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Article

The Smoluchowski Ensemble—Statistical Mechanics of Aggregation

Department of Chemical Engineering, Pennsylvania State University, University Park, PA 16802, USA
Entropy 2020, 22(10), 1181; https://doi.org/10.3390/e22101181
Received: 15 September 2020 / Revised: 12 October 2020 / Accepted: 13 October 2020 / Published: 20 October 2020
(This article belongs to the Special Issue Generalized Statistical Thermodynamics)
We present a rigorous thermodynamic treatment of irreversible binary aggregation. We construct the Smoluchowski ensemble as the set of discrete finite distributions that are reached in fixed number of merging events and define a probability measure on this ensemble, such that the mean distribution in the mean-field approximation is governed by the Smoluchowski equation. In the scaling limit this ensemble gives rise to a set of relationships identical to those of familiar statistical thermodynamics. The central element of the thermodynamic treatment is the selection functional, a functional of feasible distributions that connects the probability of distribution to the details of the aggregation model. We obtain scaling expressions for general kernels and closed-form results for the special case of the constant, sum and product kernel. We study the stability of the most probable distribution, provide criteria for the sol-gel transition and obtain the distribution in the post-gel region by simple thermodynamic arguments.
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Keywords: statistical thermodynamics; irreversible aggregation; Smoluchowski equation; gelation; phase transitions statistical thermodynamics; irreversible aggregation; Smoluchowski equation; gelation; phase transitions
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MDPI and ACS Style

Matsoukas, T. The Smoluchowski Ensemble—Statistical Mechanics of Aggregation. Entropy 2020, 22, 1181. https://doi.org/10.3390/e22101181

AMA Style

Matsoukas T. The Smoluchowski Ensemble—Statistical Mechanics of Aggregation. Entropy. 2020; 22(10):1181. https://doi.org/10.3390/e22101181

Chicago/Turabian Style

Matsoukas, Themis. 2020. "The Smoluchowski Ensemble—Statistical Mechanics of Aggregation" Entropy 22, no. 10: 1181. https://doi.org/10.3390/e22101181

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