Normal heat conductivity in two-dimensional scalar lattices

Abstract

The paper revisits recent counterintuitive results on the divergence of the heat conduction coefficient in two-dimensional lattices. It was reported that in certain lattices with on-site potential, for which a one-dimensional chain has convergent conductivity, the latter diverges in the 2D counterpart. We demonstrate that this conclusion is an artifact caused by the insufficient size of the simulated system. To overcome computational restrictions, a ribbon of relatively small width is simulated instead of a more traditional square specimen. It is further demonstrated that the heat conduction coefficient in the “long” direction of the ribbon ceases to depend on the width, as the latter achieves only 10 to 20 chains. So, one can consider the dynamics of much longer systems, than in the traditional setting, and still can gain reliable information regarding the 2D lattice. It turns out that for all considered models, for which the conductivity is convergent in the 1D case, it is convergent also in the 2D case. At the same time, however, the length of the system, necessary to reveal the convergence in the 2D case, may be much bigger than in its 1D counterpart.