We study the heat conduction problem in two-dimensional (2D) lattice models of disk shape consisting of two circular heat baths with radius $r_1$ and $r_2$ $\left(r_1<r_2\right)$, located concentrically at the center and the edge of the disk. Compared with the lattice models of rectangle shape adopted in previous studies, the main advantage of the disk models is that they have an unambiguous 2D dimensionality. The Fermi-Pasta-Ulam interaction of $\beta$ type and the ${\varphi }^4$ system are considered, respectively, as momentum conserving and nonconserving prototypes. In the former we find that in the range of the system size investigated, the heat conductivity $\kappa$ depends on the system size $L=r_2-r_1$ as $\kappa \sim {\left(\text{ln}L\right)}^{\alpha }$ with $\alpha$ being a function of $r_1/r_2$. In particular, in the limit of $r_1/r_2\to 1$ we have $\alpha \to 1$, i.e., a logarithmic dependence of $\kappa$ on $L$, which is in agreement with the prediction of existing theories. In the momentum nonconserving ${\varphi }^4$ system the heat conductivity converges to a finite value as the system size is increased.

• Received 7 June 2010

DOI:https://doi.org/10.1103/PhysRevE.82.030101