Generalized Langevin Equation and non-Markovian fluctuation-dissipation theorem for particle-bath systems in external oscillating fields

TL;DR

This paper extends the Generalized Langevin Equation and fluctuation-dissipation theorem to systems where both particles and bath oscillators respond to external oscillating fields, applicable to charged particles in AC electric fields.

Contribution

It generalizes the GLE and FDT to include external fields acting on both particles and bath oscillators, a case not previously addressed.

Findings

Derived a modified FDT incorporating external AC fields.

Found the ensemble average of stochastic force is proportional to the external field.

Presented a generalized GLE for particle-bath systems under oscillating external fields.

Abstract

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 polarisable 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 $\langle F_P(t)F_P(t')\rangle=mk_BT\nu(t-t')+(\gamma e)^2E(t)E(t')$, where $F_{p}$ is the random force, $\nu(t-t')$ is the friction memory function, and $\gamma$ is a numerical prefactor.