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Periodic Lorentz gas with small scatterers.

Péter Bálint1,2, Henk Bruin3, Dalia Terhesiu4

  • 1MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, Műegyetem rkp. 3., Budapest, 1111 Hungary.

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Summary

We establish new limit laws for periodic Lorentz gases, introducing a non-standard Central Limit Theorem and Local Limit Theorem for particle displacement. This bridges a gap between existing superdiffusive and Boltzmann-Grad regimes in statistical physics.

Keywords:
BilliardsLimit theoremsLorentz gasNagaev–Guivarc’h methodSmall scatterers

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Area of Science:

  • Statistical physics
  • Dynamical systems
  • Probability theory

Background:

  • Lorentz gases model particle behavior in random media.
  • Existing research focuses on fixed scatterer sizes or vanishing sizes in separate limits.
  • Understanding particle dynamics requires analyzing behavior across different parameter regimes.

Purpose of the Study:

  • To investigate limit laws for infinite horizon planar periodic Lorentz gases.
  • To analyze the intermediate regime where scatterer size and time tend to infinity simultaneously.
  • To derive non-standard Central Limit and Local Limit Theorems for particle displacement.

Main Methods:

  • Mathematical analysis of infinite horizon planar periodic Lorentz gases.
  • Simultaneous asymptotic analysis as time (n) and scatterer size (ρ) tend to infinity and zero, respectively.
  • Derivation of limit distributions for the displacement function.

Main Results:

  • Established limit laws for the displacement function in the specified intermediate regime.
  • Obtained a non-standard Central Limit Theorem (CLT).
  • Obtained a Local Limit Theorem (LLT).

Conclusions:

  • The study provides the first results for the intermediate regime of Lorentz gases.
  • The findings bridge the gap between well-studied superdiffusive and Boltzmann-Grad regimes.
  • The derived limit laws offer new insights into particle dynamics in periodic disordered systems.