  Dynamics of dissipation of a local phonon distribution to the substrate is a
key issue in friction between sliding surfaces as well as in boundary
lubrication. We consider a model system consisting of an excited nano-particle
which is weakly coupled with a substrate. Using three different methods we
solve the dynamics of energy dissipation for different types of coupling
between the nano-particle and the substrate, where different types of
dimensionality and phonon densities of states were also considered for the
substrate. In this paper, we present our analysis of transient properties of
energy dissipation via phonon discharge in the microscopic level towards the
substrate. Our theoretical analysis can be extended to treat realistic
lubricant molecules or asperities, and also substrates with more complex
densities of states. We found that the decay rate of the nano-particle phonons
increases as the square of the interaction constant in the harmonic
approximation.
