Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards

Area of Science:

• Statistical Physics
• Dynamical Systems
• Computational Physics

Background:

• The Lorentz gas model describes tracer particle motion in a disordered or periodic potential.
• Superdiffusion, characterized by a mean-squared displacement growing faster than linearly with time, is a key phenomenon in such systems.
• A weak form of superdiffusion, with logarithmic corrections to linear growth, is theoretically predicted for infinite Lorentz gas corridors.

Purpose of the Study:

• To numerically investigate the moments of tracer particle position in a 2D periodic Lorentz gas.
• To analyze the emergence of superdiffusion and its asymptotic behavior.
• To compare simulation results with analytical predictions for anomalous diffusion.

Main Methods:

• Numerical simulations of particle trajectories in a 2D periodic billiard model.
• Calculation of position moments, specifically the mean-squared displacement.
• Comparison of numerical data with analytical results for rescaled distributions and variances.

Main Results:

• The expected asymptotic superdiffusion behavior is obscured by dominant linear growth within accessible simulation time scales.
• Subleading linear growth significantly impacts the observed dynamics, masking the logarithmic correction.
• Simulations show good agreement with analytical predictions for the variance of anomalously rescaled limiting normal distributions.

Conclusions:

• The study highlights the challenges in numerically observing predicted asymptotic behaviors in systems exhibiting weak superdiffusion.
• Linear growth can mask subtle anomalous diffusion effects in finite-time simulations.
• Numerical and analytical approaches are complementary for understanding complex transport phenomena.