Integrable Cellular Automata on Finite Fields of Order \(2^n\)

IF 1.2 3区数学 Q3 MATHEMATICS, APPLIED

Mathematical Physics, Analysis and Geometry Pub Date : 2026-06-22 DOI:10.1007/s11040-026-09569-9

Aoi Araoka, Tetsuji Tokihiro

引用次数: 0

Abstract

This paper explores cellular automata (CA) constructed from Yang–Baxter maps over finite fields \(\mathbb {F}_{2^n}\). We define R-matrices using a map f on \(\mathbb {F}_{2^n}\) and establish necessary and sufficient conditions for f to satisfy the Yang–Baxter equation. We show that these conditions become remarkably streamlined in characteristic two. An exhaustive search for bijective solutions in fields of order 4, 8, and 16 yields 16, 736, and 269,056 maps, respectively. Analysis of the resulting CA under helical boundary conditions reveals a consistent alignment between the temporal period and the field order. We propose the conjecture that this periodic identity holds generally for \(\mathbb {F}_{2^n}\), supported by analytical proofs for \(n=2\) and \(n=3\). Our results further indicate that bijectivity is a fundamental requirement for this periodic behavior.

Abstract Image

查看原文本刊更多论文

有限阶域上的可积元胞自动机 \(2^n\)

本文探讨了有限域上由Yang-Baxter映射构造的元胞自动机（CA） \(\mathbb {F}_{2^n}\)。利用\(\mathbb {F}_{2^n}\)上的映射f定义了r -矩阵，并建立了f满足Yang-Baxter方程的充分必要条件。我们表明，在特征二中，这些条件变得非常流线型。对4阶、8阶和16阶域的双目标解进行穷举搜索，分别得到16、736和269,056个映射。对螺旋边界条件下产生的CA的分析揭示了时间周期和场序之间的一致对齐。我们提出了这个周期恒等式对于\(\mathbb {F}_{2^n}\)一般成立的猜想，并给出了\(n=2\)和\(n=3\)的解析证明。我们的结果进一步表明，双客观性是这种周期性行为的基本要求。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.