First-encounter time of two diffusing particles in confinement

Area of Science:

• Physics
• Physical Chemistry
• Statistical Mechanics

Background:

• Diffusion processes are fundamental in various scientific fields.
• Understanding particle interactions in confined spaces is critical for kinetics.
• First-encounter time and survival probability are key metrics in diffusion studies.

Purpose of the Study:

• To investigate the impact of confinement on the first-encounter time and survival probability of two diffusing particles.
• To derive analytical insights for one-dimensional diffusion in a half-line and an interval.
• To explore the effects of equal and unequal particle diffusivities.

Main Methods:

• Analytical treatment of diffusion in one-dimensional settings (half-line and interval).
• Calculation of exact survival probability and first-encounter time density for equal diffusivities.
• Evaluation of moments for the first-encounter time.
• Analysis of long-time behavior of survival probability for unequal diffusivities.

Main Results:

• Confinement drastically alters first-encounter time probability density and survival probability.
• Exact analytical results for survival probability and first-encounter time density were obtained for equal diffusivities.
• Moments of the first-encounter time were evaluated where they exist.
• Long-time behavior of survival probability was analyzed for unequal diffusivities, highlighting boundary effect impacts.

Conclusions:

• Boundary effects significantly influence diffusion-controlled kinetics, even in simple 1D systems.
• Obtaining analytical results becomes challenging when translational invariance is broken.
• The study provides a detailed understanding of confinement effects on particle diffusion dynamics.