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.

本文涉及具有记忆的随机系统的相关函数，其中一个突出的例子是分子或胶体在复杂（例如粘弹性）流体环境中移动。众所周知，基于非马尔可夫随机方程对此类系统进行分析研究非常困难。一个常见的近似值是单指数存储器的近似值，对应于一个辅助变量的引入，该辅助变量与主变量的马尔可夫动力学耦合。作为概括，我们在这里研究一类具有三个自由度的“玩具”模型，从而产生更复杂的记忆形式。具体而言，我们主要在分析的基础上考虑线性耦合到两个辅助变量的胶体颗粒的欠阻尼和过阻尼运动，其中变量之间的耦合可以是倒数或非倒数。投影出辅助变量，我们得到具有摩擦核和有色噪声的非马尔可夫Langevin方程，其结构与广义Langevin方程的结构相似。然而，对于目前的系统，非马尔可夫方程可能会违反波动-耗散关系以及详细的平衡，这表明系统是不平衡的。然后，我们系统地研究底层马尔可夫系统的耦合拓扑与各种自相关函数之间的联系。我们证明，已经有两个辅助变量可以生成令人惊讶的复杂（例如，非单调或振荡的）记忆和相关函数。最后，