Computing the optimal protocol for finite-time processes in stochastic thermodynamics

Asking for the optimal protocol of an external control parameter that minimizes the mean work required to drive a nanoscale system from one equilibrium state to another in finite time, Schmiedl and Seifert [T. Schmiedl and U. Seifert, Phys. Rev. Lett. 98, 108301 (2007)] found the Euler-Lagrange equation to be a nonlocal integrodifferential equation of correlation functions. For two linear examples, we show how this integrodifferential equation can be solved analytically. For nonlinear physical systems we show how the optimal protocol can be found numerically and demonstrate that there may exist several distinct optimal protocols simultaneously, and we present optimal protocols that have one, two, and three jumps, respectively.

Figure 1

The optimal protocol (solid line) that minimizes the average work for $V=-\text{cos}\left(x-\lambda \right)$ with $D={10}^{-5}$ and $\Delta \lambda =1.2$ is unique and jumps twice. But it is interesting to see that there exists a suboptimal protocol (dotted line) whose average work, $⟨W⟩=0.4620$, is close to the average work of the optimal protocol, $⟨W⟩=0.4619$. For better visualization we have shifted the suboptimal protocol slightly in time.

Figure 2

The optimal protocol (solid line) that minimizes the average work for $V=-\text{cos}\left(x-\lambda \right)$ with $D={10}^{-5}$ and $\Delta \lambda =2.4$ requires an average work of $⟨W⟩=1.5723$. With $⟨W⟩=1.57233$, the average work of the suboptimal protocol (dotted line) almost approaches that of the optimal protocol.

Figure 3

For $V=-\text{cos}\left(x-\lambda \right)$ with $D={10}^{-5}$ and $\Delta \lambda =3.0$ there is a whole family of optimal protocols. Two of them jump twice (solid and dashed lines). All the other optimal protocols jump at least three times. One of them is displayed by the dotted line. The average work for any member of the family of optimal protocols is $⟨W⟩=1.9802$.

Figure 4

Two members of the family of optimal protocols are displayed for $V=-\text{cos}\left(x-\lambda \right)$ with $D={10}^{-5}$ and $\Delta \lambda =3.6$. The two displayed protocols jump twice. All the other optimal protocols jump at least three times. The average work is $⟨W⟩=1.8130$.

Figure 5

For $V=-\text{cos}\left(x-\lambda \right)$ with $\Delta \lambda =\pi$ and $D={10}^{-5}$ the initial and the final jumps vanish and the optimal protocol can degenerate to one single jump that may happen at any arbitrary time, $0\le t\le t_f$. Two members of the family of optimal protocols are displayed. The average work is $⟨W⟩=1.99999$.

Figure 6

Optimal protocols that minimize the average work for $V=-\text{cos}\left(x-\lambda \right)$ with $\Delta \lambda =\pi$. For $D=0.5$ there is a family of optimal protocols. The solid line mimics one member of this family. In contrast, for $D=0.7$ the optimal protocol is unique and jumps twice as mimicked by the dashed line.

Figure 7

The two different states that drive the orientation of the dipole. The left figure displays the stable state where the system is pulled by the minimum of the potential. The right figure displays the unstable state where the system is pushed by the maximum of the potential.

Figure 8

There are two regions in the $\Delta \lambda \text{−}D$ plane. For $\Delta \lambda$ near $\pi$ and $D$ small, the optimal protocol occurs in a family and may jump several times. For $\Delta \lambda$ far away from $\pi$ or for $D$ large, the optimal protocol is unique and jumps twice.