Correlation functions and their universal connection during an extremely slow equilibration process

Heat transport in an angular-momentum-conserving lattice

It is expected that the energy-diffusion propagator in a one-dimensional nonlinear lattice with three conserved quantities: energy, momentum, and stretch, consists of a central heat mode and two sound modes. The heat mode follows a Lévy distribution. Consequently, the heat diffusion is super, i.e., the second moment of the diffusion propagator diverges as t^{β} with β>1; and the heat conduction is anomalous, i.e., the heat conductivity is size dependent and diverges with size N by N^{α}, with α>0. In this paper, we study a one-dimensional lattice with two-dimensional transverse motions, in which the total angular momentum also conserves. More importantly, the diffusion of this conserved quantity is ballistic. Surprisingly, the above pictures and the values of the mentioned power exponents keep unchanged. The universality of the scalings is then further extended. On the other hand, the detailed strengths of heat transports are largely enhanced. Such a counterintuitive finding can be explained by the change of the phonon mean-free path of the lattices.

Thermalization process in a one-dimensional lattice with two-dimensional motions: The role of angular momentum conservation

We study the thermalization process in a one-dimensional lattice with two-dimensional motions. The phonon modes in such a lattice consist of two branches. Unlike in general nonlinear Hamiltonian systems, for which the only conserved quantity is the total energy, the total angular momentum J is also conserved in this system. Consequently, the intra- and interbranch energy transports behave significantly differently. For the intrabranch transport, all the existing rules for the one-dimensional systems including the Chirikov overlap criterion apply. As for the interbranch transport, some trivial processes in one-dimensional lattices become nontrivial. During these processes, all the conservation laws can be satisfied exactly; thus the Chirikov criterion does not apply. These processes provide some fast channels for the interbranch transport, although the thermalization cannot be reached through them alone. A system with nonzero-J initial state, however, can never be thermalized to an equipartition state having zero J. Quite counterintuitively, the corresponding asymptotic mode energy distribution greatly concentrates to a few lowest-frequency modes in one branch.