Classical dynamical density functional theory (DDFT) is one of the cornerstones of modern statistical mechanics. It is an extension of the highly successful method of classical density functional theory (DFT) to nonequilibrium systems. Originally developed for the treatment of simple and complex fluids, DDFT is now applied in fields as diverse as hydrodynamics, materials science, chemistry, biology, and plasma physics. In this review, we give a broad overview over classical DDFT. We explain its theoretical foundations and the ways in which it can be derived. The relations between the different forms of deterministic and stochastic DDFT as well as between DDFT and related theories, such as quantum-mechanical time-dependent DFT, mode coupling theory, and phase field crystal models, are clarified. Moreover, we discuss the wide spectrum of extensions of DDFT, which covers methods with additional order parameters (like extended DDFT), exact approaches (like power functional theory), and systems with more complex dynamics (like active matter). Finally, the large variety of applications, ranging from fluid mechanics and polymer physics to solidification, pattern formation, biophysics, and electrochemistry, is presented.

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经典动力密度泛函理论：从基础到应用

经典动态密度泛函理论 (DDFT) 是现代统计力学的基石之一。它是非常成功的经典密度泛函理论 (DFT) 方法对非平衡系统的扩展。DDFT 最初是为处理简单和复杂的流体而开发的，现在已应用于流体动力学、材料科学、化学、生物学和等离子体物理学等多个领域。在这篇评论中，我们对经典的 DDFT 进行了广泛的概述。我们解释了它的理论基础和推导它的方式。阐明了不同形式的确定性和随机 DDFT 之间以及 DDFT 与相关理论（如量子力学时间相关 DFT、模式耦合理论和相场晶体模型）之间的关系。而且，我们讨论了 DDFT 的广泛扩展，其中涵盖了具有附加阶参数的方法（如扩展的 DDFT）、精确方法（如幂泛函理论）和具有更复杂动力学的系统（如活性物质）。最后，介绍了从流体力学和聚合物物理学到凝固、图案形成、生物物理学和电化学的大量应用。