A mean-field theory of effective normal modes in the Fermi-Pasta-Ulam-Tsingou model
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arXiv
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2025
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| author | Ponno, Antonio Gradenigo, Giacomo Baldovin, Marco Vulpiani, Angelo |
| author_facet | Ponno, Antonio Gradenigo, Giacomo Baldovin, Marco Vulpiani, Angelo |
| contents | We present a non-perturbative, mean-field theory for the Fermi-Pasta-Ulam-Tsingou model with quartic interaction, capturing the quasiperiodic features shown by the system at all energies in the thermodynamic limit. Starting from the true Hamiltonian $H$ of the system with $N$ degrees of freedom, we introduce a mean-field Hamiltonian $\mathcal{H}$ such that the difference $h_N=(H-\mathcal{H})/N$, considered as a random variable with respect to the Gibbs measure, tends to zero as $N\to\infty$, in probabilistic sense. The dynamics of the mean-field Hamiltonian $\mathcal{H}$ consists of $N$ independent oscillation modes with renormalized frequencies $Ω_k = ω_k\sqrt{1+γ(\varepsilon)}$, $ω_k$ being the frequency of the $k$-th normal mode of the linearized system, whereas $γ(\varepsilon)$ is an explicit function of the specific energy $\varepsilon$ of the system. Analytical predictions drawn from the effective Langevin equations ruling the dynamics of such oscillation modes are successfully compared with the numerical data from the original Hamiltonian dynamics. Such a simple decomposition of the true dynamics into $N$ effective normal modes holds at all energy scales, i.e. from the quasi-integrable regime to the strongly chaotic one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A mean-field theory of effective normal modes in the Fermi-Pasta-Ulam-Tsingou model Ponno, Antonio Gradenigo, Giacomo Baldovin, Marco Vulpiani, Angelo Statistical Mechanics We present a non-perturbative, mean-field theory for the Fermi-Pasta-Ulam-Tsingou model with quartic interaction, capturing the quasiperiodic features shown by the system at all energies in the thermodynamic limit. Starting from the true Hamiltonian $H$ of the system with $N$ degrees of freedom, we introduce a mean-field Hamiltonian $\mathcal{H}$ such that the difference $h_N=(H-\mathcal{H})/N$, considered as a random variable with respect to the Gibbs measure, tends to zero as $N\to\infty$, in probabilistic sense. The dynamics of the mean-field Hamiltonian $\mathcal{H}$ consists of $N$ independent oscillation modes with renormalized frequencies $Ω_k = ω_k\sqrt{1+γ(\varepsilon)}$, $ω_k$ being the frequency of the $k$-th normal mode of the linearized system, whereas $γ(\varepsilon)$ is an explicit function of the specific energy $\varepsilon$ of the system. Analytical predictions drawn from the effective Langevin equations ruling the dynamics of such oscillation modes are successfully compared with the numerical data from the original Hamiltonian dynamics. Such a simple decomposition of the true dynamics into $N$ effective normal modes holds at all energy scales, i.e. from the quasi-integrable regime to the strongly chaotic one. |
| title | A mean-field theory of effective normal modes in the Fermi-Pasta-Ulam-Tsingou model |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2512.00929 |