Numerical determination of entropy associated with excess heat in steady-state thermodynamics

We numerically determine the global entropy for heat-conducting states, which is connected to the so-called excess heat considered as a basic quantity for steady-state thermodynamics in nonequilibrium. We adopt an efficient method to estimate the global entropy from the bare heat current and find that the obtained entropy agrees with the familiar local equilibrium hypothesis well. Our method possesses a wider applicability than local equilibrium and opens a possibility to compare thermodynamic properties of complex systems in nonequilibrium with those in the local equilibrium. We further investigate the global entropy for heat-conducting states and find that it exhibits both extensive and additive properties; however, the two properties do not degenerate each other differently from those at equilibrium. The separation of the extensivity and additivity makes it difficult to apply powerful thermodynamic methods to the nonequilibrium steady states.

Figure 1

Sketch of the temperature profile before the change (initial equilibrium state) and in the final heat-conducting state. To determine the entropy in the heat-conducting state (right), we apply a protocol to change the temperature of the heat baths from $(\overline{T},\overline{T})$ to $(T_1,T_2)$. We fix $\overline{T}=\frac{T_1-T_2}{2}$.

Figure 2

Pair of protocols for estimating $\mathrm{\Delta }{S}_{\mathrm{neq}}$ by utilizing Eq. (9). The heat baths $T_1^{\mathrm{b}}$ and $T_2^{\mathrm{b}}$ are changed in a stepwise manner, as indicated by the red solid and blue dotted lines, respectively. The forward and backward protocols are exactly time-reversed. The period $\tau$ is set to be much longer than the relaxation time to the final steady state. $\tau$ should be taken longer as the size $L$ becomes larger.

Figure 3

The dependence of the degree of nonequilibrium $\varepsilon$ for the excess entropy productions $-{\mathrm{\Theta }}_{\mathrm{ex}}$ and ${\mathrm{\Theta }}_{\mathrm{ex}}^{†}$ for $L=500$. Each error bar is smaller than the size of each point. $(T_1,T_2)=(0.5,0.45), (0.45,0.4), (0.7,0.6), (0.5,0.4), (0.55,0.4), (0.6,0.4), (0.7,0.4)$. $-{\mathrm{\Theta }}_{\mathrm{ex}}$ and ${\mathrm{\Theta }}_{\mathrm{ex}}^{†}$ should coincide in the quasistatic limit to estimate the same entropy difference; however, the deviation becomes significant in the one-step protocol, especially for larger $\varepsilon$; see Eqs. (23) and (24).

Figure 4

The dependence of the degree of nonequilibrium $\varepsilon$ for the entropy difference $-\frac{1}{2}(\mathrm{\Theta }-{\mathrm{\Theta }}^{†})$ and $\mathrm{\Delta }{S}_{\mathrm{leq}}$ for $L=500$. Each error bar is smaller than the size of each point. $(T_1,T_2)=(0.5,0.45), (0.45,0.4), (0.7,0.6), (0.5,0.4), (0.55,0.4), (0.6,0.4), (0.7,0.4)$. The dotted line corresponds to the thermodynamic limit in Eq. (35). The coincidence of $\mathrm{\Delta }{S}_{\mathrm{leq}}\simeq -\frac{1}{2}(\mathrm{\Theta }-{\mathrm{\Theta }}^{†})$ has been maintained over the range of $\varepsilon$ in spite of the large deviation between ${\mathrm{\Theta }}_{\mathrm{ex}}$ and ${\mathrm{\Theta }}_{\mathrm{ex}}^{†}$ in Fig. 3.

Figure 5

System-size dependence of the entropy difference $-\frac{1}{2}(\mathrm{\Theta }-{\mathrm{\Theta }}^{†})$ and $\mathrm{\Delta }{S}_{\mathrm{leq}}$ for the change $(\overline{T},\overline{T})$ to $(T_1,T_2)$. $(T_1,T_2)$ is fixed at $(0.5,0.4)$, whereas $L=10$, 50, 100, 250, 500, 750. Each error bar is smaller than the size of each point. The dotted line corresponds to the thermodynamic limit in Eq. (35). Note that $\mathrm{\Delta }{S}_{\mathrm{neq}}$ agrees with $\mathrm{\Delta }{S}_{\mathrm{leq}}$, even for the smallest $L$.

Figure 6

Temperature profiles at $L=100$ and 1000 in the scaled space $x=\frac{i}{L}-\frac{1}{2}$. $(T_1,T_2)=(0.5,0.4)$, which corresponds to $(\overline{T},\varepsilon )=(0.45,0.222)$. For small lattices such as $L=100$, greater gaps between the temperature at both boundaries exist, and the slope in the bulk is lower. The gaps become smaller, and the slope becomes almost $(T_1-T_2)/L$ for larger lattices, as exemplified by $L=1000$.

Figure 7

Schematic illustrating the difference between the extensivity and the additivity in the heat-conducting states. In the separation of the system in (a) into (b), we first attach heat baths at an appropriate temperature of $T_{\mathrm{m}}$ at $i=L$ and $L+1$, conserving the value of the heat current $J$. Then, we remove the interaction between the two. We have the additivity of the entropy as $2S=S_1+S_2$; however, $S_1\ne S_2\ne S$. We have the extensivity of the system in (a) from (c), where $J$ is not conserved.

Figure 8

The examination of the extensivity when conserving the heat current $J$. The systems in (a) and (b) are assumed to have the same mean temperature $\overline{T}$ and heat current $J$. $T_1=(1+\frac{\varepsilon }{2})\overline{T}$ and $T_2=(1-\frac{\varepsilon }{2})\overline{T}$, whereas $T_1^{\prime }=(1+\frac{\varepsilon }{4})\overline{T}$ and $T_2^{\prime }=(1-\frac{\varepsilon }{4})\overline{T}$. We have $S\left(2L\right)>2S\left(L\right)$, which means the failure of the extensivity for the conditions of constant $J$ and $\overline{T}$.