Derivation of the generalized Langevin equation in nonstationary environments

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. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3561065]