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Boltzmann Equation in Aggregation Kinetics

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Nonequilibrium Thermodynamics and Fluctuation Kinetics

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 208))

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Abstract

We consider the application of the Boltzmann equation to aggregation kinetics, where the transport mechanism is the ballistic motion of particles. This refers to molecular gases, granular gases, and, hypothetically, dark matter. Two aggregation models are analyzed—random and impact energy-dependent aggregation. The latter is associated with different interparticle forces responsible for agglomeration. We start from the Boltzmann equation governing the evolution of the mass–velocity distribution functions of different species—the agglomerates of different sizes and derive generalized Smoluchowski equations. These describe the time dependence of the agglomerates densities and their mean kinetic energy (partial temperatures). We obtain exact solutions to these equations for simplified cases and develop a scaling theory for the asymptotic behavior of the system. We explore numerically, the agglomeration kinetics and observe a very rich behavior of the system. We reveal new surprising regimes and construct the according kinetic phase diagram. The scaling theory is in excellent agreement with the simulation results.

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Notes

  1. 1.

    We assume that the aggregates are spherical and compact; the generalization for fractal aggregates is straightforward.

  2. 2.

    Note that the constant may be always set to one using the appropriate time units.

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Acknowledgements

The study was supported by a grant from the Russian Science Foundation No. 21-11-00363, https://rscf.ru/project/21-11-00363/. We also thank the German Science Foundation (DFG) for funding through Grant P0472/40-1 and the Interdisciplinary Center for Nanostructured Films (IZNP). Support from ZISC and IZ-FPS at FAU Erlangen–Nürnberg is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Skolkovo Institute of Science and Technology, 30 Bolshoi Boulevard, Moscow, 121205, Russia

    Nikolai V. Brilliantov & Alexander I. Osinsky

  2. Institute for Multiscale Simulation, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 3, 91058, Erlangen, Germany

    Thorsten Pöschel

Authors
  1. Nikolai V. Brilliantov
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  2. Alexander I. Osinsky
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  3. Thorsten Pöschel
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Corresponding author

Correspondence to Nikolai V. Brilliantov .

Editor information

Editors and Affiliations

  1. Physics Department, Université Libre de Bruxelles (U.L.B.), Bruxelles, Belgium

    Léon Brenig

  2. Department of Mathematics, University of Leicester, Leicester, UK

    Nikolai Brilliantov

  3. Physics Department, Université Libre de Bruxelles (U.L.B), Bruxelles, Belgium

    Mustapha Tlidi

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Brilliantov, N.V., Osinsky, A.I., Pöschel, T. (2022). Boltzmann Equation in Aggregation Kinetics. In: Brenig, L., Brilliantov, N., Tlidi, M. (eds) Nonequilibrium Thermodynamics and Fluctuation Kinetics. Fundamental Theories of Physics, vol 208. Springer, Cham. https://doi.org/10.1007/978-3-031-04458-8_10

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