Heat conduction in two-dimensional disk models

We study the heat conduction problem in two-dimensional (2D) lattice models of disk shape consisting of two circular heat baths with radius r₁ and r₂ (r₁<r₂), 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 β type and the φ⁴ 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 κ depends on the system size L=r₂-r₁ as κ~(ln L)^α with α being a function of r₁/r₂. In particular, in the limit of r₁/r₂→1 we have α→1, i.e., a logarithmic dependence of κ on L, which is in agreement with the prediction of existing theories. In the momentum nonconserving φ⁴ system the heat conductivity converges to a finite value as the system size is increased.