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. The major conclusion is that the Fokker-Planck description refers to the stochastic dynamics of a sort of dressed variable rather than that of the merely bare velocity. Furthermore, we conclude that whereas the decay of the autocorrelation function of the bare velocity cannot be exponential in accordance with the general remarks by Lee [Phys. Rev. Lett. 51, 1227 (1983)], the decay of the dressed variable is an exact exponential. It is argued that to establish a contact with real experiments, the Kubo approach to the linear response to an external excitation should be reformulated and that this might naturally lead to the basic tenets of the subdynamics of Prigogine and co-workers. The relations between the present and a former approach to the Fokker-Planck equation to be associated to non-Markovian processes are studied.

SUBDYNAMICS, FOKKER-PLANCK EQUATION, AND EXPONENTIAL DECAY OF RELAXATION PROCESSES

VITALI, David;
1989

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. The major conclusion is that the Fokker-Planck description refers to the stochastic dynamics of a sort of dressed variable rather than that of the merely bare velocity. Furthermore, we conclude that whereas the decay of the autocorrelation function of the bare velocity cannot be exponential in accordance with the general remarks by Lee [Phys. Rev. Lett. 51, 1227 (1983)], the decay of the dressed variable is an exact exponential. It is argued that to establish a contact with real experiments, the Kubo approach to the linear response to an external excitation should be reformulated and that this might naturally lead to the basic tenets of the subdynamics of Prigogine and co-workers. The relations between the present and a former approach to the Fokker-Planck equation to be associated to non-Markovian processes are studied.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11581/208670
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