Four-body interactions in the long-range Hamiltonian mean-field model

doi:10.1088/1674-1056/addccf

中国物理B ›› 2025, Vol. 34 ›› Issue (11): 110501-110501.doi: 10.1088/1674-1056/addccf

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

收稿日期:2025-03-10 修回日期:2025-05-19 接受日期:2025-05-26 发布日期:2025-11-10
通讯作者: Yu Xue E-mail:yuxuegxu@gxu.edu.cn
基金资助:
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).

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

Received:2025-03-10 Revised:2025-05-19 Accepted:2025-05-26 Published:2025-11-10
Contact: Yu Xue E-mail:yuxuegxu@gxu.edu.cn
About author:2025-110501-250391.pdf
Supported by:
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).

摘要/Abstract

摘要： 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.

关键词: long-range interactions, equilibrium statistical mechanics, Hamiltonian meanfield, Lyapunov exponents

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.

Key words: long-range interactions, equilibrium statistical mechanics, Hamiltonian meanfield, Lyapunov exponents

中图分类号: (Classical statistical mechanics)

05.20.-y

05.70.Fh (Phase transitions: general studies) 05.45.-a (Nonlinear dynamics and chaos)

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[J]. 中国物理B, 2025, 34(11): 110501-110501.

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[J]. Chin. Phys. B, 2025, 34(11): 110501-110501.

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Four-body interactions in the long-range Hamiltonian mean-field model

Four-body interactions in the long-range Hamiltonian mean-field model

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