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