Study On Thermalization Of One-Dimensional Lattices

The famous ergodic hypothesis proposed by L.E.Boltzmann in the second half of the 19th century is at the foundation of statistical physics.A main result derived from this hypothesis is that the time average over the trajectory of a representative point in phase space and the ensemble average in phase space are equivalent,which,in par-ticular,means that the energy equipartition among different degrees of freedom.In 1955,E.Fermi et al conducted the first numerical experiment to verify this hypothesis by observing the rate of thermalization in microscopic reversible dynamical systems.However,the result is contrary to general expectations:the system far from equilibri-um does not enter the expected thermalized state,but exhibits a phenomenon named the FPUT recurrence later.More than half a century has passed,although it has been clear that the perturbation is large enough to achieve the equipartition of energy,but the original sense of the equipartition theorem,that is,the assumption that infinitesimal nonlinear perturbations can lead to the energy equipartition of macroscopic systems,is still uncertain.In this thesis,the thermalization problem of one-dimensional(1D)non-linear lattices in the thermodynamic limit is studied by means of the wave turbulence(WT)theory and extensive numerical experiments.We propose that the perturbation strength as the nonintegrability by using the nonlinear but integrable model as a tem-plate to measure the capacity of the nonlinear lattice for thermalization instead of a conventional practice to take the nonlinearity strength of the Hamiltonian.Within this framework,by means of the WT theory and numerical experiments,it is shown that,in the thermodynamic limit,the thermalization time of the nonintegrable system in the near-integrable region is inversely proportional to the square of perturbation strength.Specific examples include the nonlinear perturbed Toda model,the generalized FPUT model,the diatomic Toda chain and the diatomic gas model.Therefore,this paper finds and proves the universal law of thermalization for 1D lattices across different models in the thermodynamic limit.This law(power law relationship)shows that the infinites-imal perturbation of 1D nonlinear lattices in the thermodynamic limit can achieve the energy equipartition because there is always a finite time of equipartition under a given perturbation strength.