Statistical Properties Of One-dimensional Nonlinear Lattices:Heat Conduction And Thermalization

The theoretical framework of equilibrium statistical physics has been basically established dating back to the beginning of the twentieth century.However,the nonequilibrium statistical physics,on the other hand,is still far from being established and understood.Its basic contradiction lies in the connection between the reversibility of microscopic dynamics and the irreversibility of macroscopic thermodynamics,thus making it impossible to derive macroscopic thermodynamics from microscopic dynamics.With the development of nonlinear dynamics and chaotic theory in the second half of the twentieth century,a new way of thinking was given to study this old problem.But the existence of nonlinearity still creates difficulties for further development of the related theories.In this dissertation,we developed the self-consistent phonon theory for one-dimensional nonlinear lattices in terms of the Bogoliubov’s inequality.Then,we applied the self-consistent phonon theory to investigate the statistical properties of nonlinear lattices,mainly on thermal conduction and relaxation behavior.Firstly,we investigated the statistical properties of nonlinear lattices.By means of the self-consistent phonon theory,we analytically calculated the temperature of the nonlinear system in the microcanonical ensemble,which has been solved in a numerical manner in previous studies,and verified it by numerical simulations.With this technique,one can solve the thermodynamic quantities in the microcanonical ensemble by means of the canonical ensemble,under the condition that the ensemble equivalence holds,which has been numerically verified in our studies.Secondly,we investigated heat conduction in nonlinear lattices.We proposed a method for numerically solving the phonon relaxation time and the mean free path based on the self-consistent phonon theory and the Lorentzian function fitting.The obtained results are in agreement with previous studies.In comparison with other methods,our method does not require the introduction of external driving forces,which makes the intrinsic properties of the system uncorrupted and can be well applied to nonhomogeneous systems with,e.g.,interfaces and disorder.For heat conduction in interfacial systems,we explained the anomalous phenomenon of negative temperature gradient by innovatively proposing the temperature spectrum decomposition and combining it with phonon mode analysis.Then we analytically calculated the largest Lyapunov exponent,which quantitatively characterizes chaos for nonlinear lattices,based on the self-consistent phonon theory and the geometrization theory of dynamics.We verified it numerically in the exemplified FPUT-like lattices.We found analytically and numerically that the lattice exhibits scaling behavior both in the quasi-integrable and strong nonintegrable regions.And the turning point of the scaling behaviors has been obtained.This part can provide the groundwork for the next study on the relationship between relaxation behavior and chaos.Finally,we investigated the relaxation behavior of nonlinear lattices.Based on the self-consistent phonon theory,we used the effective phonon frequency in order to calculate the relaxation properties in the strong nonintegrable region.Through numerical studies,we found that in addition to scaling law found previously in the quasiintegrable region,the thermalization time also exhibits scaling behavior in the strong nonintegrable region.And both scaling laws are consistent with the inverse of the largest Lyapunov exponent.That is,the thermalization time is proportional to the inverse of the largest Lyapunov exponent.We interpreted this conclusion by introducing the concept of coarse-grained entropy.Our study provides a new way of thinking to study the statistical properties of nonlinear lattices and a new way to solve the fundamental contradictions of nonequilibrium statistical physics.