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Four-body interactions in the long-range Hamiltonian mean-field model |
| Qiang Zhang(张强), Haojie Luo(罗浩杰), Bingling Cen(岑炳玲), and Yu Xue(薛郁)† |
| Institute of Physical Science and Technology, Guangxi University, Nanning 530004, China |
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Abstract A Hamiltonian mean-field model with long-range four-body interactions is proposed. The model describes a long-range mean-field system in which $N$ unit-mass particles move on a unit circle. Each particle $\theta_{i}$ interacts with any three other particles through an infinite-range cosine potential with an attractive interaction ($\varepsilon > 0$). By applying a method that remaps the average phase of global particle pairs onto a new unit circle, and using the saddle-point technique, the partition function is solved analytically after introducing four-body interactions, yielding expressions for the free energy $f$ and the energy per particle $U$. These results were further validated through numerical simulations. The results show that the system undergoes a second-order phase transition at the critical energy $U_{\rm c}$. Specifically, the critical energy corresponds to $U_{\rm c}=0.32$ when the coupling constant $\varepsilon =5$, and $U_{\rm c}=0.63$ when $\varepsilon =10$. Finally, we calculated the system's largest Lyapunov exponent $\lambda $ and kinetic energy fluctuations $\varSigma $ through numerical simulations. It is found that the peak of the largest Lyapunov exponent $\lambda $ occurs slightly below the critical energy $U_{\rm c}$, which is consistent with the point of maximum kinetic energy fluctuations $\varSigma $. And there is a scaling law of $\varSigma /N^{1/2}\propto \lambda $ between them.
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Received: 10 March 2025
Revised: 19 May 2025
Accepted manuscript online: 26 May 2025
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PACS:
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05.20.-y
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(Classical statistical mechanics)
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05.70.Fh
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(Phase transitions: general studies)
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05.45.-a
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(Nonlinear dynamics and chaos)
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| Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11962002) and the Innovation Project of the Guangxi Graduate Education (Grant Nos. YCBZ2021021 and YCSW2022070). |
Corresponding Authors:
Yu Xue
E-mail: yuxuegxu@gxu.edu.cn
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| About author: 2025-110501-250391.pdf |
Cite this article:
Qiang Zhang(张强), Haojie Luo(罗浩杰), Bingling Cen(岑炳玲), and Yu Xue(薛郁) Four-body interactions in the long-range Hamiltonian mean-field model 2025 Chin. Phys. B 34 110501
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