Application of the frozen-modes approximation to classical harmonic oscillator systems

The problems of open classical systems usually correspond to a motion of a test particle that interacts with a large number of bath oscillators. Often, the test particle itself can be considered a harmonic oscillator. For such composite systems, exact numerical solutions are available, but they can become increasingly costly for a large number of bath oscillators. Here we take inspiration from the recent work on open quantum systems and investigate the applicability of the frozen-modes approximation to such classical systems. This approach assumes that some part of the low-frequency bath modes are frozen, thus only their initial values need to be considered. We show that by applying the frozen-modes approximation one can significantly increase the accuracy of the perturbative multiple-scales solution, especially for slow baths. This approach provides a good accuracy even for strong system–bath couplings, a regime that is not accessible to straightforward applications of the perturbation theory. We also suggest a rule for the splitting of spectral density to the fast and slow bath modes. We find that our approach gives excellent results for the ohmic spectral density, but it could be applied for other similar spectral densities as well.