Transition from ballistic to diffusive heat transfer in a chain with breaks

The transition from a ballistic to a diffusive regime of heat transfer is studied using two models. The first model is a one-dimensional chain with bonds, capable of dissociation. Interparticle forces in the chain are harmonic for bond deformations below a critical value, corresponding to the dissociation, and zero above this value. A kinetic description of heat transfer in the chain is proposed using the second model, namely, a gas of noninteracting quasiparticles, reflecting from randomly occurring barriers. The motion of quasiparticles mimicks heat (energy) transfer in the chain, while the barriers mimic dissociated bonds. For the gas, a kinetic equation is derived and solved analytically. The solution demonstrates the transition from the ballistic regime at small times to the diffusive regime at large times. In the diffusive limit, the distance traveled by a heat obeys square-root asymptotics as in the case of classical diffusion. However, the shape of the fundamental solution for temperature differs from the Gaussian function and therefore the Fourier law is not satisfied. Two examples are considered to demonstrate that the presented kinetic model is in good qualitative agreement with the results of the numerical solution of the chain dynamics. The presented results show that bond dissociation is an important mechanism underlying the transition from ballistic to diffusive heat transfer in one-dimensional chains.