Figure 1

Schematics for two kinds of CPEA. Here ${\eta }^{*}$ is the time-reversed microstate of $\eta$. The arrows show path of the system.Reuse & Permissions

Figure 2

Verification of the expression (16) for the stationary distribution ${\rho }_{\mathrm{st}}(\eta )$ in NESS. (a) The contours of $\Phi (\eta )=-\mathit{\beta }·\mathcal{E}(\eta )$ (circles) and $\ln {\rho }_{\mathrm{st}}(\eta )$ (solid lines) are plotted for $({\beta }_x,{\beta }_y,f)=(1,10,0.8)$. Equations (16) seem to work rather well even though ${\beta }_y/{\beta }_x=10$. Contours are drawn at $\{0.5,0,-1,-2,-3\}$, and most probable states are marked with “H.” The values of $\Phi (\eta )$ are shifted with some constant values which correspond to a normalization constant. (b) Error estimation by the difference $|\Delta ⟨H⟩|≔|\sum _{\eta }H(\eta )[{\rho }_{\mathrm{st}}(\eta )-\exp [\Phi (\eta )]/Z]|$ where $Z$ is a normalization constant. Horizontal axis corresponds to the thermodynamic force $ϵ$. Triangles show data set varying $f$ with ${\beta }_x={\beta }_y=2$. Circles show data set varying ${\beta }_y$ with ${\beta }_x=2$, $f=0$. The drawn guideline is ${ϵ}^3$. For the calculation of ${\rho }_{\mathrm{st}}(\eta )$, we numerically solved Fokker-Planck equations corresponding to the Langevin Eqs. (18). For $\Phi (\eta )$, we constructed and solved a set of difference equations for CPEAs of the excess heat transfer. We have checked that these results are consistent with results obtained by direct simulation of Langevin equations with stochastic energetics [15].Reuse & Permissions