Generalized mode-coupling theory (GMCT) is a first-principles-based and systematically correctable framework to predict the complex relaxation dynamics of glass-forming materials. The formal theory amounts to a hierarchy of infinitely many coupled integro-differential equations, which may be approximated using a suitable finite-order closure relation. Although previous studies have suggested that finite-order GMCT leads to well-defined solutions, and that the hierarchy converges as the closure level increases, no rigorous and general result in this direction is known. Here we unambiguously establish the existence and uniqueness of solutions to generic, schematic GMCT hierarchies that are closed at arbitrary order. We consider two types of commonly invoked closure approximations, namely mean-field and exponential closures. We also distinguish explicitly between overdamped and underdamped glassy dynamics, corresponding to hierarchies of first-order and second-order integro-differential equations, respectively. We find that truncated GMCT hierarchies closed under an exponential closure conform to previously developed mathematical theories, such that the existence of a unique solution can be readily inferred. Self-consistent mean-field closures, however, of which the well-known standard-MCT closure approximation is a special case, warrant additional arguments for mathematical rigour. We demonstrate that the existence of a priori bounds on the solution is sufficient to also prove that unique solutions exist for such self-consistent hierarchies. To complete our analysis, we present simple arguments to show that these a priori bounds must exist, motivated by the physical interpretation of the GMCT solutions as density correlation functions. Overall, our work contributes to the theoretical justification of GMCT for studies of the glass transition, placing GMCT on a firmer mathematical footing.

中文翻译：

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广义模式耦合理论的玻璃动力学：层次耦合积分微分方程解的存在性和唯一性

广义模式耦合理论 (GMCT) 是一种基于第一性原理且可系统校正的框架，用于预测玻璃成型材料的复杂弛豫动力学。形式理论相当于无限多个耦合积分微分方程的层次结构，可以使用合适的有限阶闭合关系来近似。尽管之前的研究表明有限阶 GMCT 会导致定义明确的解决方案，并且层次结构随着闭包级别的增加而收敛，但在这个方向上没有严格和一般的结果。在这里，我们明确地建立了以任意顺序封闭的通用、示意性 GMCT 层次结构的解的存在性和唯一性。我们考虑两种常用的闭包近似，即平均场闭包和指数闭包。我们还明确区分过阻尼和欠阻尼玻璃动力学，分别对应于一阶和二阶积分微分方程的层次结构。我们发现在指数闭包下封闭的截断 GMCT 层次结构符合先前发展的数学理论，因此可以很容易地推断出唯一解的存在。然而，自洽平均场闭包，其中众所周知的标准 MCT 闭包近似是一个特例，保证了数学严谨性的额外论证。我们证明了解的先验界限的存在也足以证明对于这种自洽的层次结构存在唯一的解。为了完成我们的分析，我们提出了简单的论据来证明这些先验界限必须存在，受 GMCT 解作为密度相关函数的物理解释的启发。总体而言，我们的工作有助于 GMCT 在玻璃化转变研究中的理论依据，使 GMCT 具有更牢固的数学基础。