  The paper suggests a resolution for recent controversy over convergence of
heat conductivity in one-dimensional chains with asymmetric nearest-neighbor
potential. We conjecture that the convergence is promoted not by the mere
asymmetry of the potential, but due to ability of the chain to dissociate. In
other terms, the attractive part of the potential function should approach a
finite value as the distance between the neighbors grows. To clarify this
point, we study the simplest model of this sort -- a chain of linearly elastic
disks with finite diameter. If the distance between the disk centers exceeds
their diameter, the disks cease to interact. Formation of gaps between the
disks is the only possible mechanism for scattering of the oscillatory waves.
Heat conduction in this system turns out to be convergent. Moreover, an
asymptotic behavior of the heat conduction coefficient for the case of large
densities and relatively low temperatures obeys simple Arrhenius-type law. In
the limit of low densities, the heat conduction coefficient converges due to
triple disk collisions. Numeric observations in both limits are grounded by
analytic arguments. In a chain with Lennard-Jones nearest-neighbor potential
the heat conductivity also saturates in a thermodynamic limit and the
coefficient also scales according to the Arrhenius law for low temperatures.
This finding points on a universal role played by the possibility of
dissociation, as convergence of the heat conduction coefficient is considered.
