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On the Gradient Flow Formulation of the Lohe Matrix Model with High-Order Polynomial Couplings

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Abstract

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.

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Acknowledgements

The work of S.-Y. Ha is supported by NRF-2020R1A2C3A01003881.

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Correspondence to Hansol Park.

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Communicated by Alessandro Giuliani.

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Ha, SY., Park, H. On the Gradient Flow Formulation of the Lohe Matrix Model with High-Order Polynomial Couplings. J Stat Phys 184, 19 (2021). https://doi.org/10.1007/s10955-021-02804-3

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