Memory effects in the energy dissipation of anharmonic solids

Dissipation is commonly described by a system-bath coupling. However, in systems with a macroscopic number of degrees of freedom, the bath can be the system itself, i.e., for excitations that only involve a subset of degrees of freedom, the dissipation and decoherence is produced by interactions with the remaining degrees of freedom. We use a simple model of coupled harmonic oscillators to explore the effects of local dissipation that can arise from large isolated systems, without the need to invoke a bath. When the oscillator potentials are purely harmonic, the energy of the system is transmitted by ballistic phonons that can propagate through the crystal lattice without scattering. Impurities and anharmonicities introduce memory effects in the time evolution of the phonons with the consequent loss of coherence. The energy transmission goes from ballistic to diffusive. We use a Mori-Zwanzig projection to encode anharmonic terms to all orders into the memory function. The result is an analytical expression for energy dissipation from a localized initial perturbation that describes the transition from ballistic to diffusive. A thermal conductivity coefficient can be extracted from the analysis.