Capture of particles undergoing discrete random walks

Area of Science:

• Physics
• Physical Chemistry
• Computational Science

Background:

• Understanding particle dynamics near surfaces is crucial for modeling chemical reactions and aggregation processes.
• Discrete-time jump processes offer a more general framework than continuous Brownian motion for simulating particle behavior.
• Adsorbing spheres serve as a fundamental model for interaction sites in various physical and chemical systems.

Purpose of the Study:

• To determine the capture probability of particles undergoing discrete-time jumps by an adsorbing sphere.
• To investigate the asymptotic survival probability of particles starting on the sphere's surface.
• To provide insights applicable to computer simulations of reaction and aggregation.

Main Methods:

• Analytical calculation of capture probability for particles starting at a distance r(0) from the sphere's center (r(0)>>R).
• Analysis of asymptotic survival probability for particles starting on the sphere's surface.
• Relating capture and survival probabilities to the jump distribution's properties (Fourier transform, root-mean square jump length).

Main Results:

• Capture probability is given by (R-c sigma)/r(0) for r(0)>>R, where c depends on the jump distribution's Fourier transform and sigma is the root-mean square jump length.
• Asymptotic survival probability for particles starting on the surface is nonzero and universally depends on sigma/R as sigma/(R square root(6)).
• This behavior contrasts with Brownian diffusion, where survival probability is typically zero.

Conclusions:

• Discrete-time jumps lead to distinct capture and survival dynamics compared to Brownian diffusion.
• The derived probabilities are directly applicable to enhancing the accuracy and efficiency of computational simulations for reaction and aggregation.
• The universal behavior of survival probability highlights the importance of the ratio sigma/R in determining particle fate near surfaces.