Equilibration of quasi-integrable systems

Abstract

We study the slow relaxation of isolated quasi-integrable systems, focusing on the classical problem of Fermi-Pasta-Ulam-Tsingou (FPU) chain. It is well-known that the initial energy sharing between different linear-modes can be inferred by the integrable Toda chain. Using numerical simulations, we show explicitly how the relaxation of the FPU chain toward equilibration is determined by a slow drift within the space of Toda's integrals of motion. We analyze the whole spectrum of Toda-modes and show how they dictate, via a Generalized Gibbs Ensemble (GGE), the quasi-static states along the FPU evolution. This picture is employed to devise a fast numerical integration, which can be generalized to other quasi-integrable models. In addition, the GGE description leads to a fluctuation theorem, describing the large deviations as the system flows in the entropy landscape. *

Universal law of thermalization for one-dimensional perturbed Toda lattices

“…It can be seen that thermalization is faster approached in the former case as a consequence of the tiny difference of ò 4 , suggesting that the thermalization rate depends on the perturbation strength sensitively. It can also be seen that the energy of low-frequency modes remains for a long time and the energy of high-frequency modes increases very slowly, known as a signature of the metastable state [35,36,[58][59][60]. Nevertheless, the system will be thermalized eventually.…”

Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

“…Due to that generally we should expect that, in contrast to the above estimates and numerical data the dynamics of the α-FPU problem will be integrable'. In recent years, there has been increasing attention to Toda as a reference integrable model close to the FPU [5,6,14,25,37]. Already in 1982 the pioneering work of Ferguson et al [19] put forward the idea that the higher polynomial order contact between FPU and Toda allows to associate the FPU's integrable-like behavior (e.g.…”

Fluctuations of work in realistic equilibrium states of quantum systems with conserved quantities

“…(10)- (11). This points to the importance of the role of charges in realistic non-equilibrium processes, such as equilibration in quasi-integrable systems [24], and dissipation and relaxation in driven systems with conservation laws [42,43]. A case of particular theoretical interest for future exploration arises when the charges supported by the Hamiltonian do not commute with each other [25,[44][45][46][47].…”

“…In particular, our formalism is explicitly able to deal with processes where the number of charges of the initial and final Hamiltonians differ (cf. [23]), and thus enables one to address fundamental open questions on the thermalization of integrable systems when perturbed away from integrability [1,5,7,24,25].…”