The renormalization group plays an essential role in many areas of physics, both conceptually and as a practical tool to determine the long-distance low-energy properties of many systems on the one hand and on the other hand search for viable ultraviolet completions in fundamental physics. It provides us with a natural framework to study theoretical models where degrees of freedom are correlated over long distances and that may exhibit very distinct behavior on different energy scales. The nonperturbative functional renormalization-group (FRG) approach is a modern implementation of Wilson's RG, which allows one to set up nonperturbative approximation schemes that go beyond the standard perturbative RG approaches. The FRG is based on an exact functional flow equation of a coarse-grained effective action (or Gibbs free energy in the language of statistical mechanics). We review the main approximation schemes that are commonly used to solve this flow equation and discuss applications in equilibrium and out-of-equilibrium statistical physics, quantum many-particle systems, high-energy physics and quantum gravity.

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非微扰泛函重整化群及其应用

重整化群在物理学的许多领域都起着至关重要的作用，一方面是在概念上，另一方面是作为一种实用工具来确定许多系统的长距离低能特性，另一方面在基础物理学中寻找可行的紫外线完成. 它为我们提供了一个自然的框架来研究理论模型，其中自由度在长距离上是相关的，并且在不同的能量尺度上可能表现出非常不同的行为。非微扰函数重整化群 (FRG) 方法是威尔逊 RG 的现代实现，它允许建立超越标准微扰 RG 方法的非微扰近似方案。FRG 基于粗粒度有效作用（或统计力学语言中的吉布斯自由能）的精确函数流方程。我们回顾了常用于求解该流动方程的主要近似方案，并讨论在平衡和非平衡统计物理学、量子多粒子系统、高能物理学和量子引力中的应用。