Derivation of the generalized Langevin equation in nonstationary environments

Abstract

The generalized Langevin equation (GLE) is extended to the case of nonstationary bath. The derivation starts with the Hamiltonian equation of motion of the total system including the bath, without any assumption on the form of Hamiltonian or the distribution of the initial condition. Then the projection operator formulation is utilized to obtain a low-dimensional description of the system dynamics surrounded by the nonstationary bath modes. In contrast to the ordinary GLE, the mean force becomes a time-dependent function of the position and the velocity of the system. The friction kernel is found to depend on both the past and the current times, in contrast to the stationary case where it only depends on their difference. The fluctuation-dissipation theorem, which relates the statistical property of the random force to the friction kernel, is also derived for general nonstationary cases. The resulting equation of motion is as simple as the ordinary GLE, and is expected to give a powerful framework to analyze the dynamics of the system surrounded by a nonstationary bath.

Vibrational spectroscopy via the Caldeira-Leggett model with anharmonic system potentials

“…Equation (15) shows that the error in the spectral density can cancel or accumulate depending on the phase difference ∆φ(ω). Let us assume that the error magnitudes, r pp (ω) and r pF (ω) are similar.…”

“…Another approach to reduced EOMs is to employ formal projection operator techniques to project out the bath from the system's EOM resulting in linear or nonlinear GLE forms [5,[13][14][15]. In the former, the resulting system potential is effectively harmonic, whereas in the latter the system potential is formed by a (non-linear) meanfield potential.…”

Applicability of the Caldeira–Leggett Model to Vibrational Spectroscopy in Solution

“…In this formalism the memory kernel carries a functional dependence on the system's positions and momenta and the fluctuating force has more complicated statistical properties than given by the FDT in Eq. (4) [26]. This functional dependence is hardly tractable and an apparent simplification is to linearize the kernel with respect to positions and momenta.…”

“…The third route employs a non-linear projection. Here, the system potential is given by the potential of the mean force, V mean (x), felt by the system at coordinate x, averaged over all bath coordinates Q [8,26,27]. In this formalism the memory kernel carries a functional dependence on the system's positions and momenta and the fluctuating force has more complicated statistical properties than given by the FDT in Eq.…”

“…In this case the contribution proportional to x vanishes and the resulting integral coincides with that in Eq. (2) [26].…”