Nonequilibrium identities and response theory for dissipative particles

We derive some nonequilibrium identities such as the integral fluctuation theorem and the Jarzynski equality starting from a nonequilibrium state for dissipative classical systems. Thanks to the existence of the integral fluctuation theorem we can naturally introduce an entropy-like quantity for dissipative classical systems in far from equilibrium states. We also derive the generalized Green-Kubo formula as a nonlinear response theory for a steady dynamics around a nonequilibrium state. We numerically verify the validity of the derived formulas for sheared frictionless granular particles.

Figure 1

The numerical verification of the inequality Eq. (17) for $t_0=0.0$. Here, $\langle {\Omega }^{*}\left(t\right)\rangle ={\langle \Omega \left(t\right)\rangle }_{\mathrm{NE}}/\sqrt{k/m}$ is plotted as a function of the scaled time $t^{*}=t/\sqrt{m/k}$.

Figure 2

The numerical verification of the inequality Eq. (17) for $t_0=1.0\sqrt{k/m}$. Here, $\langle {\Omega }^{*}\left(t\right)\rangle ={\langle \Omega \left(t\right)\rangle }_{\mathrm{NE}}/\sqrt{k/m}$ is plotted as a function of the scaled time $t^{*}=t/\sqrt{m/k}$.

Figure 3

The numerical verification of the integral fluctuation theorem Eq. (19) for $t_0=0.0$, where ${\omega }^{*}\left(t\right)=\omega \left(t\right)\sqrt{m/k}$ is plotted against the scaled time $t^{*}=t\sqrt{k/m}$.

Figure 4

The numerical verification of the integral fluctuation theorem Eq. (19) for $t_0=1.0\sqrt{m/k}$, where ${\omega }^{*}\left(t\right)=\omega \left(t\right)\sqrt{m/k}$ is plotted against the scaled time $t^{*}=t\sqrt{k/m}$.

Figure 5

The numerical verification of the generalized Green-Kubo formula in Eq. (28) for $t_0^{*}=t_0\sqrt{k/m}=0.0$ and $5.0$ through the plot of $\langle {\sigma }_{xy}^{*}\left(t\right)\rangle =\langle {\sigma }_{xy}\left(t\right)\rangle /(k/d)$ against the scaled time $t^{*}=t\sqrt{k/m}$, where DS and GK, respectively, denote the numerical result in terms of the direct simulation of granular particles and the generalized Green-Kubo formula Eq. (28).