Generalized Langevin equation and fluctuation-dissipation theorem for particle-bath systems in external oscillating fields

The generalized Langevin equation (GLE) can be derived from a particle-bath Hamiltonian, in both classical and quantum dynamics, and provides a route to the (both Markovian and non-Markovian) fluctuation-dissipation theorem (FDT). All previous studies have focused either on particle-bath systems with time-independent external forces only, or on the simplified case where only the tagged particle is subject to the external time-dependent oscillatory field. Here we extend the GLE and the corresponding FDT for the more general case where both the tagged particle and the bath oscillators respond to an external oscillatory field. This is the example of a charged or polarizable particle immersed in a bath of other particles that are also charged or polarizable, under an external ac electric field. For this Hamiltonian, we find that the ensemble average of the stochastic force is not zero, but proportional to the ac field. The associated FDT reads as $\langle F_P\left(t\right)F_P\left(t^{\prime }\right)\rangle =m{k}_{B}T\nu (t-t^{\prime })+{\left(\gamma e\right)}^{2}E\left(t\right)E\left(t^{\prime }\right)$, where $F_p$ is the random force, $\nu (t-t^{\prime })$ is the friction memory function, and $\gamma$ is a numerical prefactor.

Figure 1

Schematic example of system of charged (solid circles) or polarizable molecules. In the former case the particles could be ions in a plasma or ions and electrons in a liquid metal. In the latter case, pairs of negatively and positively charged particles represent the electron cloud and the molecular ion of a polarized neutral molecule as in, e.g., dielectric relaxation of molecular liquids. A particle-bath Hamiltonian like the one of Eqs. (1) and (2) can be applied to these systems where a tagged particle (green) interacts with the local environment via an interaction potential $V\left(Q\right)$, which may represent the interaction potential with neighbors. The tagged particle interacts also with all other degrees of freedom in the system which can be effectively represented as a bath of harmonic oscillators to which the tagged particle is coupled via a set of coupling constants $c_{\alpha }$, where $\alpha$ runs over all other bath oscillators in the system. In traditional models of bath-oscillator dynamics, only the tagged particle is subjected to the external ac field (see, e.g., [5]), whereas the other particles are not. In the proposed model, both the tagged particle and also all the other particles (forming the bath) are responding to the ac electric field, which leads to new physics.