Energy transport and diffusion in low-dimensional lattices

Scientific studies in many disciplines have entered the low-dimensional mesoscopic world. This creates a valuable chance for not only improving near-equilibrium statistical mechanics, but also investigating the foundations of statistical mechanics, establishing thermodynamics and statistical mechanics for small systems, and developing the theory of nonequilibrium systems far from the equilibrium state. Predicting transport coefficients, as a main task of the near-equilibrium statistical mechanics, was formally fulfilled in the 1960-1970s, Green-Kubo formula which bridges the states of equilibrium and nonequilibrium. The Green-Kubo formula has played a crucial role in understanding nonequilibrium transport processes though so far only qualitatively, as the exact values of a systems transport coefficients cannot be worked out explicitly given the systems Hamiltonian. A quantitative description of transport behavior, therefore, is still required. Specifically, the “last mile” problem remains unsolved, and is still the bottleneck of near-equilibrium theory. Encouragingly, though, achievements in recent years may lead to a breakthrough in the near future. On one hand, the rapid development of computing technology has made it possible to effectively investigate the dynamics of nonlinear lattices, providing an important method of studying transports; on the other hand, various low-dimensional materials have now been synthesized in laboratories. With the help of advanced measurement and testing techniques, it has also become possible to perform experimental studies of transport in low-dimensional systems; theoretical results can be checked directly, which could in turn stimulate new ideas for improving theories. In addition, some theoretical approaches, such as nonlinear fluctuation hydrodynamics and wave turbulence theories, have been successfully applied to transport and diffusion studies—there are more effective theoretical tools available now than ever before. We therefore anticipate a first-principle formula that gives transport coefficients explicitly by solving the Green-Kubo formula based on microscopic dynamics. In the last four decades, thermal transport and diffusion in one-dimensional lattices has been studied intensively and extensively by researchers all over the world, and many insights have been obtained. In this work, we review the advances made by the group in Xiamen University—the most important of which being the discovered relation between heat conduction behavior and symmetry of internal interaction potential. We find that the latter both affects the exponent of the power-law decaying tail of the thermal current correlation function (TCCF) that characterizes hydrodynamic transport and dominates rapid decay behavior due to the kinetic process. When the potential is asymmetric, at certain temperatures and nonlinear interaction strength regions, the TCCF may decay by several orders due to the kinetic process exclusively, resulting in system size-independent heat conductivity such that the Fourier heat conduction law is obeyed formally. In contrast, when the potential is symmetric, though the TCCF still undergoes rapid decay in the kinetic process stage, the rate of decay rate can be much lower than exponential, leading to a system size-dependent heat conductivity. If, besides internal interactions, the system is also subject to any on-site potentials, its heat conductivity will converge to a finite constant, as the total momentum is not conserved in this case, and the TCCF in general only decays exponentially. These findings demonstrate that the kinetic process plays a crucial role not only in a fluid, but also in a lattice, which must be taken into account for calculating transport coefficients; in other words, a complete transport theory must be established on the basis of both kinetics and hydrodynamics. In addition, we review the general laws followed by the energy equipartition process in lattices and their deep connections with the transport behavior, in addition to several novel effects observed in coupled transport.