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SUMMARY:A tribute to Marian Smoluchowski's legacy on colloid type matter a
ggregation\, and related issues
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UID:indico-contribution-162@zakopane.if.uj.edu.pl
DESCRIPTION:Speakers: Adam Gadomski (UTP University of Science and Technol
ogy Bydgoszcz\, Poland)\nIn 1916 Marian Smoluchowski proposed a case of co
nstant-kernel cluster cluster aggregation\, for which it is manageable to
find analitycally by employing scaling arguments\, a solution in terms of
the cluster size (k) distribution function\, n(k) [1\,2].\nBy applying thi
s scaling function it is then possible to get\, within the long times' lim
it\, the results for the mean cluster size {k} and the total number of the
clusters N\, both scalable in terms of time t with a single exponent\, g
[2].\nThe clustering arguments\, first introduced by Smoluchowski [1]\, ar
e easily applicable to statistical description of physical-metallurgical p
rocesses and ceramic-polycrystalline evolutions\, termed the normal grain
growth\, in which bigger clusters grow at the expense of their smaller nei
ghboring counterparts due to capillary conditions [3].\nThe normal grain g
rowth\, and its dynamics\, can be expressed in d-dimensional space (d - Eu
clidean dimension of the space). Upon identifying {k} from the Smoluchowsk
i description with {R}\, the mean cluster radius' size from the normal gra
in growth approach\, and by taking the "extreme" condition of k >> 0\, one
is able to embark on their equivalence by stating rigorously that g=1/(d+
1)\, since the asymptotic scaling rule for N (here: the number of grains)
goes via a simple logarithmic depiction as: ln[N]~-ln[g].\nThe crucial ass
umption\, however\, that assures the equivalence claimed\, appears to be f
ully feasible when rearranging the time domain by substituting t in a way
such that a new rescaled time variable T(t) is given by a definite integra
l in [0\,t] upon dT(t)=dt/f(t)\, with an adjustable function f\, coming fr
om the dispersive or long-tail\, or fractal kinetics' arguments [4].\nThe
arguments may at least qualitatively concern biomembranes dynamics\; they
can also contribute to nucleation-growth processes in (psychodynamic-clus
tering) living matter conditions [5-7].\n\n$ $\n\n[1] M. von Smoluchowski\
, Physikalische Zeitschrift **17**\, 585 (1916).\n\n[2] R. Jullien\, Croat
ica Chemica Acta **65**(2)\, pp. 215-235 (1992).\n\n[3] P.A. Mulheran\, J.
H. Harding\, Materials Science Forum **94-96**\, pp. 367-372\, 1992.\n\n[4
] A. Plonka\, Dispersive Kinetics\, Kluwer\, Dordrecht\, 2002.\n\n[5] A. G
adomski\, A. Gadomski\, European Physical Journal B **9**\, 569 - 571 (199
9).\n\n[6] A. Gadomski\, M. Ausloos\, T. Casey\, Nonlinear Dynamics in Psy
chology & Life Sciences **21**/2\, 129-141 (2017).\n\n[7] A. Gadomski\, P
hilosophical Magazine Letters **70**\, 335 (1994).\n\nhttps://zakopane.if.
uj.edu.pl/event/4/contributions/162/
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URL:https://zakopane.if.uj.edu.pl/event/4/contributions/162/
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