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First-passage time: lattice versus continuum.

Kamal Sharma¹, N Kumar

¹Raman Research Institute, Bangalore 560080, India.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|October 4, 2012

Summary

This summary is machine-generated.

This study revises classical diffusion calculations for discrete lattices, modifying a key boundary condition for accurate first-passage probability density. The findings offer a new analytical method for barrier escape problems.

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Area of Science:

Statistical Physics
Computational Physics
Quantum Mechanics

Background:

Classical diffusion on a continuum uses Schrödinger's integral equation for first-passage probability density.
Diffusion by hopping on discrete lattices presents unique challenges for this calculation.
Existing boundary conditions, derived for continuous systems, require re-evaluation for discrete models.

Purpose of the Study:

To revisit the Schrödinger's integral equation approach for calculating first-passage probability density in discrete lattice diffusion.
To identify and modify the boundary condition necessary for accurate calculations in discrete systems.
To derive an analytical expression for first-passage density in a specific discrete model.

Main Methods:

Revisiting the integral equation approach for discrete lattice diffusion.
Modifying the boundary condition based on normalization requirements.
Deriving an analytical expression for a three-site barrier escape model.

Main Results:

A modified boundary condition is identified as crucial for discrete lattice diffusion.
The normalization condition uniquely determines the required boundary condition.
An explicit analytical expression for first-passage density is derived for a three-site model.

Conclusions:

The standard boundary condition for continuous diffusion needs modification in discrete lattice hopping.
The normalization condition is sufficient to determine the correct boundary condition for discrete systems.
The derived analytical expression provides a method for modeling escape over barriers in discrete systems.