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The Diffusion of Passive Tracers in Laminar Shear Flow

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Published on: May 1, 2018

Variance of residence time spent by diffusing particle in a sub-domain. Path integral based approach.

A M Berezhkovskii¹

¹Mathematical and Statistical Computing Laboratory, Division of Computational Bioscience, Center for Information Technology, National Institutes of Health, Bethesda, MD 20892, USA.

Chemical Physics

|June 10, 2010

Summary

This study analyzes particle diffusion in complex systems using a path integral approach. We found that the variance of residence time grows linearly with observation time, providing a method to determine this growth rate.

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Area of Science:

Statistical Mechanics
Physical Chemistry
Computational Physics

Background:

Understanding particle diffusion is crucial in various scientific fields.
Quantifying residence time and its fluctuations in specific regions is a key challenge.
Existing models may not fully capture behavior in complex potentials.

Purpose of the Study:

To develop a theoretical framework for analyzing residence time variance in diffusing systems.
To investigate the behavior of residence time variance under arbitrary potentials.
To provide a method for calculating the linear growth rate of variance.

Main Methods:

Utilizing a path integral based approach.
Assuming no particle absorption within the domain or at boundaries.
Leveraging the ergodicity property of the system.

Main Results:

Demonstrated that the variance of residence time grows linearly with observation time at large times.
Established a method to determine the slope of this linear dependence.
Derived explicit formulas for variance in illustrative examples.

Conclusions:

The path integral method offers a robust way to analyze residence time variance.
The linear growth of variance with observation time is a general characteristic of such diffusion processes.
The derived methodology allows for quantitative predictions in systems with arbitrary potentials.