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Finite-size effect in persistence in random walks.

D Chakraborty¹, J K Bhattacharjee

¹Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India. tpdc2@mahendra.iacs.res.in

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

|March 16, 2007

Summary

We studied random walks in finite systems, finding that system size affects persistence probability. The scaling function shows exponential decay, confirmed by analytical and numerical methods.

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Area of Science:

Statistical Physics
Condensed Matter Physics
Computational Physics

Background:

The random walk model is fundamental in statistical physics, describing diffusion and transport phenomena.
Understanding behavior in finite systems is crucial for realistic modeling, differing from infinite systems.
Persistence probability, the likelihood of a system variable remaining positive, is key in non-equilibrium systems.

Purpose of the Study:

To investigate the random walk problem within finite systems.
To analyze the influence of system size on the crossover of persistence probability.
To explore trapping mechanisms and their impact on particle dynamics.

Main Methods:

Analytical calculations using mathematical derivations.
Numerical simulations to validate theoretical findings.
Analysis of two trapping scenarios: a finite box and a harmonic trap.

Main Results:

The scaling function for persistence probability exhibits exponential decay.
System size induces a crossover in persistence probability.
Numerical results were obtained for randomly accelerated particles with and without viscous drag.

Conclusions:

Finite system size significantly alters random walk dynamics, particularly persistence probability.
Exponential decay characterizes the scaling function, providing a universal behavior.
The study provides a comprehensive analysis of random walk and related particle dynamics in confined environments.