Survey on Lattice Gas Models on 2D Lattices: Critical Behavior of Closed Trajectories
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909038180040704 |
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| author | Zhou, Tianyi |
| author_facet | Zhou, Tianyi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23129 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Survey on Lattice Gas Models on 2D Lattices: Critical Behavior of Closed Trajectories Zhou, Tianyi Statistical Mechanics 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. |
| title | Survey on Lattice Gas Models on 2D Lattices: Critical Behavior of Closed Trajectories |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2512.23129 |