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\noindent{\bf Marian SMOLUCHOWSKI}\\
b. 28 May 1872 - d. 5 September 1917
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\noindent{\bf Summary} The investigations of the physicist Smoluchowski on
the theory of Brownian movement concided with those of Einstein. The
work on statistical mechanics contains probably the first manifestation of a
branching process with immigration, and of statistical inference for a random process.
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Smoluchowski (pronounced Smo-loo-khov-ski) was born in Vorderbr\"{u}hl,
near Vienna to a Polish family and died in Cracow (Polish: Krak\'{o}w ;
\ German: Krakau). The times were those of the Austro-Hungarian Monarchy
of which the north-east part was Galicia (German: Galizien) with its
university cities of Krakau (the old capital of Poland); and Lemberg
(Ukrainian: L'viv; Polish: Lw\'{o}w; Russian: L'vov), a city in Poland
between World Wars 1 and 2, and now in Ukraine. Smoluchowski spent his
youth in Vienna, where his father was a senior official of Emperor Franz
Josef, and attended the Collegium Theresianium from 1880 to 1890. From
1890 to 1894 Smoluchowski studied physics at the University of Vienna under the
direction of Josef Stefan and F. Exner. His studies were followed by a year of military
service in the Austrian army, 1894/95, with his PhD in physics from Vienna
University being conferred with the highest distinction in 1895. After some
years of research in Paris in the
laboratory of Gabriel Lippmann, in Glasgow under Lord Kelvin, in London, and
in Berlin under Emil Warburg (this period being most significant for his future
research), he worked at the University of Lemberg from 1899. Here he held the Chair
of Mathematical Physics from 1903 to 1913. In 1913 he became Professor
of Experimental Physics at the Jagiellonian University in Cracow.
He died of dysentery (doubtless due to wartime conditions)
in the summer of 1917 at the age of 45 before he could take up his
duties as rector of this university. His most important work was done
during the period in L'vov, and is heavily probabilistic and
statistical. It is regrettable, as the later Nobel laureate
Chandrasekhar (1943) makes clear several times, that Smoluchowski's
lucid contributions to the theory of Brownian motion, molecular
fluctuations and laws of thermodynamics, have beem neglected. One might
add to this list the theory of branching processes, where his
contributions have long occupied a neglected position similar to that of
I.J. Bienaym\'{e} (q.v.).
From about 1900 Smoluchowski worked on Brownian movement, and obtained a
theory starting from the standpoint of examining the effects of
successive collisions between a Brownian particle and successive
molecules. Einstein started from general relations of statistical physics,
and presented a theory of the Brownian movement in papers of 1905 and
1906. Smoluchowski had waited to use experimental data for verification
of his own theory, which was little different, but in view of Einstein's first
paper, published his findings in 1906. When Einstein was awarded the Nobel Prize
in Physics in 1921, his work on Brownian motion is cited, before the discussion
of the main reason for the prize : the discovery of the law of photoelectric
effect.
Smoluchowski's theory of density fluctuations is within the context of the
Brownian motion work, as the title of the 1914 paper testifies.
Chandrasekhar (1943) described
it as ``one of the most outstanding achievements in
molecular physics". Smoluchowski here modelled the fluctuation in
the number of particles contained in a geometrically well-defined small
element of volume, {\it v}\,, \ within a much larger volume of solution
containing particles
exhibiting random motion, the system being in
equilibrium. Observations $X_1 , X_2 , \cdots$ on the number of particles
in \ $v$ \ are made at points of time at equal intervals, \ $\tau$\,, \
apart. During such a time interval some particles present in \ $v$ \ at
its beginning will have left it (have ``died") by its end while some
external to \ $v$ \ at the beginning of the interval may be present within
\ $v$ \ at its end (have ``immigrated"). The probability \ $P$ \ that a
particle present at the beginning will be external at the end depends on
the precise
circumstances of the problem. An explicit {\it theoretical}
solution was obtained by
Smoluchowski (1914) when the motions are governed
by the laws of Brownian movement, in terms of the various physical
parameters. Since the system is in equilibrium, and \ $v$ \ is small, the
probability distribution describing each \ $X_n$ \ is Poisson, with
mean/variance parameter \ $\mu$, \ say : $Pr(X_n = x) = e^{-\mu} \mu^x
/ x! , \ x = 0, 1, 2, \cdots$ . Both $P$ and $\mu$ can be
estimated {\it statistically} from the numerical observations on $X_1 ,
X_2 , \cdots$ . \ A comparison of the predictions of the theory of
colloid statistics with the data is therefore made possible, and was
in fact carried out on the data of Th. Svedberg by Smoluchowski himself.
In regard to the statistical estimation procedure, Smoluchowski derived
the elegant probabilistic expression $E((X_{n+1} - X_{n})^2) = 2\mu \, P$.
Its left-hand side was estimated, using observations $X_1, \cdots,
X_{N+1}$ , naturally by
$$\sum_{i=1}^N (X_{i+1} - X_{i})^2 / N$$
\noindent while \ $\mu$ \ was estimated by ${\hat \mu} = \sum_{i=1}^N
(X_i - {\bar X})^2 / N$ where ${\bar X} = \sum_{i=1}^N X_i / N$, thus
giving \ $\hat P$ . \ Shortly after, it became common to use the
estimator ${\hat \mu} = {\bar X} \ \mbox{for} \ \mu$ (F\"{u}rth, 1918).
The striking advance here on earlier fluctuation theory was the
introduction of the {\it probability after-effect
(``Wahrscheinlichkeitsnachwirkung") $P$} , \ which is a manifestation
of the assumed Markovian (``given the present, the future is independent
of the past") structure of the assumed model, and is of great
significance in other physical contexts. Indeed the random process
\ $\{X_t\}$ \ used as a model by Smoluchowski is a Bernoulli-Poisson
branching process with immigration, apparently the first time a
branching process with immigration of any kind occurs in the literature.
Branching processes with immigration are examples of Markov (q.v.)
chains, which arose, at least as a general concept, circa 1906. This
Markovian aspect of Smoluchowski's work was overlooked in
German-language literature. The transition probabilities of the
Bernoulli-Poisson Markov chain, which are of simple explicit kind, were
however already present, both as a physical idea and in explicit form in
Smoluchowski (1914).
It is worth reflecting on the fact that at the time of
Smoluchowski's work, statistical estimation theory, even for independent
samples, was in its infancy, and yet he used totally appropriate
estimators in the more complex setting of a Markov chain.
Sommerfeld (1917) in his obituary lists 53 works by Smoluchowski
over the period 1893-1917, mainly
in the official language of the Austro-Hungarian Monarchy, German, but some in French and
English; and mentions the existence of some 36 further works in Polish.
Smoluchowski married Zofia Baraniecka, the daughter of a mathematics
professor at the Jagiellonian University, in 1901 and lived contentedly in
Lemberg, taking an active part in the life of the university and the
community, until the move to Cracow. He was fond of mountaineering, skiing
and music
in the company of his close friend Hasen\"orhl, who was killed in
World War 1, in 1915.
Smoluchowski's wife outlived him
by some 40 years, and died in the U.S.A. as did his son, Roman Smoluchowski
(1910 - 1996), a notable physicist. There is also family in Poland.
Sommerfeld(1917) refers to him as: Marian Ritter von
Smolan-Smoluchowski (``Ritter" translates into ``knight".)
\vspace{.5cm}
\begin{thebibliography}{3}
\bibitem{1}
Chandrasekhar, S. (1943). Stochastic problems in physics and astronomy.
{\it Reviews of
Modern Physics} {\bf 15}, 1-89. {[Also in N. Wax (Ed.)
{\it Selected Problems on Noise and
Stochastic Processes.} Dover, New
York, 1954, pp.3-91.]}
\bibitem{2}
F\"{u}rth, R. (1918). Statistik und Wahrscheinlichkeitsnachwirkung. {\it
Physikalische Zeitschrift}
{\bf 19}, 421-426, {\bf 20} (1919) 21.
\bibitem{3}
Smoluchowski, M. von (1914). Studien \"{u}ber Molekularstatistik von
Emulsionen und deren
Zusammenhang mit der Brown'schen Bewegung. {\it
Sitzungsberichte. Akademie der Wissenschaften}.
Mathematisch-Naturwissenschaftliche Klasse. CXXIII Band. X Heft.
Dez. 1914. Abteilung II A. Wien. pp.2381-2405.
\bibitem{4}
Smoluchowski, M. (1924-1928). {\it Pisma Mariana Smoluchowskiego
(Collected Works of
Marian Smoluchowski)}. 3 vols., W. Natanson (Ed.).
PAU, Krak\'{o}w.
\bibitem{5}
Sredniawa, B. (1993). The collaboration of M. Smoluchowski and Theodor
Svedberg on
Brownian motion and density fluctuations. {\it Centaurus}
(Copenhagen) {\bf 41}, 325-355.
\bibitem{6}
Sommerfeld, A. (1917). Zum Andenken an Marian von Smoluchowski. {\it Physikalische
Zeitschrift,}, No.22, {\bf 18}, 533-539.
\bibitem{7}
Teske, A. (1970). An outline account of the work of Marian Smoluchowski
1872-1917.
In: {\it \'{E}tudes consacr\'{e}es \`{a} Maria
Sklodowska-Curie et \`{a} Marian Smoluchowski}.
Monografie z
Dziej\'{o}w Nauki i Techniki, {\bf 51}, PAN, Wroclaw. {[See also the
entry by
his son A.A. Teske (1975) on Smoluchowski in
{\it Dictionary of Scientific Biography},
{\bf 12}, 496-8]}.
\bibitem{8}
Ulam, S. (1957). Marian Smoluchowski and the theory of probabilities in
physics.
{\it American Journal of Physics}, {\bf 25}, 475-481.
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\hfill E. Seneta
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