In previous work, the present author presented calculations in support of the contention that the long-time average rate of energy dissipation that occurs when two sufficiently small nonmetallic solid bodies slide in contact with each other in steady state is nearly zero. This phenomenon is believed to be due to the lack of ergodicity found in model calculations by Fermi, Pasta, and Ulam. In this paper, similar calculations will be presented on a square and a triangular lattice, whose atoms interact with a truncated Lennard-Jones potential, sliding with one edge in contact with both periodic and disordered potentials (due to a second two-dimensional solid in which it is in contact) at both zero and nonzero temperature, which support this idea. These new results are discussed using scaling arguments and in conjunction with results on the interaction of the sample with a solid which supports it, as well as with the atmosphere, in order to make estimates of the experimental conditions under which extremely low dissipation should be observable in real mesoscopic solids. The calculations at nonzero temperature support the contention that this phenomenon could occur at reasonably high fraction of the melting temperature if the solid is sufficiently small.