### Speaker

Adam Gadomski
(UTP University of Science and Technology Bydgoszcz, Poland)

### Description

In 1916 Marian Smoluchowski proposed a case of constant-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].
By applying this scaling function it is then possible to get, within the long times' limit, 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].
The clustering arguments, first introduced by Smoluchowski [1], are easily applicable to statistical description of physical-metallurgical processes and ceramic-polycrystalline evolutions, termed the normal grain growth, in which bigger clusters grow at the expense of their smaller neighboring counterparts due to capillary conditions [3].
The normal grain growth, and its dynamics, can be expressed in d-dimensional space (d - Euclidean dimension of the space). Upon identifying {k} from the Smoluchowski description with {R}, the mean cluster radius' size from the normal grain 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].
The crucial assumption, however, that assures the equivalence claimed, appears to be fully 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 integral in [0,t] upon dT(t)=dt/f(t), with an adjustable function f, coming from the dispersive or long-tail, or fractal kinetics' arguments [4].
The arguments may at least qualitatively concern biomembranes dynamics; they can also contribute to nucleation-growth processes in (psychodynamic-clustering) living matter conditions [5-7].
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[1] M. von Smoluchowski, Physikalische Zeitschrift **17**, 585 (1916).
[2] R. Jullien, Croatica Chemica Acta **65**(2), pp. 215-235 (1992).
[3] P.A. Mulheran, J.H. Harding, Materials Science Forum **94-96**, pp. 367-372, 1992.
[4] A. Plonka, Dispersive Kinetics, Kluwer, Dordrecht, 2002.
[5] A. Gadomski, A. Gadomski, European Physical Journal B **9**, 569 - 571 (1999).
[6] A. Gadomski, M. Ausloos, T. Casey, Nonlinear Dynamics in Psychology & Life Sciences **21**/2, 129-141 (2017).
[7] A. Gadomski, Philosophical Magazine Letters **70**, 335 (1994).

### Primary author

Adam Gadomski
(UTP University of Science and Technology Bydgoszcz, Poland)

### Co-author

Marcel Ausloos
(GRAPES, Liège, Belgium & University of Leicester, UK)