Extracting work from a single reservoir in the non-Markovian underdamped regime

We derive optimal-work finite time protocols for a colloidal particle in a harmonic well in the general non-Markovian underdamped regime in contact with a single reservoir. Optimal-work protocols with and without measurements of position and velocity are shown to be linear in time. In order to treat the underdamped regime one must address forcing the particle at the start and at the end of a protocol, conditions which dominate the short time behavior of the colloidal particle. We find that for protocols without measurement the least work by an external agent decreases linearly for forced start-stop conditions while those only forced at starting conditions are quadratic (slower) at short times, while both decrease asymptotically to zero for quasistatic processes. When measurements are performed, protocols with start-end forcing are still more efficient at short times but can be overtaken by start-only protocols at a threshold time. Measurement protocols derive work from the reservoir but always below that predicted by Sagawa's generalization of the second law. Velocity measurement protocols are more efficient in deriving work than position measurements.

Figure 1

The inertial function $G\left(t_f\right)$ for different values of the memory parameter $\tilde{\alpha }$. As $\tilde{\alpha }$ increases we approach the Markovian limit. Note the discontinuity at $t_f=0$ when $\tilde{\alpha }\to \infty$.

Figure 2

Comparison between ${\mathbf{W}}_{\mathrm{S}}$ and ${\mathbf{W}}_{\mathrm{G}}$ for the Markovian limit ($\tilde{\alpha }\to \infty$) for different particle mass values. As the mass increases, the protocol $\mathrm{G}$ is less efficient and is always improved upon by the $\mathrm{S}$ protocol. We consider parameters ${\tilde{\lambda }}_f=2$, and work values are normalized to the ${\mathbf{W}}_{\mathrm{Ins}}$.

Figure 3

Comparison between ${\mathbf{W}}_{\mathrm{S}}$ and ${\mathbf{W}}_{\mathrm{G}}$ for a range of memory parameters $\tilde{\alpha }$. As the memory increases, the $\mathrm{G}$ protocol becomes more efficient but never improves on the $\mathrm{S}$ protocol. Here we take ${\tilde{\lambda }}_f=2$ and $\tilde{m}=2$.

Figure 4

Comparison of work performed for the Markovian system ($\tilde{\alpha }\to \infty$), using the optimal protocols $\mathrm{G}$ and $\mathrm{S}$, compared to the optimal after measuring position $x$. While ${\mathbf{W}}_{\mathrm{S}}$ and ${\mathbf{W}}_{\mathrm{G}}$ have asymptotes at zero work, ${\mathbf{W}}_x$ can return work to the external agent (shaded region). ${\mathbf{W}}_x$ becomes more efficient than any of the nonmeasuring protocols at a threshold value of time. Here the parameters take on the values ${\tilde{\mathrm{\Delta }}}_x^2=0.2, {\tilde{\lambda }}_f=2$, and $\tilde{m}=2$.

Figure 5

Comparison between ${\overline{\mathbf{W}}}_x$ for different values of the memory parameter $\tilde{\alpha }$. As the memory increases, the protocol becomes more efficient in extracting work. We selected the values ${\tilde{\mathrm{\Delta }}}_x^2=0.2, {\tilde{\lambda }}_f=2$, and $\tilde{m}=2$.

Figure 6

Comparison of the work performed for the protocols $\mathbf{W}, {\overline{\mathbf{W}}}_x, {\overline{\mathbf{W}}}_v$, and ${\overline{\mathbf{W}}}_{xv}$ for the non-Markovian evolution ($\tilde{\alpha }=0.5$). All measured protocols are able to return work to the external agent. The information derived from the velocity is more advantageous than that of the position. The parameter values used are ${\tilde{\mathrm{\Delta }}}_x^2=0.2, {\tilde{\mathrm{\Delta }}}_v^2=0.2, {\tilde{\lambda }}_f=2$, and $\tilde{m}=2$.

Figure 7

${\lambda }_{ov}$ plots for the parameters in the legend. The dotted lines show the Markovian limit, while the solid lines are for $\tilde{\alpha }=0.5$.