Subdynamics, Fokker-Planck equation, and exponential decay of relaxation processes

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Abstract
The problem of relaxation is studied via the microscopic Hamiltonian model of an impurity (or particle of interest) embedded in a linear chain of harmonic oscillators. When the mass of the particle of interest is sufficiently larger than that of the ‘‘bath’’ particle and the system is classical, the velocity autocorrelation function of the particle of interest is known to consist of the sum of an exponentially decaying term and a nonexponential contribution with a slow tail of oscillatory nature. The damping of the exponential decay is determined by using a renormalization procedure within the context of the generalized Langevin equation. By expressing the ‘‘bath’’ coordinates in terms of normal modes and using a scalar product of the Kubo type, it is shown that the classical Liouvillian becomes formally equivalent to the quantum-mechanical Hamiltonian introduced by Friedrichs to study unstable quantum-mechanical states. In the case of a finite number N of particles (or 2N normal modes) the excited state of the Friedrichs model largely overlaps an ‘‘eigenstate’’ ‖S〉, the ‘‘eigenvalue’’ of which is straightforwardly expressed in terms of the model parameters. It is shown that in the continuum limit (N=∞) this ‘‘eigenvalue’’ becomes complex and its imaginary part coincides with the renormalized damping above. It is also shown that the projection approach to the Fokker-Planck equation leads precisely to the same renormalized damping coefficient.