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Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards.

Giampaolo Cristadoro1, Thomas Gilbert2, Marco Lenci3

  • 1Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy.

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Summary

Numerical simulations of a Lorentz gas model reveal that the expected superdiffusion behavior is masked by linear growth in accessible time ranges. Analytical comparisons confirm these findings for particle position moments.

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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.