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

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.