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On Marian Smoluchowski's Life and Contribution to Physics Vol. 29 (1998) ACTA PHYSICA POLONICA B No 6 ON MARIAN SMOLUCHOWSKI’S LIFE AND CONTRIBUTION TO PHYSICS , ∗ ∗∗ A. Fuliński M. Smoluchowski Institute of Physics, Jagellonian University Reymonta 4, 30-059 Kraków, Poland e-mail: [email protected]; http://zfs.if.uj.edu.pl (Received March 18, 1998) The first part presents the life and historical contributions of Marian Smoluchowski to the foundations of atomic theory of matter. Remaining parts are devoted to the presentation of the original derivation and present applications of the Smoluchowski equations. Commonly used in various physical, chemical, etc., problems is the kinetic Smoluchowski equation, being the high-friction limit of the full Fokker-Planck equation. Of less use so far has been the functional Smoluchowski (called also Chapman-Enskog or Bachelier) equation. Within the scope of modern theory of stochas- tic processes this equation is an identity equivalent to the definition of a Markovian process. It is argued that, in the view of the growing interest in non-Markovian stochastic processes, the functional Smoluchowski equation may become the most natural and effective tool for checking the Marko- vian vs. non-Markovian character of a considered process, especially when applied to experimental data. Some examples are given. PACS numbers: 05.40. +j, 02.50. Ga, 02.50. Ey ∗ Introductory lecture presented at the Marian Smoluchowski Symposium on Statistical Physics, Zakopane, Poland, September 1–10, 1997. ∗∗ o This work was partially supported by the Polish KBN grant N 2 P03B 005 13. (1523) 1524 A. Fuliński 1. Introduction Marian Smoluchowski1 was born May 28th, 1872 in Vorderbrüchl near Vienna. His father was a high official in the chancellery of emperor Franz Joseph I. Smoluchowski studied at Vienna University, where he received his Ph.D. — with highest honours (sub auspiciis imperatoris) — in 1894. After that, he spent two years at universities in Paris, Glasgow and Berlin. In 1898 he returned to Vienna, where obtained veniam legendi (“Privatdozent”). In 1900, at the age of 28, Marian Smoluchowski got the chair of the- oretical physics at the Lvov University, becoming the youngest university professor (from 1903 — full professor) in the whole Habsburg Empire. He married Zofia Baraniecka in 1901. In 1913 he took the chair of experimental physics at the Jagellonian University. Just before his premature death he was elected for the Rector of our University. Died September 5th, 1917 in Kraków. One of his lifelong passions was musics, other one — mountains. I am devoting a few words to his achievements in that field, because “To Smolu- chowski, mountain climbing was more than a sport, it was an essential part of life. It was in the mountains that he could face unspoiled nature and ac- quire strength for his scientific endeavours. Mountains meant to him three things, he wrote to his alpine club: They accustomed him to undertaking difficult tasks, they gave him satisfaction when he overcame the challenge and they embellished his life with what he believed to be the highest poetry, the poetry of mountains” [7]. During his studies at Vienna University he was an outstanding alpinist and rock climber, one of the pioneers of modern alpinism. In their students’ years, in 1890’s, brothers Tadeusz and Marian Smoluchowski traced 24 new record ascents in Dolomite and Ortler Alps. Their generation revolutionized alpinism — inaugurated the climbing without the help of local highlander guides. Later on, till 1909, Marian Smoluchowski spent several climbing sea- sons in Central Alps, ascending several highest and difficult peaks, including legendary in that time Matterhorn. He climbed also in Scotland. 1 Material about life and work of Marian Smoluchowski can be found in several books. His biography by A. Teske was published in Polish and in German [1]. Ref. [2] con- tains the reproduction of all Smoluchowski’s published papers. In Ref. [3] are repro- duced (in original and in English translation): his fundamental paper about Brownian motion [4], and “Three lectures [given at Göttingen Univ.] on diffusion, Brownian motion and coagulation of colloids” [5], being a recapitulation of his work on these problems. Besides, this book gives also the chronological table of Smoluchowski’s life, and full bibliography of his publications as well as a bibliography of works about Mar- ian Smoluchowski. Ref. [6] presents essays discussing Smoluchowski’s contributions to experiments, to statistical physics, and to didactics. On Marian Smoluchowski’s Life and Contribution to Physics 1525 In the Lvov times Smoluchowski brothers took part in pioneer ski expe- ditions in East Carpathians. Later on, Marian Smoluchowski was an active member of Polish mountaineering (Climbing) Club in Tatra Mts. In the history of physics Marian Smoluchowski will be remembered mainly for his work on the theory of Brownian motion and related problems. In his obituary, Arnold Sommerfeld [8] wrote: “His name will, forever, be associ- ated with the first flowering of atomic theory”. Let us remember that at the beginning of our century the “molecular hypothesis” was just a hypothesis, which needed proofs, and that the first experimental proofs — honoured in 1926 by Nobel prizes in physics (for Perrin) and in chemistry (for Sved- berg) — were based on Smoluchowski’s theory of Brownian motion, critical opalescence, sedimentation and coagulation. However, Smoluchowski’s name remains still alive in the modern physics. His papers are frequently cited in the literature. The theoretical basis and the methods established in his work, including the Smoluchowski equations derived within theories of Brownian motion, etc., became the founding blocks of modern statistical physics, especially of the theory of stochastic processes. This aspect of Smoluchowski work has been appreciated earlier by Mark Kac and Subrahmanyan Chandrasekhar, who were certainly better equipped than me to judge these results. Therefore I shall use their words [9,10]. Mark Kac wrote [9]: “... while directed toward the same goal how differ- ent the Smoluchowski approach is from Boltzmann’s. There is no dynamics, no phase space, no Liouville theorem — in short none of the usual under- pinnings of Statistical Mechanics. Smoluchowski may not have been aware of it but he begun writing a new chapter of Statistical Physics which in our times goes by the name of Stochastic Processes. It is the probabilistic point of view in contradistinction to the statistico-mechanical one that is also clearly present in Smoluchowski’s first paper on Brownian motion [11] and in a paper on the mean free path which just preceded it [4]. [...] This is a remarkable paper in which as far as I know the relation between random walk and diffusion is first established (L. Bachelier who is usually credited with this discovery comes a little later). Equally remarkable is the fact that Smoluchowski uses here Fourier transforms to handle convolutions of prob- ability distributions and is thus first to prove a special case of the central limit theorem of probability theory by means of characteristic functions. [...] The novelty and originality of the Smoluchowski approach lie in his bold replacement of an impossibly difficult dynamical problem (a Brownian particle in a gas or liquid) with the statistical element coming in through the lack of specification of the initial state by a relatively simple stochastic process. A dynamical event like a collision, for example, is thus treated as if it were the result of a toss of a coin or a die with the laws of mechanics determining (to some extent at least) the probabilities of various outcomes. 1526 A. Fuliński The underlying idea proved enormously fruitful and it gradually permeated much of statistical physics; permeated it, in fact, so well that few of us re- alize today that much of ‘modern’ problematics (notably that related to so called master equations) is directly traceable to ideas first promulgated by Smoluchowski in the early years of this century”. Kac mentions also (quoting Chandrasekhar’s paper [12]) “... that it was Smoluchowski who first introduced fluctuations into Physics (as early as 1904 in the Boltzmann Festschrift) [13]”. According to Chandrasekhar [7], “... a series of papers which Smolu- chowski wrote during last five years of his life, [are] papers in which the foundations of the modern theory of stochastic processes were laid [...] De- tailed comparisons of the predictions of the foregoing theory [of Brownian motion in a colloidal solution] with experimental results [...] confirm the basic premises of the theory in a most striking way. But what is perhaps of even greater significance is that we have here the first example of a case in which it has been possible to follow in all detail, both theoretically and ex- perimentally, the transition between the macroscopically irreversible nature of diffusion and the microscopically reversible nature of molecular fluctua- tions”. 2. Smoluchowski’s equations Marian Smoluchowski’s legacy for modern physics are equations bearing his name, and being still the useful tools, though these equations are nowa- days derived in another manner, and different is the reconstruction of the general scheme of the theory. The main point in the Einstein-Smoluchowski [14,11] theory of Brownian motion is the observation that the proper measure of the motion is not the mean velocity of the particle of the suspension, but the mean square of its 2 shift ∆x = x x0 from initial position x0. In these first papers (∆x) was calculated:− by Einstein from general laws of diffusion and statisticalh i mechanics, by Smoluchowski — by detailed analysis of the mechanism of motion of the Brownian particle. It is worth to mention that they both assumed the same property of probabilities (discussed in more detail below). The Smoluchowski equations were formulated in his later papers, con- taining the theory of Brownian motion, and of coagulation, in external force fields [5,15,16].
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