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Uninfected random walkers in one dimension.

S J O'Donoghue¹, A J Bray

¹Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom.

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

|June 13, 2002

Summary

This study numerically investigates diffusing walkers and their encounters in one dimension. The fraction of uninfected walkers relates to unvisited sites by U(t) approximately [P(t)]gamma, with gamma near 1.39.

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Area of Science:

Statistical Mechanics
Physical Chemistry
Computational Physics

Background:

Understanding particle interactions and diffusion is crucial in various scientific fields.
The behavior of diffusing systems, especially in one dimension, presents unique challenges.
Previous studies often focus on specific reaction kinetics or initial conditions.

Purpose of the Study:

To numerically investigate the relationship between uninfected walkers and unvisited sites in a 1D diffusion system.
To analyze the impact of random initial conditions on walker encounters.
To extend the analysis to include a diffusion-limited reaction (A+B-->Phi) model.

Main Methods:

Numerical simulations of unbiased diffusing walkers in one dimension.
Calculation of the fraction of uninfected walkers U(t) over time.
Calculation of the fraction of unvisited sites P(t) over time.
Extension of simulations to include a one-dimensional diffusion-limited reaction model with equal initial densities of A and B particles.

Main Results:

A power-law relationship was found between uninfected walkers and unvisited sites: U(t) approximately [P(t)]gamma.
The exponent gamma was found to be approximately 1.39 in both the walker encounter and reaction models.
Evidence suggests that a smaller value of gamma may be required as time approaches infinity.

Conclusions:

The relationship U(t) approximately [P(t)]gamma holds for both walker encounters and diffusion-limited reactions in 1D.
The exponent gamma provides insight into the spatial-temporal dynamics of diffusion-limited processes.
Further investigation is needed to determine the asymptotic value of gamma for infinite time.

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