The low-temperature properties of a generalized Frenkel-Kontorova model were investigated analytically within the framework of a phenomenological approach. The model took account of a realistic (anharmonic) interaction of particles that were subjected to a periodic substrate potential. A system of strongly interacting atoms was treated as a system of weakly interacting quasi-particles (kinks). By using the phenomenology of the ideal kink gas, in which the low-temperature ground state of the chain was described as one that consisted of so-called residual kinks supplemented by thermally excited kinks, it was possible to describe the ground state of the system as a hierarchy of sequentially melted kink lattices. The system dynamics were then described in terms of kink dynamics. The equation of motion for a single kink was reduced to one of Langevin type, and was investigated with the help of Kramers theory. This permitted a qualitative analysis to be made of the dependence of the susceptibility, conductivity, and chemical diffusivity of the chain upon the concentration of atoms in the chain. The model predicted a series of effects which were probably related to experimentally observed phenomena in quasi 1-dimensional systems such as super-ionic conductors and anisotropic layers of adsorbed atoms at crystal surfaces.

O.M.Braun, J.S.Kivshar: Physical Review B, 1994, 50[18], 13388-400