Example of the description of dissipative processes in terms of reversible dynamic equations and some comments on the fluctuation-dissipation theorem

The dynamics of a macroscopic oscillator which is interacting with a heat reservoir, which also consists of oscillators, is analyzed. This problem, which can be solved exactly in its general form in both the classical and quantum-mechanic cases, is used as an example for a study of the transition from a purely dynamic description to a statistical description. Since the system of linear oscillators is not ergodic, an averaging procedure must be regarded as taking an average over the time or over repeated measurements on a unique dynamic trajectory. Depending on the nature of the quadratic form of the potential energy, the oscillations of a macroscopic oscillator can decay in various ways, including exponentially, in the initial stage of the evolution. After a Poincare cycle, the system returns to its initial state, and the damping of the oscillations gives way to a growth. The reversibility of the motion means that the Green's function of the system of oscillators is of odd parity in the time. Equilibrium fluctuations of a macroscopic oscillator are examined. In the classical case the Callen-Welton fluctuation-dissipation theorem can be formulated as follows: The derivative of the coordinate correlation function is proportional to the Green's function of the macroscopic oscillator. In a description in terms of frequencies, the odd parity of the Green's function gives rise to an imaginary part of the Fourier transform of this function in the fluctuation-dissipation theorem. This result is a consequence of the reversibility of the motion in time. The fluctuation-dissipation theorem is proved for Hamiltonian systems without dissipation, but it also applies to systems with dissipation. The exact microscopic Green's function is replaced in this case by the Green's function of a simplified phenomenological description, which explicitly contains dissipative parameters. In the quantum-mechanical case, the results are analogous. The classical and quantum-mechanical versions of the Nyquist relation which follow from the fluctuation-dissipation theorem when the Green's function is approximated by an exponentially damped sinusoidal oscillation are discussed.