Driven Lorentz gas model in the discrete time domain

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Computational Physics

Background:

• Tracer particle dynamics are fundamental in statistical mechanics.
• Understanding particle behavior in disordered systems is crucial.
• Immobile obstacles significantly alter random walk dynamics.

Purpose of the Study:

• To investigate the discrete-time random walk of a tracer particle on a 2D lattice with obstacles under a constant driving force.
• To calculate displacement moments and analyze deviations from linear response theory.
• To characterize the nature of diffusion (normal vs. superdiffusion) and fluctuation behavior.

Main Methods:

• Analytical calculation of displacement moments to first order in obstacle density.
• Analysis of the approach to terminal velocity for small driving forces.
• Investigation of fluctuation variance and diffusion regimes (normal and superdiffusion).
• Validation through computer simulations.

Main Results:

• The approach to terminal velocity scales as ~N^{-1}exp(-NF^{2}/16) for small forces, deviating from linear response.
• Obstacles enhance fluctuations around the mean displacement.
• Superdiffusion (variance ~N^{3}) is observed for large forces at intermediate steps, transitioning to normal diffusion (~N) at larger steps.
• Superdiffusion begins at N=1 in this discrete-time model.

Conclusions:

• Einstein's linear response theory breaks down for this system.
• The presence of obstacles and a driving force leads to complex diffusive behaviors, including superdiffusion.
• The developed framework accommodates various waiting-time distributions and continuous-time transitions via subordination.