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Dispersion in Rectangular Networks: Effective Diffusivity and Large-Deviation Rate Function.

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We developed a large-deviation theory for scalar dispersion in fluid networks, extending predictions beyond standard Gaussian limits. This approach accurately models concentration distribution over large distances and times.

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Area of Science:

  • Fluid dynamics
  • Statistical physics
  • Network theory

Background:

  • Scalar dispersion in fluid networks is crucial for understanding biological flows, porous media, water supply, and urban pollution.
  • Existing models often rely on Gaussian approximations with limited validity range.

Purpose of the Study:

  • To develop a large-deviation theory for predicting scalar concentration evolution in rectangular networks.
  • To extend the range of validity for scalar concentration approximations beyond standard Gaussian limits.
  • To derive an effective diffusivity tensor for Gaussian approximations.

Main Methods:

  • Development of a large-deviation theory for scalar dispersion.
  • Mathematical analysis for large time (t≫1) limits.
  • Monte Carlo simulations using Brownian particles for validation.

Main Results:

  • The large-deviation theory provides accurate concentration predictions for distances up to O(t), surpassing the O(t^{1/2}) range of Gaussian approximations.
  • A closed-form expression for the effective diffusivity tensor was derived.
  • Monte Carlo simulations confirmed the theory's effectiveness, even for moderately large times.

Conclusions:

  • The developed large-deviation theory offers a more accurate and extended description of scalar dispersion in fluid networks.
  • The findings are applicable to various real-world scenarios involving scalar transport in complex flow systems.
  • The study validates theoretical predictions with robust numerical simulations.