Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Survey on Lattice Gas Models on 2D Lattices: Critical Behavior of Closed Trajectories

Zoey Zhou

TL;DR

The work surveys how two-dimensional Lorentz lattice gases with quenched scatterers exhibit critical, scale-free statistics for closed trajectories at special scatterer concentrations. It formalizes a scaling framework where loop-length distributions follow a power law at criticality with exponents consistent with 2D percolation hulls, notably $\\tau=15/7$, $d_f=7/4$, and $\\sigma=3/7$, and links these to universal geometric and winding properties. A key finding is that partially occupied systems define new universality classes, with distinct imbalance exponents $\\alpha'\\approx 0.39$, while mirror models and triangular lattices often share percolation-like exponents. The paper also provides a practical minimal simulation protocol using virtual-lattice sampling, and outlines scaling-collapse procedures and future directions, including conformal methods and dynamical extensions, to deepen understanding of transport in disordered 2D media.

Abstract

Lorentz lattice gases (LLGs) are discrete-time transport models in which a point particle moves ballistically between lattice sites and is scattered by randomly placed, quenched local scatterers such as ``rotators'' or ``mirrors.'' Despite the elementary update rules, LLGs exhibit rich dynamical regimes: typically, trajectories close quickly and the distribution of loop lengths has exponential tails, but at special concentrations of scatterers one observes critical behavior with scale-free statistics and fractal geometry. This survey focuses on the critical behavior of closed trajectories in two-dimensional LLGs, starting from the numerical study of Cao and Cohen, and its relation to percolation-hull scaling and kinetic hull-generating walks. We highlight the scaling hypothesis for loop-length distributions, the emergence of critical exponents $τ=15/7$, $d_f=7/4$, and $σ=3/7$ in several universality classes, and the appearance of alternative exponents in partially occupied models.

Survey on Lattice Gas Models on 2D Lattices: Critical Behavior of Closed Trajectories

TL;DR

The work surveys how two-dimensional Lorentz lattice gases with quenched scatterers exhibit critical, scale-free statistics for closed trajectories at special scatterer concentrations. It formalizes a scaling framework where loop-length distributions follow a power law at criticality with exponents consistent with 2D percolation hulls, notably , , and , and links these to universal geometric and winding properties. A key finding is that partially occupied systems define new universality classes, with distinct imbalance exponents , while mirror models and triangular lattices often share percolation-like exponents. The paper also provides a practical minimal simulation protocol using virtual-lattice sampling, and outlines scaling-collapse procedures and future directions, including conformal methods and dynamical extensions, to deepen understanding of transport in disordered 2D media.

Abstract

Lorentz lattice gases (LLGs) are discrete-time transport models in which a point particle moves ballistically between lattice sites and is scattered by randomly placed, quenched local scatterers such as ``rotators'' or ``mirrors.'' Despite the elementary update rules, LLGs exhibit rich dynamical regimes: typically, trajectories close quickly and the distribution of loop lengths has exponential tails, but at special concentrations of scatterers one observes critical behavior with scale-free statistics and fractal geometry. This survey focuses on the critical behavior of closed trajectories in two-dimensional LLGs, starting from the numerical study of Cao and Cohen, and its relation to percolation-hull scaling and kinetic hull-generating walks. We highlight the scaling hypothesis for loop-length distributions, the emergence of critical exponents , , and in several universality classes, and the appearance of alternative exponents in partially occupied models.
Paper Structure (28 sections, 19 equations, 1 figure, 1 table)

This paper contains 28 sections, 19 equations, 1 figure, 1 table.

Table of Contents

  1. Lorentz lattice gases: models, observables, and numerical protocol
  2. General definition
  3. Ensembles and control parameters
  4. Closed trajectories and basic statistics
  5. Geometric observables: size, fractal dimension, winding
  6. Numerical protocol: "virtual lattice" sampling and finite-size control
  7. Critical behavior and scaling of closed trajectories
  8. Scaling hypothesis for loop-length distributions
  9. Fully occupied square-lattice rotator model and percolation-hull universality
  10. What is critical here?
  11. Structural scaling at criticality: imbalance and visit multiplicities
  12. Winding angle statistics and stretched-exponential deviations
  13. Partially occupied square-lattice rotator model: new universality
  14. Mirror models and triangular lattices: critical lines and shared exponents
  15. Summary table of key exponents (critical and near-critical)
  16. ...and 13 more sections

Figures (1)

  • Figure 1: Simulated trajectories from Lorentz mirror model ($p=0.3,0.5,0.7,1.0$) conditioning on the light reaches the boundary of $N$ times $N$ box, rescaled to unit length. One color corresponds to one trajectory.