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

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.