Linear response theory in stochastic many-body systems revisited

The Green–Kubo relation, the Einstein relation, and the fluctuation–response relation are representative universal relations among measurable quantities that are valid in the linear response regime. We provide pedagogical proofs of these universal relations for stochastic many-body systems. Through these simple proofs, we characterize the three relations as follows. The Green–Kubo relation is a direct result of the local detailed balance condition, the fluctuation–response relation represents the dynamic extension of both the Green–Kubo relation and the fluctuation relation in equilibrium statistical mechanics, and the Einstein relation can be understood by considering thermodynamics. We also clarify the interrelationships among the universal relations.