon the interpretation of quantum physics and he wrote his first article on Brownian movement in1929 with Goudsmit, with whom he had already worked on the spin a few years earlier, while thetwo young doctors were professors at Michigan University. This paper discussed the rotationalBrownian motion of small suspended mirrors used in a 1927 experiment by Walther Gerlach; wedo not talk about it.Uhlenbeck and Ornstein’s paths finally crossed in their common article in 1930 (Uhlenbeckand Ornstein, 1930), which represented the birth of Ornstein-Uhlenbeck process. In this article,the authors attempted to compute moments of order greater than 2, for displacements andfor velocity, in order to obtain complete probability distributions. Indeed, in his 1917 article,Ornstein had only calculated the moments of order 1 and 2, which was enough to generalizeEinstein’s results, but which was insufficient to find the whole probability distribution, valid atall times. They also tried to get the Fokker-Planck equation associated with the displacementdistribution, valid for any time, but failed.Finally, in 1934 this difficulty was overcome when Ornstein published an article with vanWijk, a former student of his who obtained his thesis under Ornstein’s supervision in 1930 atUtrecht university. In this article (Ornstein and Wijk, 1934) they derived the Fokker-Planckequation for the distribution of displacements. The long time limit of this equation restored thediffusion equation, and the short time limit gave an equation accounting for the hydrodynamicaspects of the motion.In the first part we analyse the 1917 article, in which the fundamental ideas used later wereexposed. In a second part we study the articles of 1930 and 1934, which completed the linessuggested in 1917. These papers marked major advances in the theorization of Brownian motionin the presence of external forces, such as harmonic potential or coupled potentials, but we do nottackle these points because they reduce to the study of Langevin’s equation with an additionalterm and bring nothing to the general understanding of the theory.
Ornstein exposed the objectives of his article as follows: from the relation used by de Haas-Lorentz in her 1912 thesis, and used to derive the relation connecting the mean of the squaresof the deviations to time (eq. (3.8)), the probability function of Brownian motion can be de-termined. In addition, Ornstein claimed a new method for the calculation of averages, and toobtain the distribution in velocity, in addition to that in position.What was de Haas-Lorentz’s relation? This was actually Langevin’s equation (eq. (3.14)),but Ornstein did not cite him at any time. It is interesting to note that in his next article36Ornstein and Zernike, 1919), he quoted Langevin twice in the expression ‘Langevin-EinsteinFormula’ referring to the preceding article, without further details on this formula. We learna little more in the 1934 article (Ornstein and Wijk, 1934) in which the authors spoke ofLangevin equation in these terms ‘This equation is generally known as the equation of Einstein.According to F. Zernike, Langevin used it before Einstein.’ For Ornstein, it appears that DeHaas-Lorentz’s relation and that of Langevin-Einstein are the same, so we decided to name itLangevin’s equation, as it was the case until now.It is true that Ornstein presented a new way of calculating the averages, based on the prop-erties imposed on Langevin’s stochastic force X , and deduced the first two moments of theposition and velocity distributions, valid at any time, but complete distributions were obtainedonly in 1930 with Uhlenbeck.Ornstein started with the relation d v d t = − βv + F , (5.1)where β = 6 πµa/m (it is the same definition as Wiener’s one in his article (Wiener, 1921b)) andwhere F was Langevin’s stochastic force X divided by mass m . By formally integrating thisequation, Ornstein got v = v e − βt + e − βt (cid:90) t e βt (cid:48) F ( t (cid:48) )d t (cid:48) , (5.2)where v was the initial velocity of the particle. He then averaged at a given time over manyparticles having the same initial velocity, using the fact that (cid:104) F (cid:105) = 0 because collisions areirregular and uncorrelated, and got (cid:104) v (cid:105) = v e − βt . (5.3)Thus, velocity decreases exponentially because of viscous drag.Ornstein repeated the operation by first squaring eq. (5.2) before averaging to obtain (cid:104) v (cid:105) ,which made a term in (cid:104) F ( t (cid:48) ) F ( t (cid:48)(cid:48) ) (cid:105) appear (cid:104) v (cid:105) = v e − βt + e − βt (cid:90) t (cid:90) t e β ( t (cid:48) + t (cid:48)(cid:48) ) (cid:104) F ( t (cid:48) ) F ( t (cid:48)(cid:48) ) (cid:105) d t (cid:48) d t (cid:48)(cid:48) . (5.4)Ornstein was the first to propose a quantitative treatment for the term (cid:104) F ( t (cid:48) ) F ( t (cid:48)(cid:48) ) (cid:105) . Accordingto him, there were correlations between collisions only for very short periods, i.e. when t (cid:48) and t (cid:48)(cid:48) were very close, otherwise F ( t (cid:48) ) and F ( t (cid:48)(cid:48) ) would be independent and therefore the average oftheir product would be null. He defined ψ as t (cid:48)(cid:48) = t (cid:48) + ψ , hence (cid:104) F ( t (cid:48) ) F ( t (cid:48)(cid:48) ) (cid:105) was non-null onlyfor values of ψ very close to 0. From this statement, he used a series of approximations. He firstreplaced t (cid:48)(cid:48) by t (cid:48) in the exponential term and was then able to factor the double integral into aproduct of two integrals as (cid:90) t dt (cid:48) e βt (cid:48) (cid:20)(cid:90) t (cid:104) F ( t (cid:48) ) F ( t (cid:48) + ψ ) (cid:105) d ψ (cid:21) . (5.5)Since the term in the second integral is non-null only for values of ψ close to 0, he extended theintegration bounds of this integral to ] − ∞ , + ∞ [, without changing the result significantly. Hefinally defined γ , a constant which depended on the parameters of the problem and which wouldbe computed later, as (cid:90) + ∞−∞ (cid:104) F ( t (cid:48) ) F ( t (cid:48) + ψ ) (cid:105) d ψ = γ . (5.6)After replacing this in the main expression, Ornstein obtained (cid:104) v (cid:105) = v e − βt + γ − e − βt β . (5.7)37t remained only to explain the constant γ according to the parameters of the problem, for thatOrnstein used the equipartition of energy once the equilibrium reached, i.e. in the long timelimit lim t →∞ (cid:104) v ( t ) (cid:105) = k B Tm ⇒ γ = 2 βk B Tm . (5.8)Finally, Ornstein obtained an expression for (cid:104) v (cid:105) , analogous to the one for the displacements(eq. (3.8)), but true at all times (cid:104) v (cid:105) = v e − βt + k B T − e − βt m . (5.9)He did not discuss this expression, or test his behaviour in the regime of small times, butwent directly to the calculations on displacements. By integrating again eq. (5.2), he made thedisplacement s appear, defined as s = x − x where x is the position and x the initial position βs = v − v + (cid:90) t F ( t (cid:48) )d t (cid:48) . (5.10)Following the same logic as for the velocity, he squared this expression, and used the results ofeqs. (5.3), (5.6) and (5.9) to obtain β (cid:104) s (cid:105) = v (cid:16) − e − βt + e − βt (cid:17) + γ β (cid:16) − − e − βt + 4 e − βt (cid:17) + γt . (5.11)This equality was exactly the one Wiener would get (eq. (4.23)) four years later, independently.It should be noted that contrary to Einstein’s formula, the initial velocity v appeared explic-itly. In the long time limit, Ornstein’s formula gave back Einstein’s formula, and cancelled thedependency on the initial velocity (cid:104) s ( t ) (cid:105) ∼ t →∞ γβ t = k B T πµa t . (5.12)To obtain a result which was completely comparable to that of Einstein, even at short times, itremained to average eq. (5.11) over the different initial velocities v , by replacing the value of v by the one predicted by the equipartition of energy: (cid:104) v (cid:105) = k B T /m . We from now on write (cid:104)·(cid:105) v for the statistical average on all particles and on all initial velocities. Therefore the equationread β (cid:104) s (cid:105) v = γβ (cid:16) βt − e − βt (cid:17) . (5.13)It appears that Einstein’s formula is valid when the first term in the parenthesis dominates, i.e.when βt > t > m/ πµa , which is consistent with the range of validitydefined by Langevin.Ornstein did not discuss in this article the short time limit of this formula, but he did in the1930 article with Uhlenbeck.In the last part of his article, Ornstein established the Fokker-Planck equation whose solutionwas the velocity distribution. The aim was to give an expression, similar to eq. (3.5), for thedistribution f ( v, v , t ), in the case where f ( v, v , t )d v represented the number of particles havinga velocity between v and v + d v at time t , and an initial velocity v . If the first part of thearticle was strongly based on Langevin’s approach, Ornstein followed from there the logic usedby Einstein to obtain the diffusion equation, discussed in the section 3.1.1.Ornstein integrated eq. (5.1) over a very short time τ , and got v = u (1 − βτ ) + ξ , (5.14)38here u is the velocity at the beginning of the time interval, and where he defined ξ = (cid:90) τ F ( t )d t . (5.15)He used the statistical properties of F to define a probability distribution φ τ ( ξ ) so that (cid:104) ξ (cid:105) = (cid:82) ∞ ξφ τ ( ξ )d ξ = 0 because (cid:104) F (cid:105) = 0, and (cid:104) ξ (cid:105) = (cid:82) ∞ ξ φ τ ( ξ )d ξ = γτ , by definition of γ (eq. (5.6)).Thus, following Einstein’s idea, Ornstein wrote that the number of particles f ( v, v , t + τ )d v having a velocity between v and v + d v at time t + τ was equal to the number of particles f ( u, v , t )d uφ τ ( ξ )d ξ having a velocity between u and u + d u at time t and having received avelocity increment of value ξ due to the force F , integrated over all the values of ξf ( v, v , t + τ )d v = (cid:90) + ∞−∞ f ( u, v , t )d u φ τ ( ξ )d ξ = (1 + βτ )d v (cid:90) + ∞−∞ f ( u, v , t ) φ τ ( ξ )d ξ . (5.16)Ornstein took the Taylor expansion of f to the first order in τ and to the second order in u about v . Using the properties of φ τ he got ∂f∂t = βf + βv (1 + βτ ) ∂f∂v + (1 + βτ ) γ ∂ f∂v . (5.17)The last step was to make τ tend toward 0. This was one of the main differences with Einstein’sderivation for which a physical hypothesis set the acceptable value of τ , whereas in our presentcase, no independence was imposed and therefore the time τ can be taken arbitrarily small. Inthis limit, we have ∂f∂t = β ∂∂v ( v · f ) + γ ∂ f∂v . (5.18)This equation is the diffusion equation for velocity, analogous to eq. (3.5) for position, but validat all times. One can note, as Ornstein did, that the parameter γ , defined by eq. (5.6), andwhose value is given by eq. (5.8), plays the role of the diffusion coefficient in velocity space,and is therefore analogous to the diffusion coefficient D in position space. Combining eqs. (3.1),(4.18) and (5.8) we find the relation γ = 2 β D . (5.19)Ornstein did not fully solve the problem in this article, as he only solved this equation in steadystate, that is to say with the left-hand side null. Doing so, he found the Maxwell-Boltzmanndistribution for the steady state velocity distribution.Let us take a few moments to list fundamental differences between the speed distribution andthe position distribution. Although the physical phenomenon is the same: a Brownian particlemoves and one chooses to study either its position or its speed, Fokker-Planck equations whosesolutions are the two distributions in question differ by a convective term. Such a term can alsoappear in the Fokker-Planck equation for position if one considers the case of a particle subjectedto a force, which we do not study here. The two phenomena are said to be diffusive, in a broadsense in the case of velocity because of the additional convective term, however the diffusioncoefficients of the two phenomena are different. Finally, the velocity distribution reaches asteady state: the Maxwell-Boltzmann distribution, while the position distribution is explicitlytime-dependent at any time and does not approach a stable distribution; it is a Gaussian lawthat flattens and expands indefinitely. Even if the physical experiment is the same, owing to thedifferences listed above, it makes sense to give to the two stochastic processes different names.When one is interested in the diffusion of the velocity of a Brownian particle, one speaks of aOrnstein-Uhlenbeck process, and when one is interested in the diffusion of the position of theBrownian particle, one speaks of a Wiener process.39 .1.2. Probability laws at short times - Ornstein & Uhlenbeck 1930
In 1930, Ornstein and Uhlenbeck wrote an article together, in continuity with Ornstein’s 1917article. Back then, he had generalized Einstein’s formula, giving the average square of displace-ments as a function of time, for any time and had given a similar formula for the average squareof velocities. He had also obtained the Fokker-Planck equation associated with the velocitydiffusion process. The two authors’ aim was therefore to finish the construction by determiningthe complete distributions for displacements and velocities, valid at all times, and to obtain theFokker-Planck equations of the two processes, with a general method. In this article we canrecognize Uhlenbeck’s style, known to be very clear and orderly (Dresden, 1989): he began bygiving a detailed reminder of all the results obtained so far, in the form of a review article, thenestablished clearly the points he aimed to deal with, and in the end explained the limits of hiswork and the directions to be pursued.For each of the objectives they set, the authors used a different method: they determinedthe probability distributions by the method of moments, that is by calculating all the moments {(cid:104) q k (cid:105)} k ∈ N , which fully characterize the distribution; and obtained the Fokker-Planck equationsby a general method similar to that of Ornstein in 1917.Moments 1 and 2 had already been computed by Ornstein, so the authors recalled the resultsand proposed an interpretation of eq. (5.13) in the short time limit (cid:104) s (cid:105) v ∼ t → k B Tm t ∼ t → (cid:104) v (cid:105) t . (5.20)This relation was fundamental, as it was the main difference with Einstein’s formula (eq. (3.8)).Indeed, Einstein’s prediction took the form (cid:104) s (cid:105) ∝ t , leading to the impossibility to define avelocity; whereas Ornstein and Uhlenbeck obtained a law written in the form (cid:104) s (cid:105) v ∝ t → t which allowed the interpretation of a uniform displacement at velocity (cid:104) v (cid:105) . It should also benoted that this law was independent of the viscosity of the medium.For higher moments, they started by the case of velocity. The two authors announced that theGaussian distribution held for all times, but for the modified variable V = v − v e − βt = v − (cid:104) v (cid:105) .If V has a Gaussian distribution, then its moments must satisfy the following properties (cid:104)V n +1 (cid:105) = 0 , (cid:104)V n (cid:105) = 1 · · · ... · (2 n − (cid:104)V (cid:105) n . (5.21)Thus, one just needs to compute the moments of V and to check that they satisfy the aboverelations, to be sure that V follows a Gaussian law. From the first two moments already computedby Ornstein (eqs. (5.3) and (5.9)), they knew that (cid:104)V(cid:105) = 0 , (cid:104)V (cid:105) = k B Tm (cid:0) − e − βt (cid:1) . (5.22)The method they used to compute the other moments was essentially the same as for the first two:they used the statistical properties of F to calculate the averages and then used the equipartitionof energy at equilibrium to determine the constants. Doing so, they obtained the moments (cid:104)V (cid:105) and (cid:104)V (cid:105) , and left the general case to the reader. Let us analyse these two calculations. Theyfirst put eq. (5.2) to the desired power, then took the average and used the assumptions on thedistribution of F . We do not detail this step which was technical and required the introductionof new assumptions, such as the one made in eq. (5.6) for the average of a product of two terms40 , but for products of over three terms F , which gave rise to new constants C i , similar to γ but for larger products. There was no additional physical ingredient in this step, the principlewas still to consider that the function F was very sharp around 0 and that, as a consequence,products of type F ( t ) ...F ( t n ) were non-zero only close to the domain t = ... = t n . We ratherfocus on the second step in which the physical reasoning appeared.Ornstein and Uhlenbeck obtained (cid:104)V (cid:105) = C β (cid:16) − e − βt (cid:17) . (5.23)To determine the constant, it must be noted that V and v have the same distribution when t → + ∞ . Since they supposed that steady state velocities v followed a Maxwell-Boltzmanndistribution by hypothesis, then lim t → + ∞ (cid:104)V (cid:105) = lim t → + ∞ (cid:104) v (cid:105) = 0 , (5.24)so C = 0 and (cid:104)V (cid:105) = 0, which was required by eq. (5.21).Let us look at their calculation for the fourth moment now (cid:104)V (cid:105) = 3 γ β (cid:16) − e − βt (cid:17) + C β (cid:16) − e − βt (cid:17) . (5.25)Once again, they determined the constant C using the Gaussian distribution of velocities v once the steady state was reached lim t → + ∞ (cid:104)V (cid:105) = lim t → + ∞ (cid:104) v (cid:105) = 3 lim t → + ∞ (cid:104) v (cid:105) = 3 γ β . (5.26)It followed that C = 0 and (cid:104)V (cid:105) = 3 (cid:104)V (cid:105) .The moments of higher orders were computed in the same way and eventually the two authorsdeduced that V followed a Gaussian law, as announced, which took the form f ( v, v , t ) = (cid:114) m πk B T (1 − e − βt ) exp (cid:34) − m k B T (cid:0) v − v e − βt (cid:1) − e − βt (cid:35) . (5.27)This transformed into Maxwell-Boltzmann distribution in the long-time limit, and into a Diracdistribution at initial velocity v for the t = 0 limit, as expected.The problem at the level of displacements is to find the probability for a particle that startedat time t = 0 at position x with velocity v to lie between positions x and x + d x at time t . Thecomputations were performed in a similar manner and made constants C i , already determined,appear. Ornstein and Uhlenbeck showed that the Gaussian law for displacements also held forany time, but for the modified variable S = s − v (cid:0) − e − βt (cid:1) /β = s − (cid:104) s (cid:105) , and could be written f ( x, x , t ) = (cid:115) mβ πk B T (2 βt − e − βt − e − βt ) exp − mβ k B T (cid:104) x − x − v β (cid:0) − e − βt (cid:1)(cid:105) βt − e − βt − e − βt . (5.28)41nterestingly, the velocity distribution depends only on the initial velocity v , whereas the dis-placement distribution depends on both v and the initial position x . In the short time limit,this distribution converges to a Dirac function at x , as expected. In the long time limit however,at the numerator of the fraction inside the exponential remains the term x − x − v /β , whereasonly x − x appears in Einstein’s formula eq. (3.6). The explanation given in the following article(Ornstein and Wijk, 1934) was that ‘For time intervals large in comparison with [1 /β ], we mayneglect [ v β (cid:0) − e − βt (cid:1) ] compared with x − x ; since | x − x | will not remain finite for large valuesof t’ whereas v /β remains constant. In fact this argument is a bit incorrect because position x and time t in the probability distribution have to be considered as independent variables takingany values, but not supposing a dependency of x on t . The result is correct though, and thedifference with Einstein’s formula is accounted for by the fact that Ornstein and Uhlenbeck con-sidered the case of a particle moving with initial velocity v , unlike Einstein. Their probabilitydistribution should then be written rigorously f ( x, x , v , t ).In this article, the authors also offered a general method to obtain Fokker-Planck equations,which they applied to the particular cases of the velocity distribution, to rediscover Ornstein’s1917 result, and of the displacement distribution. The derivation followed the same scheme asEinstein’s one, detailed in section 3.1.1, and as Ornstein’s one, presented in section 5.1.1, butwith some significant differences in the hypotheses, which we examine now.Let f ( q, q , t ) be the distribution of a variable q , with initial value q at time t = 0. Theassociated Fokker-Planck equation is the partial differential equation whose f is solution. Letus consider that during a time ∆ t the variable q changes by an amount ∆ q , with a probabilitydistribution φ (∆ q, q, t ), depending on the value q at the beginning of the time interval ∆ t , butwhich we suppose to be independent of the initial value q . We write with a prime symbol thevalue of q after the increment: q (cid:48) = q + ∆ q . Following the same reasoning as before, they got f ( q (cid:48) , q , t + ∆ t ) = (cid:90) + ∞−∞ f ( q (cid:48) − ∆ q, q , t ) φ (∆ q, q (cid:48) − ∆ q, t )d(∆ q ) . (5.29)Like their predecessors, they went on with a Taylor expansion in ∆ t for the left-hand side andin ∆ q for the right-hand side. Both expansions were for the moment complete, that is to saythey did not choose to cut the Taylor series at a particular order, and consequently they had aninfinity of terms. The right-hand side expansion gave rise to the apparition of all the momentsof ∆ q : (cid:8) (cid:104) ∆ q k (cid:105) (cid:9) k ∈ N . The equation contained two infinitesimal quantities: ∆ t and ∆ q , and inorder to get a useful equation for physicists, they had to neglect terms from a particular orderand therefore to choose where to stop the expansion for both infinitesimal quantities coherently.For that, the authors defined two functions lim ∆ t → (cid:104) ∆ q (cid:105) ∆ t = g ( q, t ) , lim ∆ t → (cid:104) ∆ q (cid:105) ∆ t = g ( q, t ) , (5.30)(5.31)and assumed that for higher orderslim ∆ t → (cid:104) ∆ q k (cid:105) ∆ t = 0 k ≥ . (5.32)This hypothesis was in fact equivalent to keeping terms of order 1 in ∆ t and terms of order 2in ∆ q , in the case where ∆ t tended to 0. Indeed, dividing the expression that contained thefull expansions by ∆ t and letting ∆ t tend to 0 made all terms of order greater than or equal to2 in ∆ t null, and all terms of order greater than or equal to 3 in ∆ q null as well, according toeq. (5.32). This was the very choice that Einstein made to obtain the diffusion equation.42sing these definitions and letting ∆ t tend toward 0, they finally obtained ∂f∂t = g ∂ f∂q + (cid:18) ∂g ∂q − g (cid:19) ∂f∂q + (cid:18) ∂ g ∂q − ∂g ∂q (cid:19) f . (5.33)This is a general Fokker-Planck equation for any system that satisfies the hypothesis given byeq. (5.32) and for which increments ∆ q are independent of the initial value q . In order to useit for their problem, the authors needed to determine the functions g and g and to make surethat eq. (5.32) was satisfied.From all the previous developments, it was easy to check that g ( v, t ) = − vβ , g ( v, t ) = γ forthe case of velocity, and that the condition of eq. (5.32) was satisfied. By replacing these valuesin eq. (5.33), they found the Fokker-Planck equation associated with the diffusion of velocities,already obtained by Ornstein in 1917 (eq. (5.18)).A difficulty arose when it came to applying this method to displacements. They needed tocompute (cid:104) ∆ x (cid:105) and (cid:104) ∆ x (cid:105) to get g and g , but when averaging eq. (5.10) they obtained − β (cid:104) ∆ x (cid:105) = (cid:104) v (cid:48) (cid:105) − (cid:104) v (cid:105) = v e − βt (cid:16) e β ∆ t − (cid:17) , (5.34)thus in the ∆ t → (cid:104) ∆ x (cid:105) = v e − βt ∆ t . (5.35)Following the same calculations by putting first the equation to the square, they got (cid:104) ∆ x (cid:105) ∝ ∆ t and so g was null. Then Fokker-Planck equation read ∂f∂t = − v e − βt ∂f∂x . (5.36)This equation could not be the general Fokker-Planck equation of the problem since it did nottransform into Einstein’s diffusion equation in the t (cid:29) β − limit, as it should have.Ornstein and Uhlenbeck noted this difficulty and attributed it to the hypothesis they madewhen deriving the Fokker-Planck equation in the general case (eq. (5.33)) and which seemeddefective in this case. The hypothesis in question was the independence of the increments ∆ x with respect to the initial values x and v . Indeed, we can see in the calculation that thishypothesis was not verified since the average of the increments (cid:104) ∆ x (cid:105) depended explicitly on v according to eq. (5.35)This difficulty was overcome only four years later, by the joint efforts of Ornstein and Van Wijkin their 1934 article, in which they obtained the true Fokker-Planck equation without using thepreviously mentioned hypothesis, therefore in a more general case, which allowed its applicationto the case of the displacements. In reality, it was not the relaxation of this hypothesis that madeit possible to obtain the announced Fokker-Planck equation. Following the reasoning of the twoauthors, we however explicitly note the dependence on the initial condition q in the probabilitydistribution of the increments φ (∆ q, q, q , t ). The derivation performed in the previous article isstill valid because the q dependency did not play a direct role. The only modifications to takeinto account are the dependencies in q of all the quantities derived from φ (∆ q, q, q , t ). This isthe case of the moments (cid:104) ∆ q k (cid:105) of this distribution and consequently of the functions g and g which we thus explicitly note g ( q, q , t ) and g ( q, q , t ). The general equation (eq. (5.33)) wastherefore still valid but Ornstein and van Wijk admitted they could not deduce the equation forthe case of displacements from this reasoning. Their next idea was quite surprising and couldappear as a mathematical sleight of hand. They assumed that the functions g and g were only43unctions of time, and called them n ( t ) and m ( t ) respectively. This assumption was obviouslynot verified a priori but allowed them to eliminate the derivatives of these functions with respectto q in eq. (5.33), which gave ∂f∂t = m ( t )2 ∂ f∂q − n ( t ) ∂f∂q . (5.37)By defining N ( t ) = (cid:90) t n ( t (cid:48) ) dt (cid:48) ,M ( t ) = 2 (cid:90) t m ( t (cid:48) ) dt (cid:48) , (5.38)(5.39)the exact solution read f ( q, q , t ) = 1 (cid:112) πM ( t ) exp (cid:34) − ( q − N ( t )) M ( t ) (cid:35) . (5.40)Hence, f ( q, q , t ) followed a Gaussian distribution for the variable q , centred on N ( t ) and ofvariance M ( t ) / (cid:40) N ( t ) = (cid:104) q (cid:105) ,M ( t ) = 2 (cid:2) (cid:104) q (cid:105) − (cid:104) q (cid:105) (cid:3) . (5.41)(5.42)Functions n ( t ) and m ( t ) were given by the derivatives of the above results n ( t ) = ddt (cid:104) q (cid:105) ,m ( t ) = ddt (cid:2) (cid:104) q (cid:105) − (cid:104) q (cid:105) (cid:3) . (5.43)(5.44)To finish the calculation, they used the value of the moments (cid:104) q (cid:105) and (cid:104) q (cid:105) given by thedisplacement distribution (eq. (5.28)) n ( t ) = v e − βt ,m ( t ) = 2 k B Tmβ (cid:16) − e − βt (cid:17) . (5.45)(5.46)Replacing these values in the simplified Fokker-Planck equation (eq. (5.37)), they had ∂f∂t = (cid:16) − e − βt (cid:17) k B Tmβ ∂ f∂x − v e − βt ∂f∂x . (5.47)This reasoning may seem illegitimate because of the unrealistic assumption made but it ulti-mately gave the expected result, as clear-mindedly explained by the authors The mode of reasoning by which [eq. (5.47)] has been obtained from [eq. (5.33)] may seemanything but stringent, because [ g ] and [ g ] will certainly depend on x too in the case inquestion but however this may be.. [eq. (5.47)] is a differential equation possessing [eq. (5.28)]as its fundamental solution and transforming into the diffusion equation for t = ∞ . On theother hand, if t = 0, it gives the equation of hydrodynamics [eq. (5.36)] which describes theflow of an ensemble of particles. Thus, the whole range of t values is covered by [eq. (5.47)]and it is very instructive to see from [eq. (5.47)] in what way the equation of motion for anensemble of particles all having the same position x o and velocity v at t = 0 is transformedinto the diffusion equation by the action of the unsystematic impulses of the surrounding liquid.(Ornstein and Wijk, 1934, p. 253-254) . Comparison between Brownian motion theories We have studied in detail the theory proposed by Ornstein and Uhlenbeck to account for Brow-nian motion. This theory, like that of Wiener, was based on the theories developed by Ein-stein, Smoluchowski and Langevin in the 1900s. The question of the comparison of Ornstein-Uhlenbeck’s theory with Wiener’s one can then be asked since they have been developed inparallel over the same period but in different directions. It also seems wise to compare thesetwo theories with the first physical theories from which they derived.We can already mention the fundamental difference between these two theories: Ornstein-Uhlenbeck theory assumed by construction the existence of the velocity of Brownian particleswhile Wiener dealt with an idealized Brownian motion in which the Gaussian distribution fordisplacements was valid at times, as small as desired, which implied the non differentiabilityof the trajectories. One may wonder whether Ornstein and Uhlenbeck were aware of Wiener’swork, and if so, what justified for them the use of a model where velocity existed. To providesome answers, we must look later in time. Although we do not analyse the articles publishedafter 1934, we pick some elements that offer useful insights to our problem.
The years 1930-1940 were marked by an important development of modern probability theory,that is to say probability based on measure theory coming from analysis; in particular withL´evy’s, Kolmogorov’s and Doob’s works. This modern theory gave birth to stochastic processes,on the boundary between mathematics and theoretical physics. A new vocabulary appeared fornew mathematical objects.Doob was one of the first mathematicians to formulate a theory of stochastic processes withcontinuous parameters (e.g. time) at a time when most probabilists were not fond of measuretheory (Getoor, 2009). Doob wrote his first article on the subject in 1937, but it is his funda-mental article published in 1942 that we are particularly interested in. In this article appearedthe first rigorous mathematical theory of the Ornstein-Uhlenbeck process, expressed in modernterms stemming from the theory of stochastic processes. Doob’s stated objectives were: to usemodern probability methods to analyse Ornstein-Uhlenbeck distributions and to give the ab-solute probability distributions, as opposed to the conditional distributions that Ornstein andUhlenbeck offered. These distributions are said to be conditional because they depend on theinitial values q of the concerned parameters. In the sense of modern probabilities, it is said thatthey are conditional probability of q knowing q . It was thanks to this modern analysis thatDoob demonstrated that the velocity v , considered in Ornstein-Uhlenbeck model, did not admita derivative, that is to say that Brownian particles had no finite acceleration. He then came toan essential point of his article: the proper writing of Langevin equation to avoid writing d v/ d t since this quantity did not exist. It was on this occasion that Doob introduced a new differentialwriting, which gave rise to stochastic integrals, such as those initiated by Wiener, of which wespoke at the end of section 4.2.5.Doob showed that the variance of the velocity increment over a time t was proportional to t for short times. Indeed, he demonstrated that (cid:104) [ v ( t + s ) − v ( s )] (cid:105) = 2 k B Tm (cid:16) − e − βt (cid:17) ∼ t → k B Tm βt . (6.1)Consequently, (cid:113) (cid:104) [ v ( t + s ) − v ( s )] (cid:105) was of the order of magnitude of √ t and therefore the veloc-ity was not differentiable. It was an argument quite similar to the one of the non differentiability45f displacements in the case of Gaussian increments.Hence, Doob needed a new writing for Langevin equation, in order to avoid writing d v/ d t .He proposed the following formulationd v ( t ) = − βv ( t )d t + d B ( t ) , (6.2)where B ( t ) was a stochastic noise which must be specified. The above equation was simplyLangevin equation multiplied by d t , in which Doob set d B ( t ) = F ( t )d t . Therefore, the force F was formally the derivative of B .Doob then devoted one part of his article to make sense of eq. (6.2) and the different termsthat composed it. According to him, we must understand this equation as equivalent to anyequation of the form (cid:90) ba f ( t )d v ( t ) = − β (cid:90) ba f ( t ) v ( t )d t + (cid:90) ba f ( t )d B ( t ) . (6.3)for all real a and b and for all continuous function f . The first two integrals were classicalintegrals while the third one was a stochastic integral.We do not discuss further this breakthrough, which gave rise to the development of stochasticcalculus, which is a very rich subject that we can not deal with in our history of Brownianmotion. It is interesting to note though, that Doob showed that this function B had the sameproperties as a Wiener process. He also demonstrated that the Gaussian white noise F wasformally the derivative of a Wiener process, although this one did not admit a derivative in thestrict sense.By choosing f ( t ) = exp ( βt ) in eq. (6.3) and by taking v = 0, one can integrate the first twomembers making use of classic integration by parts methods, which gives rise to the stochasticintegral v ( t ) = e − βt (cid:90) t e βt (cid:48) d B ( t (cid:48) ) , (6.4)whose solution is, according to (Kahane, 1998), v ( t ) = e − βt β W ( e βt ) , (6.5)where W is a Wiener process, whose argument has been rescaled.As Jean-Pierre Kahane pointed out in his lecture on Paul Langevin at the Biblioth`eque Na-tionale de France (Kahane, 2014), the physicists’ Brownian motion presupposed the existenceof the velocity while that of Wiener demonstrated that velocity did not exist. According toKahane, they were therefore incompatible, but complementary. He further explained that thetwo echo each other by construction. Indeed, one can start from the Langevin equation, whichrelies on a well-defined velocity of the particle, to obtain Einstein’s relation on the mean of thesquares of displacements in function of the time. Then Wiener started from this last formulato construct an idealized Brownian motion theory, in which velocity did not exist. With Doobwe complete the circle in the other direction because the Langevin equation, that controlled theevolution of the velocity of a particle, can be written formally with the help of the mathemati-cians’ Brownian motion, that is the Wiener process, as we saw above.
In 1945, Uhlenbeck wrote a new article (Uhlenbeck and Wang, 1945) on Brownian movement incollaboration with Ming Chen Wang, a colleague of Michigan University, with whom he co-wrote46leven articles on kinetic-theory problems. The authors proposed to redo a review article on thedifferent Brownian movement theories, with the new vocabulary developed in the years 1930-1940. It was in this publication that Ornstein-Uhlenbeck process was expressed as a stochasticprocess for the first time.It was also in this 1945 article, twelve years after Wiener’s article in which he demonstratedthe non-differentiability of Brownian trajectories in the framework of the Wiener process, andthree years after Doob’s article in which he proved the non-differentiability of the velocity ofBrownian particles in the framework of the Ornstein-Uhlenbeck process, that Uhlenbeck referredto these two works for the first time. In the words of the authors
The authors are aware of the fact that in the mathematical literature (especially in papers byN. Wiener, J. L. Doob, and others; cf. for instance [(Doob, 1942)], also for further references)the notion of a random (or stochastic) process has been defined in a much more refined way.This allows for instance to determine in certain cases the probability that the random function y ( t ) is of bounded variation, or continuous or differentiable, etc. However, it seems to us thatthese investigations have not helped in the solution of problems of direct physical interest, andwe will, therefore, not try to give an account of them. (Uhlenbeck and Wang, 1945, p. 324,footnote 9) We thus see that the authors knew the results of the mathematicians, moreover they quotedsome of Wiener’s articles explicitly in the rest of their article. They believed, however, that themodels constructed by mathematicians were not of physical interest and did not contribute tothe analysis of experimental results. Indeed, even though the non-differentiability of trajectorieshad given rise to many developments in mathematics, it seemed hardly acceptable from a phys-ical point of view, although evoked by Perrin in the first place. Despite the extreme sinuosity ofthe trajectories, there was a scale below which the particle moved in straight line at a constantvelocity between two collisions, which was accounted for by Ornstein and Uhlenbeck’s extensionto short times of the first physical theories. The contributions for physics of the mathematicalmodels of Brownian motion rather lay in the development of mathematical tools that wouldbe used by physicists, and even chemists and biologists, to deal with other stochastic processesappearing in natural sciences.
We can also ask the question of the comparison between the Ornstein-Uhlenbeck theory andthe Einstein-Smoluchowski theory. A good summary of both theories and their relationship wasgiven in (Nelson, 1967):
For ordinary Brownian motion (e.g., carmine particles in water) the predictions of the Ornstein-Uhlenbeck theory are numerically indistinguishable from those of the Einstein-Smoluchowskitheory. However, the Ornstein-Uhlenbeck theory is a truly dynamical theory and representsgreat progress in the understanding of Brownian motion. Also, as we shall see later (Chapter 10),there is a Brownian motion where the Einstein-Smoluchowski theory breaks down completelyand the Ornstein-Uhlenbeck theory is successful. (Nelson, 1967, p. 45)
Let us examine why Edward Nelson claimed that both theories were indistinguishable, and inwhich particular case they could in fact give different results. According to him , the varianceof displacements in Ornstein-Uhlenbeck’s theory was given by (cid:104) x (cid:105) − (cid:104) x (cid:105) = 2 Dt + Dβ (cid:16) − e − βt − e − βt (cid:17) , (6.6)where the second term in the right-hand side was the deviation from the value 2 Dt predictedby Einstein’s theory, and could be estimated numerically. Nelson gave the value 3 · − for the This is straightforward from eq. (5.11) or from eq. (4.23). β − = 10 − s and t = 1 / t ≥ β − .Nevertheless, there was a case where the predictions of the exact theory and of the approx-imation differed significantly: Brownian motion in the presence of an external force, which wehave not discussed until now. It seems important to briefly present this argument because itgives legitimacy to the Ornstein-Uhlenbeck theory, which otherwise would be a great theoreticaladvance without any real experimental application, since Einstein’s theory already accounts foreverything. The argument was detailed in (Nelson, 1967), and read as follows. Nelson consid-ered a Brownian particle in a harmonic potential of pulsation ω , then Langevin equation readin the following coupled form (cid:40) d x ( t ) = v ( t )d t , d v ( t ) = − ω x ( t )d t − βv ( t )d t + d B ( t ) , (6.7)(6.8)where B was a Wiener process of variance β D . There were then three cases:(i) β > ω : over-damped, the viscous friction force is stronger than the harmonic force,(ii) β < ω : under-damped, the viscous friction force is weaker than the harmonic force,(iii) β = 2 ω : critically damped.This experiment was carried out by Eugen Kappler in 1931 in the three cases. Brownian particleswere small suspended mirrors that were impacted by surrounding gas molecules. This was indeeda one-dimensional Brownian motion, described by the angle that the mirrors made with theirequilibrium positions around their axis, while they were subjected to a torsion force of harmonicform. In case (i) the result was expected to be very close from the Brownian motion of a freeparticle, and therefore we expect Einstein’s approximation to be still valid. The plot of the angleversus time in case (i) was very similar to a Wiener trajectory (noisy, random, without a regularpattern), although it never deviated much from its median position because of the force. Thisis the curve of a Markov process, satisfyingly approximated by a Wiener process, which is itselfa Markov process. In case (ii) however, because of the dominance of the harmonic potential,the trajectory was much smoother and approached a sinusoid, which could not be accountedfor by a Markov process. In this case, Einstein-Smoluchowski’s theory, or equivalently Wienerprocesses, which were Markov processes, failed to account for this experimental behaviour.
7. Conclusion
In this article we were interested in the history of Brownian motion from Einstein’s first articlein 1905 to Ornstein’s last article in 1934. We investigated a transition period that has often beenoverlooked, when Brownian motion became an object of interest for mathematicians. Throughthe concept of velocity, we tested the links between theories and experiments and betweentheories themselves, thus exploring the relationship between physics and mathematics. Thiswas the occasion to clarify the status of velocity in Brownian motion theories, to have a glimpseat the birth of stochastic processes and the different ways to introduce randomness into physics,and to present in a simple way Wiener’s results to the physicists. This Langevin equation written in differential form follows the formalism developed by Doob in eq. (6.2), whichwe discussed in section 6.1. The variance of the Wiener process was set to the value 2 β D , which was what Ornstein called γ , according toEquation (5.19). This is therefore in agreement with the primary definition of γ , given by eq. (5.6) (varianceof F ).
48y first looking at the first physical theories of Brownian motion, we deduced that Einsteinand Smoluchowski developed a quite similar model in which displacement was preferred to veloc-ity, which was not properly defined; and that they introduced the stochasticity in their theoriesby considering quantities coming from probability theory, namely probability distributions oraverage quantities. On the contrary, Langevin considered an equation in which velocity ap-peared, and brought randomness into physics thanks to an additional stochastic term, addedto the deterministic equation. Those two ways of thinking the stochastic aspect of a problemare still employed today and named Fokker-Planck equation and Langevin equation. They allagreed however on the fundamental formula (cid:104) x (cid:105) ∝ t , which was a consequence of the Gaussiandistribution of displacements.We focused on Wiener’s work to draw the reasons of his interest for Brownian motion, untilthen a playground for physicists. We found out that Wiener’s taste for physics can be partiallyaccounted for by the lectures his Cambridge’s professors suggested, in which he learnt aboutBrownian motion. Moreover, he found in Brownian motion a good test ground for his ideason integration theory, which he investigated in the light of Perrin’s quotation on the similaritybetween Brownian trajectories and functions without tangent. He built the first mathematicaltheory of Brownian motion, taking Einstein’s theory as a starting point but knowingly workingon an idealised version of it, in which the Gaussian distribution discussed above held for anytime. This choice led to the proof of the non-differentiability of Brownian trajectories, thusechoing Perrin’s intuition.Alongside Wiener’s work, Ornstein, later joined by Uhlenbeck, took another road. WhereasWiener decided to extend Einstein’s Gaussian distribution to short times, even if the physicalreality was much more complex, Ornstein was interested in completing Einstein’s results with atreatment of the short time limit that respected the physical reality. Moreover, he chose to workwith the velocity, following Langevin’s early calculations. For these two reasons, Wiener’s andOrnstein-Uhlenbeck’s theories were conflicting although complementary, as shown by modernstochastic calculus. Ornstein and Uhlenbeck showed that in the short time limit the Gaussiandistribution no longer held, and that a relation of the type (cid:104) x (cid:105) ∝ t held instead, thus leadingto a proper definition of the velocity in this limit.During this walk through history, we did not observe any dialogue between mathematiciansand physicists, for they worked on the same subject in parallel without interacting, expect forthe small mention of Wiener’s work by Uhlenbeck in which he denied its usefulness for physicalpurposes. Indeed, Wiener’s theory did not improve our understanding of physical phenomenarelated to Brownian motion, but instead contributed to the birth of the field of stochasticprocesses, which is now highly used by physicists.In the end, velocity is a concept whose existence and meaning depend on the model oneconsiders. When talking about experiments, velocity has the classic definition used in physicsbut cannot be measured, which explains the failure of early attempted measurements. Whentalking about Wiener process, velocity is not defined, and trajectories are described by con-tinuous functions without tangent at any point, leading to the development of fractal theory.If we consider the Ornstein-Uhlenbeck process, velocity is well defined and linked to physicalparameters, whereas it is not part of the Einstein-Smoluchowski theory. Appendix A Results overview
We intend in this appendix to give a one-page synthesis of the different theories examined inthis article, for the reader to grab at once all the ideas and results. The first table comparesEinstein’s, Smoluchowski’s and Langevin’s theories as for their physical ingredients, mathemat-ical contents and results. The second figure provides a guideline to Wiener’s work, following thesteps of his construction. The last table gathers the articles in which Ornstein’s and Uhlenbeck’s49esults are contained. Einstein Smoluchowski Langevin P h y s i c s Equipartition of energy Indirect (cid:88) (cid:88)
Stokes’ law (cid:88) (cid:88) (cid:88)
Osmotic pressure (cid:88) (cid:55) (cid:55)
Treatment of collisions (cid:55) (cid:88)
Through X M a t h s Stochasticity Distributions Averages Noise X Existence of velocity (cid:55) (cid:55) (cid:88) R e s u l t s λ x = f ( t ) (cid:88) (cid:88) (cid:88) D = f ( T, µ, a ) (cid:88) (cid:88) (cid:55) Diffusion equation (cid:88) (cid:55) (cid:55)
Table 1: Comparison of Einstein’s, Smoluchowski’s and Langevin’ theories of Brownian motion.
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