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Exact combinatorial approach to finite coagulating systems.

Agata Fronczak1, Anna Chmiel1, Piotr Fronczak1

  • 1Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland.

Physical Review. E
|March 18, 2018
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Summary

This study presents an exact combinatorial method for finite coagulating systems, offering a new way to calculate cluster probabilities and distributions over time. The approach accurately models system evolution, outperforming mean-field approximations.

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Area of Science:

  • Physical Chemistry
  • Statistical Mechanics
  • Computational Physics

Background:

  • Coagulating systems describe particle aggregation, crucial in fields like colloid science and atmospheric chemistry.
  • Existing models, such as the Smoluchowski equation, often rely on mean-field approximations that can limit accuracy.
  • Discrete-time and discrete-size models offer an alternative for precise system analysis.

Purpose of the Study:

  • To develop an exact combinatorial approach for finite coagulating systems.
  • To derive a general expression for system probability based on cluster growth histories.
  • To calculate time-dependent cluster distributions and their statistical properties.

Main Methods:

  • Discrete modeling of cluster sizes and time.
  • Analysis of binary aggregation processes.
  • Derivation of exact probability expressions from cluster growth histories.
  • Validation using systems with a constant aggregation kernel.

Main Results:

  • An exact expression for the probability of a given cluster configuration was derived.
  • Time-dependent distributions for cluster numbers and sizes were calculated.
  • The approach's accuracy was validated against analytical and numerical results.
  • Comparison with the Smoluchowski equation highlighted limitations of mean-field approximations.

Conclusions:

  • The exact combinatorial approach provides a robust framework for finite coagulating systems.
  • This method offers higher accuracy than mean-field approximations for modeling aggregation dynamics.
  • The approach is potentially extensible to systems with arbitrary initial conditions and kernels.