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

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 θ_i interacts with any three other particles through an infinite-range cosine potential with an attractive interaction (ε > 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_c. Specifically, the critical energy corresponds to U_c = 0.32 when the coupling constant ε = 5, and U_c = 0.63 when ε = 10. Finally, we calculated the system’s largest Lyapunov exponent λ and kinetic energy fluctuations Σ through numerical simulations. It is found that the peak of the largest Lyapunov exponent λ occurs slightly below the critical energy U_c, which is consistent with the point of maximum kinetic energy fluctuations Σ. And there is a scaling law of Σ/N^1/2 ∝ λ between them.