Abstract
A new expression of the Harada–Sasa equality is derived by multiple-scale analysis. The new expression unifies the equality for the underdamped and overdamped Langevin models in special cases. In addition, the expression shows that the equality is available in a new time region, which differs from that in the underdamped or overdamped model. The expression is obtained by the expansion of the fluctuation response relation (FRR) violation in the underdamped model in powers of ϵ = m/γ, where γ is the friction coefficient and m is the mass of a Brownian particle. The violation of the FRR is in agreement with the energy dissipation rate up to the second order of ϵ.
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