弱记忆区微观与纳米尺度系统的动力学：超越马尔可夫近似的数学框架
Dynamics of micro and nanoscale systems in the weak-memory regime: A mathematical framework beyond the Markov approximation
AI总结：
本文针对微观与纳米尺度系统，在弱记忆区（时间尺度相近）证明存在忠实的局域时间近似，通过收敛微扰理论确定时间无关生成元，并给出三个示例验证，超越标准马尔可夫近似。
摘要：
小尺度系统的可见动力学受到不可观测自由度的强烈影响，这些自由度可能属于外部环境或内部子系统，并且几乎不可避免地引发记忆效应。形式上，通过投影算子技术，这些不可达的自由度可以从几乎任何微观模型中系统地消除，从而得到非局域的时间演化方程。本文研究在何种条件下，超越标准马尔可夫近似（通常要求可达与不可达自由度的特征时间尺度被显著分离）能够严格恢复时间上的局域性。具体而言，我们考虑对感兴趣变量为自治且线性的非局域时间演化方程。对于这类模型，我们证明了一个数学定理，确立了良定义的弱记忆区，在该区域即使相关时间尺度具有相近的数量级，也存在忠实的局域近似。这些局域近似的生成元在长时间极限下变得精确，它们是时间无关的，并且可以通过关于记忆强度的收敛微扰理论以任意精度确定，其中马尔可夫生成元在一阶恢复。为说明起见，我们详细推导了三个简单但具有启发性的例子，涵盖粗粒化马尔可夫跳跃网络、半马尔可夫跳跃过程和广义朗之万方程。
英文摘要：
The visible dynamics of small-scale systems are strongly affected by unobservable degrees of freedom, which can belong either to external environments or internal subsystems and almost inevitably induce memory effects. Formally, such inaccessible degrees of freedom can be systematically eliminated from essentially any microscopic model through projection operator techniques, which result in non-local time evolution equations. This article investigates how and under what conditions locality in time can be rigorously restored beyond the standard Markov approximation, which generally requires the characteristic time scales of accessible and inaccessible degrees of freedom to be sharply separated. Specifically, we consider non-local time evolution equations that are autonomous and linear in the variables of interest. For this class of models, we prove a mathematical theorem that establishes a well-defined weak-memory regime, where faithful local approximations exists, even if the relevant time scales are of comparable order of magnitude. The generators of these local approximations, which become exact in the long-time limit, are time independent and can be determined to arbitrary accuracy through a convergent perturbation theory in the memory strength, where the Markov generator is recovered in first order. For illustration, we work out three simple, yet instructive, examples covering coarse grained Markov jump networks, semi-Markov jump processes and generalized Langevin equations.
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