Equality Connecting Energy Dissipation with a Violation of the Fluctuation-Response Relation

In systems driven away from equilibrium, the velocity correlation function and the linear-response function to a small perturbation force do not satisfy the fluctuation-response relation (FRR) due to the lack of detailed balance in contrast to equilibrium systems. In this Letter, an equality between an extent of the FRR violation and the rate of energy dissipation is proved for Langevin systems under nonequilibrium conditions. This equality enables us to calculate the rate of energy dissipation by quantifying the extent of the FRR violation, which can be measured experimentally.

Figure 1

$\tilde{C}(\omega )$ and $2T{\tilde{R}}^{\prime }(\omega )$ as functions of $\omega$ when $D=5$ and $\alpha =10$. All the quantities are dimensionless under the normalization as $\gamma =l=T=1$. The solid and dotted lines represent $\tilde{C}(\omega )$ and $2T{\tilde{R}}^{\prime }(\omega )$, respectively.

Figure 2

Energy fluxes $⟨J{⟩}_0$ and $⟨J_{\mathrm{in}}{⟩}_0$ calculated on the basis of Eqs. (5, 8), respectively, as functions of the transition rate $\alpha$. The solid line and the closed circles represent $⟨J{⟩}_0$ and $⟨J_{\mathrm{in}}{⟩}_0$, respectively, when $D=1$. The dotted line and the closed squares represent $⟨J{⟩}_0$ and $⟨J_{\mathrm{in}}{⟩}_0$, respectively, when $D=5$. As in Fig. 1, all the quantities are dimensionless.