We present a first-order aggregation model for a homogeneous Lohe matrix ensemble with higher order couplings via a gradient flow approach. For homogeneous free flow with the same Hamiltonian, it is well known that the Lohe matrix model with cubic couplings can be recast as a gradient system with a potential which is a squared Frobenius norm of of averaged state. In this paper, we further derive a generalized Lohe matrix model with higher-order couplings via gradient flow approach for a polynomial potential. For the proposed model, we also provide a sufficient framework in terms of coupling strengths and initial data leading to the emergent dynamics of a homogeneous ensemble.

我们通过梯度流方法提出了具有高阶耦合的齐次 Lohe 矩阵系综的一阶聚合模型。对于具有相同哈密顿量的均匀自由流动，众所周知，具有三次耦合的 Lohe 矩阵模型可以重铸为梯度系统，其势是平均状态的平方 Frobenius 范数。在本文中，我们通过多项式势的梯度流方法进一步推导出具有高阶耦合的广义 Lohe 矩阵模型。对于所提出的模型，我们还在耦合强度和初始数据方面提供了一个足够的框架，导致同构集成的涌现动力学。