Understanding of mechanisms for the energy transfer from and/or through nano particles in contact with the large samples have become important in various biological processes, molecular electronics and friction. In this thesis, the phononic heat conductance of an atomic wire between two reservoirs, and the vibrational relaxation of an atom adsorbed on a surface is studied. The former problem is studied using the Keldysh formalism which yields the steady state properties of the system. The dependence of the total conductance on temperature, on the number of atoms in the wire and on the coefficient is studied. It is found that the conductance shows quantal structure similar to the electronic counterpart. The reduced density matrix is used to study the latter problem. The time evolution of the reduced density matrix has been evaluated for an arbitrary system coupled to a heat bath. The formalism is then applied to study the vibrational relaxation of an atom adsorbed on a surface. The frequency dependence of the relaxation time is also determined.