We study the transition from integrability to chaos for the three-particle Fermi–Pasta–Ulam–Tsingou (FPUT) model. We can show that both the quartic β-FPUT model (α=0) and the cubic one (β=0) are integrable by introducing an appropriate Fourier representation to express the nonlinear terms of the Hamiltonian. For generic values of α and β, the model is non-integrable and displays a mixed phase space with both chaotic and regular trajectories. In the classical case, chaos is diagnosed by the investigation of Poincaré sections. In the quantum case, the level spacing statistics in the energy basis belongs to the Gaussian orthogonal ensemble in the chaotic regime, and crosses over to Poissonian behavior in the quasi-integrable low-energy limit. In the chaotic part of the spectrum, two generic observables obey the eigenstate thermalization hypothesis.
We study the transition from integrability to chaos for the three-particle Fermi–Pasta–Ulam–Tsingou (FPUT) model. We can show that both the quartic β-FPUT model (α=0) and the cubic one (β=0) are integrable by introducing an appropriate Fourier representation to express the nonlinear terms of the Hamiltonian. For generic values of α and β, the model is non-integrable and displays a mixed phase space with both chaotic and regular trajectories. In the classical case, chaos is diagnosed by the investigation of Poincaré sections. In the quantum case, the level spacing statistics in the energy basis belongs to the Gaussian orthogonal ensemble in the chaotic regime, and crosses over to Poissonian behavior in the quasi-integrable low-energy limit. In the chaotic part of the spectrum, two generic observables obey the eigenstate thermalization hypothesis.
The Fermi--Pasta--Ulam--Tsingou (FPUT) problem provides a paradigmatic framework for understanding thermalization in weakly nonlinear many-body Hamiltonian systems. This focused review summarizes major developments in near-integrable dynamics, wave resonances, and phonon kinetic theory, emphasizing recent results on thermalization-time scaling in nonlinear lattices. A coherent picture emerges in the thermodynamic limit based on the eigenmode properties of an appropriate integrable reference system. If the reference system has extended eigenmodes and the leading resonant or quasi-resonant processes form a sufficiently connected network, the thermalization time follows Teq∝g−2, where g measures the effective deviation from integrability. This scaling is found broadly in ordered and weakly disordered lattices and is robust to dimensionality, interaction potential, integrability-breaking mechanism, and multimode initial conditions. Identifying the correct integrable reference is essential; in some one-dimensional FPUT-type lattices with cubic interactions, the nearby Toda lattice, rather than the harmonic chain, provides the proper reference. If all reference eigenmodes are localized, spatial-overlap constraints progressively fragment low-order resonance networks as g decreases, and thermalization becomes controlled by higher-order processes. Numerical studies reveal successive regimes Teq∝g−γ with γ=2,4,6, together with weak system-size dependence. Whether this hierarchy persists asymptotically and whether a finite thermalization threshold exists remain open questions. We also discuss finite-size effects, strongly nonintegrable dynamics, heat transport, localization, and higher-dimensional lattices.
We will discuss various aspects of thermalization, chaos and hydrodynamics in one dimensional classical Hamiltonian systems. We study two problems. First, we will revisit the Fermi-Pasta-Ulam-Tsingou (FPUT) problem in order to understand what thermalization is and discuss what leads the system starting from a typical initial condition to a thermal state. Here we discuss the various possibilities like the role of averaging, the choice of observables, the choice of initial conditions and the role of chaos in thermalization. Then, we study the evolution of a blast wave in an alternate mass hard particle (AHP) gas and study the evolution of conserved fields in the system at short times. We find a good agreement with the Taylor-von Neumann-Sedov (TvNS) solution, which was studied during the Second World War in the context of atomic explosions, everywhere except near the core of the blast. We then model this behaviour by using the Navier-Stokes-Fourier (NSF) equations.
In this work, we investigate the semiclassical limit of a simple bosonic quantum many-body system exhibiting both integrable and chaotic behavior. A classical Hamiltonian is derived using coherent states. The transition from regularity to chaos in classical dynamics is visualized through Poincar\'e sections. Classical trajectories in phase space closely resemble the projections of the Husimi functions of eigenstates with similar energy, even in chaotic cases. It is demonstrated that this correlation is more evident when projecting the eigenstates onto the Fock states. The analysis is carried out at a critical energy where the eigenstates are maximally delocalized in the Fock basis. Despite the imperfect delocalization, its influence is present in the classical and quantum properties under investigation. The study systematically establishes quantum-classical correspondence for a bosonic many-body system with more than two wells, even within the chaotic region.
In their seminal work, Fermi, Pasta, Ulam and Tsingou explored the connection between statistical mechanics and dynamical properties, such as chaos and ergodicity. Even today, seventy years later, the topic is not fully understood: while most results of statistical mechanics require the ergodic hypothesis to be rigorously proved, there are many indications that these predictions, both in and out of equilibrium, hold even in the absence of a rigorous form of ergodicity. Motivated by the above considerations, in this work we reconsider the point of view that the relevant ingredients for the validity of statistical mechanics are the large number of degrees of freedom and the choice of extensive observables, while the details of the dynamics do not play an essential role. This is the idea behind Khinchin's famous proof of the typicality of macroscopic observables at equilibrium. We extend this perspective to the context of non equilibrium, by investigating the thermalization properties of both harmonic (integrable) and nonharmonic (chaotic) oscillator chains initially prepared in out-of-equilibrium conditions. In integrable systems, thermalization occurs, or not, depending on the observable. In the chaotic regime, instead, thermalization is reached by any observable, although the relaxation timescale might be larger than the observation time.
We show how to systematically construct weak integrability breaking perturbations (WIBs) for classical integrable models on the lattice. These perturbations, which allow quasi-conserved quantities, have mostly been explored in quantum systems, where they are expected to delay the onset of thermalization and diffusive transport to timescales far exceeding those predicted by Fermi's golden rule. However, accessing such long-time dynamics in quantum models is computationally challenging. Classical integrable lattice models offer a complementary setting for probing transport and long-time dynamics under WIBs. In this work, we specialize our general framework to construct several families of WIBs for the Ishimori model, the Toda chain, and the Harmonic Oscillator Chain (HOC). Such constructions can help quantify how WIBs contribute to anomalous transport and serve as a benchmark for thermalization studies in perturbed integrable models. An important example is the Fermi-Pasta-Ulam-Tsingou (FPUT) model: Starting from the HOC, we show that the cubic nonlinearity (the alpha-FPUT interaction) is a genuine WIB perturbation. Using the integrals of motion (IoMs) of the Toda lattice, we explicitly construct corrections to the entire hierarchy of the HOC IoMs, thereby obtaining an infinite tower of quasi-conserved quantities for the alpha-FPUT chain. We further identify the corresponding adiabatic gauge potential (AGP) as a nontrivial trilocal generator in real space, and show that, more generally, any cubic, translationally invariant, momentum-conserving perturbation of the HOC admits such a generator and is therefore a WIB. Together with our transport and AGP-variance studies, our results provide a unified classical framework for weak integrability breaking and for diagnosing anomalous thermalization and transport in nearly integrable Hamiltonian lattice systems.
We perform a series of numerically accurate tensor-train (TT) simulations of population dynamics and energy transfer in quantum β-Fermi-Pasta-Ulam-Tsingou (FPUT) chains comprising 10 to 30 nonlinear oscillators. The dynamics are propagated over timescales ranging from 18 to 60 fundamental vibrational periods. We demonstrate that the chain anharmonicity opens new energy-transfer channels, accelerates initial ballistic energy propagation, reduces (partial) recurrence periods relative to the harmonic limit, and regularizes oscillatory wavepacket motion. This work serves as the first proof-of-principle demonstration of the high efficiency of TT methods for simulating quantum FPUT dynamics.