We present a comprehensive study about the relationship between the way detailed balance is broken in non-equilibrium systems and the resulting violations of the fluctuation–dissipation theorem. Starting from stochastic dynamics with both odd and even variables under time-reversal, we derive an explicit expression for the time-reversal operator, i.e. the Markovian operator which generates the time-reversed trajectories. We then exploit the relation between entropy production and the breakdown of detailed balance to establish general constraints on the non-equilibrium steady-states (NESS), which relate the non-equilibrium character of the dynamics with symmetry properties of the NESS distribution. This provides a direct route to derive extended fluctuation–dissipation relations, expressing the linear response function in terms of NESS correlations. Such framework provides a unified way to understand the departure from equilibrium of active systems and its linear response. We then consider two paradigmatic models of interacting self-propelled particles, namely active Brownian particles and active Ornstein–Uhlenbeck particles. We analyze the non-equilibrium character of these systems (also within a Markov and a Chapman–Enskog approximation) and derive extended fluctuation–dissipation relations for them, clarifying which features of these active model systems are genuinely non-equilibrium.

我们对非平衡系统中详细平衡被打破的方式与由此产生的波动-耗散定理的违反之间的关系进行了全面的研究。从时间反转下具有奇数和偶数变量的随机动力学出发，我们推导出时间反转算子的显式表达式，即生成时间反转轨迹的马尔可夫算子。然后，我们利用熵产生与详细平衡破坏之间的关系来建立对非平衡稳态 (NESS) 的一般约束，这将动力学的非平衡特征与 NESS 分布的对称特性联系起来。这提供了导出扩展波动-耗散关系的直接途径，根据 NESS 相关性表达线性响应函数。这种框架提供了一种统一的方式来理解有源系统的平衡偏离及其线性响应。然后我们考虑相互作用的自推进粒子的两个范式模型，即活性布朗粒子和活性 Ornstein-Uhlenbeck 粒子。我们分析了这些系统的非平衡特征（也在马尔可夫和查普曼-恩斯科格近似内）并为它们推导出扩展的波动 - 耗散关系，阐明这些活动模型系统的哪些特征是真正的非平衡。