Random-walk approach to the d-dimensional disordered Lorentz gas

Area of Science:

• Physics
• Statistical Mechanics
• Computational Physics

Background:

• Diffusion in disordered media is crucial for understanding transport phenomena.
• Existing models like Boltzmann predictions have limitations at higher densities.
• The Lorentz gas serves as a fundamental model for studying transport in random systems.

Purpose of the Study:

• To derive an analytic expression for the diffusion constant in disordered nonoverlapping Lorentz gas.
• To develop an Enskog-like correction to the Boltzmann prediction for diffusion.
• To validate the theoretical predictions through numerical simulations.

Main Methods:

• Application of a correlated random walk approach to diffusion.
• Utilizing the Lu-Torquato theory for chord-length distributions in random media.
• Derivation of an analytic expression for the diffusion constant in d dimensions.
• Comparison with Boltzmann predictions and renormalized kinetic theory.
• Conducting extensive numerical simulations.

Main Results:

• An analytic expression for the diffusion constant in arbitrary dimensions (d) was obtained.
• The derived expression provides an Enskog-like correction to the Boltzmann prediction.
• The results are exact in the dilute limit.
• The predictions show improved or near-exact accuracy compared to renormalized kinetic theory for d=2,3 across all densities.
• Numerical simulations confirmed the validity of the approximations.

Conclusions:

• The correlated random walk approach, combined with chord-length distribution theory, effectively models diffusion in disordered Lorentz gases.
• The derived Enskog-like correction refines Boltzmann predictions, offering better accuracy, especially at higher densities.
• The study provides a robust theoretical framework and numerical validation for diffusion in complex media.