Reciprocal relations associated with linear responses to mechanical and thermal perturbations in nonequilibrium Langevin systems

We study universal relationships in linear responses to instances when mechanical and thermal perturbations are applied to a nonequilibrium steady state. For perturbations in nonequilibrium Langevin systems, we consider a time-dependent external force exerted on a particle along with a time-dependent temperature for the heat bath. In this system, by calculating heat currents and particle velocity, we obtain the kinetic coefficients (Onsager's coefficients) and the time correlation functions using the Fokker--Planck equations. With these quantities, we derive the reciprocal relation associated with the violation of the fluctuation response relation. We obtain the relation by integrating the extent of the violation with respect to the frequency. We also find the reciprocity in the first moment of the frequency integration. These frequency sum rules hold for both overdamped and underdamped regimes for any type of potential and any strength of the external force.