Correlation functions of non-Markovian systems out of equilibrium: analytical expressions beyond single-exponential memory

Abstract This paper is concerned with correlation functions of stochastic systems memory, a prominent example being molecule or colloid moving through complex (e.g. viscoelastic) fluid environment. Analytical investigations such based on non-Markovian equations are notoriously difficult. A common approximation that single-exponential corresponding to the introduction one auxiliary variable coupled Markovian dynamics main variable. As generalization, we here investigate class ‘toy’ models altogether three degrees freedom, giving rise more forms memory. Specifically, consider, mainly an analytical basis, under- and overdamped motion colloidal particle linearly two variables, where coupling between variables can be either reciprocal non-reciprocal. Projecting out obtain Langevin friction kernels colored noise, whose structure similar generalized equation. For present systems, however, may violate fluctuation–dissipation relation as well detailed balance, indicating equilibrium. We then study systematically connection topology underlying system various autocorrelation functions. demonstrate already generate surprisingly non-monotonic oscillatory) memory Finally, show minimal model suitable non-reciprocal yields resembling those describing hydrodynamic backflow in optical trap.