Normal heat conductivity in two-dimensional scalar lattices

Abstract

The paper revisits recent counterintuitive results on divergence of heat conduction coefficient in two-dimensional lattices. It was reported that in certain lattices with on-site potential, for which one-dimensional chain has convergent conductivity, for the 2D case it turns out to diverge. We demonstrate that this conclusion is an artifact caused by insufficient size of the simulated system. To overcome computational restrictions, a ribbon of relatively small width is simulated instead of the 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 interparticle distances. So, one can consider the dynamics of much longer systems, than in the traditional setting, and still can gain a 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 indeed convergent in the 2D case. In 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.

Anomalous energy diffusion in two-dimensional nonlinear lattices

“…In addition, it can be seen from the figure that the dimensional crossover from 1D to 2D with the varying width occurs rapidly such that the diffusion exponent β for N y = 16 almost converges to the value of 1.27. This rapidly convergent tendency is consistent with the direct simulation of thermal conductivity 33 , where the converged results are obtained when the width N y > 32.…”

“…A recent study with scalar displacements 32 reveals a power-law divergence in the 2D FPU-β nonlinear lattices and a logarithmically divergent thermal conductivity for the purely quartic lattices. Besides, normal heat conductivity was observed 33 in the 2D scalar lattices. A possible explanation for these differences in the simulation results maybe lies in the strong finite-size effects 32 within affordable computational resources.…”

“…The term U = 1 4 gr 4 denotes the on-site potential, which breaks the momentum conservation. To compare our results of energy diffusion with previous simulations 29,[31][32][33]41 of heat conduction, we consider three typical types of 2D nonlinear lattices: the FPU-β model with k = 1, β = 1, g = 0; the purely quartic lattice with k = 0, β = 1, g = 0 and the…”

“…So far, no investigation of energy diffusion or of momentum diffusion has been made in 2D nonlinear lattices. Only a few work on heat transport in 2D nonlinear lattices was performed [29][30][31][32][33]41 through the direct nonequilibrium molecular dynamics simulations or the equilibrium Green-Kubo approach. Here we consider energy diffusion in the 2D square lattices made of N x × N y atoms with a vector displacement of q i,j .…”

Multiparticle collision simulations of two-dimensional one-component plasmas: Anomalous transport and dimensional crossovers

“…In order to check the conservation of linear momentum P i within the i−th cell under the SRD rule, let us substitute the definition of particles relative velocities (17) in the r.h.s. of Eq (18).…”