The macroscopic fluctuating theory developed during the last 30 years is applied to generic systems described by continuum fields ϕ(x, t) that evolve by a Langevin equation that locally either conserves or does not conserve the field. This paper aims to review well-known basic concepts and results from a pedagogical point of view by following a general framework in a practical and self-consistent way. From the probability of a path, we study the general properties of the system’s stationary state. In particular, we focus on the study of the quasipotential that defines the stationary distribution at the small noise limit. To discriminate between equilibrium and non-equilibrium stationary states, the system’s adjoint dynamics are defined as the system’s time-reversal Markov process. The equilibrium is then defined as the unique stationary state that is dynamically time-reversible, and therefore its adjoint dynamics are equal to those of the original one. This property is confronted with the macroscopic reversibility that occurs when the most probable path to create a fluctuation from the stationary state is equal to the time-reversed path that relaxes it. The lack of this symmetry implies a nonequilibrium stationary state; however, the converse is not true. Finally, we extensively study the two-body correlations at the stationary state. We derive some generic properties at various situations, including a discussion about the equivalence of ensembles in nonequilibrium systems.

在过去30年中发展起来的宏观涨落理论被应用于由连续域ϕ（x，t）描述的通用系统，该连续域由Langevin方程演化而来，该方程局部地保存或不保存该字段。本文旨在通过以实用和自洽的方式遵循通用框架，从教学法的角度审查著名的基本概念和结果。从路径的可能性，我们研究系统静止状态的一般性质。特别地，我们专注于定义在小噪声限制下的平稳分布的准势的研究。为了区分平衡态和非平衡态，系统的伴随动力学被定义为系统的时间逆向马尔可夫过程。然后将平衡定义为动态时间可逆的唯一静态，因此其伴随动力学等于原始动力学。当从静止状态产生波动的最可能路径等于使它松弛的时间反向路径时，此属性将遇到宏观可逆性。缺乏这种对称性意味着不平衡的静止状态。但是，事实并非如此。最后，我们广泛研究了稳态下的两体相关性。我们推导了在各种情况下的一些通用属性，包括有关非平衡系统中合奏的等效性的讨论。