Simulation of the wiener sausage

Area of Science:

• Statistical Physics
• Stochastic Processes
• Computational Physics

Background:

• Brownian motion describes the random movement of particles suspended in a fluid.
• The Wiener sausage, the volume traced by a Brownian particle over time, is a key metric in characterizing this motion.
• Understanding the statistical properties of the Wiener sausage volume is crucial for various scientific disciplines.

Purpose of the Study:

• To investigate the statistical properties of the Wiener sausage volume using computational methods.
• To determine the probability distribution of the Wiener sausage volume over a range of timescales.
• To refine the understanding of the long-time asymptotic behavior of the Wiener sausage dispersion.

Main Methods:

• Employed Brownian dynamics simulations to model the particle's random walk.
• Analyzed the volume of the generated Wiener sausage for various simulation times.
• Performed statistical analysis on the simulation data to determine probability densities and dispersion.

Main Results:

• The probability density function of the Wiener sausage volume is accurately approximated by a Gaussian distribution.
• This Gaussian approximation holds true not only for asymptotically long times but also over a broad range of timescales.
• A correction term was identified and added to the existing expression for the long-time asymptotic dependence of the dispersion.

Conclusions:

• The Wiener sausage volume exhibits robust Gaussian statistics across a wide temporal spectrum.
• The refined dispersion formula provides a more accurate description of the Wiener sausage's long-time behavior.
• These findings enhance the characterization of Brownian motion and its associated geometric properties.