Harmonic Chain with Weak Dissipation

We consider finite harmonic chain (consisting of N classical particles) plus dissipative force acting on one particle (called dissipating particle) only. We want to prove that “in the generic case” the energy (per particle) for the whole system tends to zero in the large time limit and then in the large N limit. “In the generic case” means: for almost all initial conditions and for almost any choice of the dissipating particle, in the thermodynamic limit.