General achievable bound of extractable work under feedback control

A general achievable upper bound of extractable work under feedback control is given, where nonequilibrium equalities are generalized so as to be applicable to error-free measurements. The upper bound involves a term which arises from the part of the process whose information becomes unavailable and is related to the weight of the singular part of the reference probability measure. The obtained upper bound of extractable work is more stringent than the hitherto known one and sets a general achievable bound for a given feedback protocol. Guiding principles of designing the optimal protocol are also suggested. Examples are presented to illustrate our general results.

Figure 1

A single-particle classical Szilard engine. The forward (backward) protocol is indicated by dashed downward (solid upward) arrows. (a) The system is initially in thermal equilibrium at temperature $T$. (b) A barrier is inserted at position $l_{\mathrm{i}}$ and an error-free measurement is performed to find in which partition [left (L) or right (R)] the particle resides. (c) Depending on the measurement outcome L or R, the barrier is moved to the position $l_{\mathrm{L}}$ or $l_{\mathrm{R}}$. (d) The barrier is removed and the system becomes thermalized at temperature $T$. A certain amount of information becomes unavailable upon the removal of the barrier from (c) to (d). The backward process (solid upward arrows) starts from the equilibrium state at temperature $T$ in (d). The barrier is inserted at $l_{\mathrm{L}}$ or $l_{\mathrm{R}}$ depending on the measurement outcome of the forward process in (c). Then the barrier is moved to $l_{\mathrm{i}}$ in (b) and finally it is removed in (a). The leftmost (rightmost) path is a singular one which does not exist in the forward path corresponding to the measurement outcome L (R).

Figure 2

Brownian particle in a harmonic potential with the forward (indicated by the downward dashed arrows) or backward (indicated by the upward solid arrows) protocol. First, the position of the particle is measured without error (F1). Then, according to the outcome, the potential is instantaneously shifted and work is extracted from the difference in the potential energy (indicated by the downward solid arrow) (F2). Finally, the system becomes equilibrated with the heat bath (F3). The associated backward process (indicated by the upward solid arrows) starts from the equilibrium distribution for the final configuration of the forward process (B1, B2). Then the potential is instantaneously shifted back to the initial position (B3). Some amount of information gets lost and becomes unavailable in the ensuing relaxation process (from F2 to F3).

Figure 3

The criterion of experimental implementation of an error-free measurement in the model of a Brownian particle in a harmonic potential. The LHS of Eq. (12) (vertical line) is plotted against the measurement error probability $\epsilon$ (horizontal line) for $\sqrt{\beta k}\Delta =1.0$ and $N=1000$. The shaded region implies that the deviation of the value of $\langle e^{-\beta W-I+I_{\mathrm{u}}}\rangle$ from 1 is masked by the statistical error in that region so that the measurement can be regarded as error-free. The error bars show the standard deviation given in Eq. (A4).