Nonintegrability and thermalization of one-dimensional diatomic lattices

Abstract

Nonintegrability is a necessary condition for the thermalization of a generic Hamiltonian system. In practice, the integrability can be broken in various ways. As illustrating examples, we numerically studied the thermalization behaviors of two types of one-dimensional (1D) diatomic chains in the thermodynamic limit. One chain was the diatomic Toda chain whose nonintegrability was introduced by unequal masses. The other chain was the diatomic Fermi-Pasta-Ulam-Tsingou-β chain whose nonintegrability was introduced by quartic nonlinear interaction. We found that these two different methods of destroying the integrability led to qualitatively different routes to thermalization, but the thermalization time, Teq, followed the same law; Teq was inversely proportional to the square of the perturbation strength. This law also agreed with the existing results of 1D monatomic lattices. All these results imply that there is a universal law of thermalization that is independent of the method of breaking integrability.

Three-wave resonant interactions in the diatomic chain with cubic anharmonic potential: theory and simulations

“…Some years later, it has been given the evidence that, in the large box limit, the mechanism that leads to the thermalization in chains like the α-and β-FPUT is the four-wave resonant interaction process [4,5], see also [6]. Numerical confirmation of these predictions can be also found in [7,8].…”

“…a diatomic chain with cubic potential (quadratic nonlinearities in the equation of motion) and we study the properties of thermalization within the wave turbulence framework [9,10]. Numerical simulations of a diatomic β-FPUT chain and of the diatomic Toda lattice were considered in [11] and it was shown that the thermalization time followed the same scaling as the one for monoatomic chains α-and β-FPUT [5,12] and the nonlinear Klein-Gordon equation [13]. From a mathematical point of view, we point out a rigorous result in [14] for a diatomic chain where it was proved that, in the limit of small temperature and large ratio between the masses, the exchange of energy between the modes of the optical branch and those of the acoustic one is practically null for the majority of initial conditions up to some time estimated in [14] (see also [15,16]).…”

Too Close to Integrable: Crossover from Normal to Anomalous Heat Diffusion

“…At lower energies, one should estimate the soliton scattering rates due to mass inhomogeneities; however, there is no reason to expect a different scaling behavior. Indeed, the typical thermalization time is of order δ −2 in a wide energy range [64]. The conductivities taken from Ref.…”

The anti-Fermi-Pasta-Ulam-Tsingou problem in one-dimensional diatomic lattices

“…The form and the strength of H ′ completely depend on the choice of H 0 . And the existing research results show that the choice of H 0 is very important to characterize the thermalization behavior of a system [22,24,31].…”

“…Recent studies have shown that these scaling laws can be explained in the framework of wave turbulence theory [18][19][20]. Subsequently, it is shown that, in the thermodynamic limit, the thermalization behavior of a near-integrable system is universal as long as the ability of the system to be thermalized is properly measured, that is, the T eq is inversely proportional to the square of the perturbation strength [21][22][23][24][25][26]. The key to accurately describing the thermalization capability of a system is to define the perturbation strength by selecting a suitable reference integrable system [22].…”