We consider a system of hierarchical interacting spins under dynamics of spin-flip type with a ferromagnetic mean field interaction, scaling with the hierarchical distance, coupled with a system of linearly interacting hierarchical diffusions of Ornstein–Uhlenbeck type. In particular, the diffusive variables enter in the spin-flip rates, effectively acting as dynamical magnetic fields. In absence of the diffusions, the spin-flip dynamics can be thought of as a modification of the Curie–Weiss model. We study the mean field and the two-level hierarchical model, in the latter case restricting to a subcritical regime, corresponding to high temperatures, obtaining macroscopic limits at different spatio-temporal scales and studying the phase transitions in the system. We also formulate a generalization of our results to the $k$th level hierarchical case, for any $k$ finite, in the subcritical regime. We finally address the supercritical regime, in the zero-temperature limit, for the two-level hierarchical case, proceeding heuristically with the support of numerics.

我们考虑在自旋翻转型动力学下的分级相互作用自旋系统，具有铁磁平均场相互作用，随分级距离缩放，再加上 Ornstein-Uhlenbeck 型线性相互作用分级扩散系统。特别是，扩散变量进入自旋翻转率，有效地充当动态磁场。在没有扩散的情况下，自旋翻转动力学可以被认为是居里-魏斯模型的修改。我们研究平均场和两级层次模型，在后一种情况下，限于亚临界状态，对应于高温，获得不同时空尺度的宏观极限并研究系统中的相变。我们还将我们的结果概括为$克$第 层 层级情况，对于任何 $克$有限，在亚临界状态。我们最终在零温度极限下解决了两级分层情况下的超临界状态，在数值的支持下启发式地进行。