\((m,k)\) -Ary树上(1,1/2)-混合Ising模型动力学系统的Gibbs测度及稳定性分析

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED

Mathematical Physics, Analysis and Geometry Pub Date : 2025-05-04 DOI:10.1007/s11040-025-09504-4

Aminah Qawasmeh, Farrukh Mukhamedov, Hasan Akın

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引用次数: 0

摘要

本文介绍了一种新的（1,1/2）混合自旋Ising模型（简称(1,1/2)-MSIM），该模型在(m, k)-ary树上具有\(J_1\)和\(J_2\)竞争相互作用。通过构造分裂Gibbs测度，我们建立了多个Gibbs测度的存在性，这意味着(1,1 /2)-MSIM在(m, k)-树上发生相变。进一步，证明了两个平移不变吉布斯测度在（1,k）元树中的极值性。此外，还发现了无序相的极值条件，并检验了其非极值状态。众所周知，为了研究树状结构上的晶格模型，对代表模型的动力系统进行稳定性分析，并检查这些动力系统不动点的行为，可以为模型提供重要的见解。我们围绕不动点进行了稳定性分析，以研究MSIM在（m, k）元树（也称为k元树）上的行为。

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Analysis of Gibbs Measures and Stability of Dynamical System Linked to (1,1/2)-Mixed Ising Model on \((m,k)\)-Ary Trees

This paper introduces a new (1,1/2) mixed spin Ising model (shortly, (1,1/2)-MSIM) having \(J_1\) and \(J_2\) competing interactions on (m, k)-ary trees. By constructing splitting Gibbs measures, we establish the presence of multiple Gibbs measures, which implies the occurrence of the phase transition for the (1, 1/2)-MSIM on the (m, k)-ary trees. Furthermore, the extremality of the two translation-invariant Gibbs measures is demonstrated for (1, k)-ary trees. Moreover, the extremality condition for the disordered phases is found and its non-extremality regimes are examined as well. It is well known that, to investigate lattice models on tree-like structures, conducting a stability analysis of the dynamical systems that represent the model and examining the behavior of the fixed points of these dynamical systems can provide significant insights into the model. We conducted a stability analysis around fixed points to investigate the behavior of the MSIM on the (m, k)-ary trees, also referred to as k-ary trees.

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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.