Systems under colored noise: Treatment by renormalized operator cumulant expansions

The averaged properties of systems undergoing external influences simulated by colored noise may be obtained from the operator cumulant expansion (OCE) method of R. Kubo and R.F. Fox in a systematic way. However, the increasing complexity of the OCE in higher orders hinders partial summations necessary in many colored noise problems to be done conveniently. The present paper derives first a folded diagram representation of the OCE which is rearranged to obtain a highly compact closed expression for the relevant quantities of the average time evolution, e.g., the damping operator. It is formulated in terms of a kind of dissipative Heisenberg dynamics and in terms of effective interactions related to the response of the system to certain external nonrandom fields. The effective interaction is given explicitly for the low noise case with short correlation time. Nonlinear field dependencies and excitations of higher harmonics are found as specific colored noise effects. The present formulation is claimed to be much more effective than the original one both because of its physical transparency and the ready access to approximations corresponding to infinite partial sums of the OCE.