Thermalization Of Classical Lattice Systems

Whether a generic many-body system can reach energy equipartition and relax to a thermal state is a frontier topic of theoretical physics research.In the thermal state,the values of macroscopic quantities are stationary,universal concerning widely different initial conditions,and predictable using statistical mechanics.According to the theory of thermodynamic limit,a many-body system of statistical interest is expected to reach thermalization in principle.At this point,the core problem in practical terms is determining the time scale needed to reach thermalization or equipartition.The thermalization of the one-dimensional lattice system with uniform mass has been extensively studied,and a certain consensus has been reached.It is widely accepted that the multiwave resonances allow for energy redistribution between different modes,resulting in thermalization.In the weak interaction and thermodynamic limit,a research group at Xiamen University has proved on the basis of previous research that the thermalization time of a one-dimensional lattice is inversely proportional to the squared interaction strength,whose definition depends on choosing an appropriate integrable Hamiltonian as the reference point.This dissertation further studied whether this conclusion applies to lattice systems with random masses,high-dimension or strong perturbations.This dissertation studied ordered and disordered lattice models of arbitrary spatial dimensions described by a near-integrable Hamiltonian.Firstly,based on the waveturbulence theory,we derived Zakharov equation describing the irreversible evolution of the system towards thermodynamic equilibrium.Then we further proved that when the system size is large enough,through multi-wave resonance,all normal modes can be grouped into a network through which the system reaches energy equipartition and finally tends to thermalization.Under weak perturbations,we obtained the scaling law that the thermalization time is inversely proportional to the square of the nonlinear perturbation strength.The relevant theoretical predictions have been verified numerically in typical lattice models.On the one hand,the progress in this part proved that the classical lattice system obeys the universal law for thermalization time under weak interactions;on the other hand,it shows that Anderson localization is unstable against multi-wave resonances in the thermodynamic limit.For systems with strongly nonlinear perturbations,we found that they can be statistically equivalent to a weakly interacting system.Taking a generic type of onedimensional nonlinear lattice system as an example,we showed that the trivial resonances enhance the linear dispersion and meanwhile weaken the nonlinear interactions equivalently.A weakly interacting renormalized-wave system will result if one subsumes these trivial resonances into the integrable part.Based on the equivalent systems,we obtained a modified Zakharov equation.This equation predicts that thermalization time is still inversely proportional to the square of interaction strength,however,which is renormalized now.In this sense,the universal law for thermalization time holds as before for such systems under strong nonlinearities or high temperatures.Numerical simulations also verify the theoretical predictions in this part.