Susceptibility characterizes the fundamental nature of systems in and out of equilibrium. The fluctuation-dissipation relation connects the susceptibility to the fluctuation of observables near equilibrium. As a measure of the fluctuation, the precision of observables, expressed as the signal-to-noise ratio, has been upper bounded by a class of inequalities, such as the thermodynamic uncertainty relation (TUR) and the kinetic uncertainty relation (KUR) out of equilibrium. On the other hand, the kinetic susceptibility equality (KSE) connects the precision to the dynamical activity and contains KUR as a special limiting case. In light of KSE, we first derive a general susceptibility equality (SE) that connects susceptibilities to a well-defined generalized dynamical activity. Thereby, we show an entropic and kinetic susceptibility equality (EKSE) that relates the fluctuating currents out of equilibrium to the pseudoentropy production and the dynamical activity in terms of the precision. The pseudoentropy production and the dynamical activity emerge as symmetric and antisymmetric cases of the generalized dynamical activity. For charge transfers, EKSE contains TUR and KUR. Therefore, EKSE is considered as a master equality of the fluctuation of charge transfers out of equilibrium.

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• Nonequilibrium statistical mechanics
• Stochastic thermodynamics