Work fluctuation and total entropy production in nonequilibrium processes

Work fluctuation and total entropy production play crucial roles in small thermodynamic systems subject to large thermal fluctuations. We investigate a trade-off relation between them in a nonequilibrium situation in which a system starts from an arbitrary nonequilibrium state. We apply a variational method to study this problem and find a stationary solution against variations over protocols that describe the time dependence of the Hamiltonian of the system. Using the stationary solution, we find the minimum of the total entropy production for a given amount of work fluctuation. An explicit protocol that achieves this is constructed from an adiabatic process followed by a quasistatic process. The obtained results suggest how one can control the nonequilibrium dynamics of the system while suppressing its work fluctuation and total entropy production.

Figure 1

Normalized total entropy production versus normalized work fluctuation plotted along the stationary solution (25). Here, $\mathrm{\Delta }D(p_{\mathrm{ini}}\parallel p_{{\lambda }_0}^{\mathrm{can}}):=D(p_{\mathrm{ini}}\parallel p_{{\lambda }_0}^{\mathrm{can}})-D_0(p_{\mathrm{ini}}\parallel p_{{\lambda }_0}^{\mathrm{can}})$ with $D_0$ defined in Eq. (47) [see also Eq. (49)], and ${\mathcal{F}}_{{\lambda }_0}\left(x_0\right)$ is the nonequilibrium free energy (44). The green curve gives the lower bound of the total entropy production for a given work fluctuation, thereby showing the boundary of the trade-off relation between the work fluctuation and the total entropy production (see Sec. 4). The two blue dots at both ends of the curve show the values of $\mathrm{Var}\left[W\right]$ and that of $\langle \sigma \rangle$ obtained from the deterministic work extraction protocol (49) and the thermodynamically reversible protocol (43), respectively.

Figure 2

Protocol achieving the stationary solution (26). A change in the state of the system is shown vertically, and a change in the Hamiltonian of the system is shown horizontally. The forward protocol consists of an adiabatic process followed by a quasistatic process. Here, by the quasistatic process, we mean that a change in the Hamiltonian of the system is slow compared with the relaxation of the system via the interaction with the heat bath. At the beginning of the quasistatic process, the state of the system changes from $p_{\mathrm{ini}}$ to a thermalized state $q$ given in Eq. (27). In $(q,H_q),H_q$ represents the Hamiltonian whose canonical distribution is $q$. The backward protocol consists of a quasistatic process followed by an adiabatic process. Because the total entropy production of the forward process is nonvanishing, the final state of the backward process $q$ differs from the initial state of the forward process.

Figure 3

Lorenz curve and the thermomajorization criterion. The Lorenz curve shows a nonuniformity of the state of the system with respect to the canonical distribution and gives a graphical representation of the quasiordering ${\succ }_{\mathrm{th}}$, i.e., the thermomajorization. We plot $\left\{{\sum }_{i=1}^k\frac{e^{-\beta E\left(i\right)}}{Z}\right\}$ and $\left\{{\sum }_{i=1}^{k}p\left(i\right)\right\}$ on the $(X,Y)$ plane where the curve $p=\rho$ is shown by the orange curve, $p=\sigma$ is shown by the green dotted curve, and $p={\rho }_{\mathrm{can}}=\frac{e^{-\beta H}}{Z}$ is shown by the black dashed line. The thermomajorization criterion tells us that state $\rho$ can be transformed into $\sigma$ via a thermal operation if the Lorenz curve $\left(\sigma ,\frac{e^{-\beta H}}{Z}\right)$ is below that of $\left(\rho ,\frac{e^{-\beta H}}{Z}\right)$. Here, the Lorenz curves are plotted for $\rho$ and $\sigma$ having the same rearranged orderings as in Eq. (A4).

Figure 4

(a) Lorenz curves corresponding to the process shown in Eq. (A21). If $w-u\le 0$, the blue line is below the orange curve, and the transition (A21) is possible. (b) Lorenz curves corresponding to the process shown in Eq. (A23). If $v-w\le kTln\frac{Z_{{\lambda }_N}}{Z_{{\lambda }_0}^{*}}$, the blue curve is below the orange curve, and therefore the transition (A23) is possible.

Figure 5

(a) Difference between Eqs. (27) and (B3) in the normalized total entropy production for different choices of the initial probability. The orange dot represents a point that satisfies $p_{\mathrm{ini}}=p_{{\lambda }_0}^{\mathrm{can}}$. We find that there is a wide region in which $p_{\mathrm{ini}}$ and $p_{{\lambda }_0}^{\mathrm{can}}$ are very different but the difference in the total entropy production is small. (b) Plot of $\delta (p_{\mathrm{ini}}^{*}\parallel p_{{\lambda }_0}^{*\mathrm{can}})$ for different choices of the initial probability. Here, we plot for a three-level system and fix $p_{{\lambda }_0}^{\mathrm{can}}=\{0.2,0.5,0.3\}$.

Figure 6

Normalized difference in the total entropy production $\frac{\mathrm{\Delta }\sigma }{D(p_{\mathrm{ini}}\parallel p_{{\lambda }_0}^{\mathrm{can}})}$ versus $\delta (p_{\mathrm{ini}}^{*}\parallel p_{{\lambda }_0}^{*\mathrm{can}})$. The data are obtained for a three-level system and $p_{{\lambda }_0}^{\mathrm{can}}=\{0.2,0.5,0.3\}$. Each dot is obtained for different choices of $p_{\mathrm{ini}}$. If the distance $\delta (p_{\mathrm{ini}}^{*}\parallel p_{{\lambda }_0}^{*\mathrm{can}})$ is small, two lower bounds of the total entropy production in Eqs. (32) and (B3) give approximately equal numerical values. Each dot is obtained by randomly generating $p_{\mathrm{ini}}$.