Experimental Observation of the Role of Mutual Information in the Nonequilibrium Dynamics of a Maxwell Demon

We validate experimentally a fluctuation relation known as generalized Jarzynski equality governing the work distribution in a feedback-controlled system. The feedback control is performed on a single electron box analogously to the original Szilard engine. In the generalized Jarzynski equality, mutual information is treated on an equal footing with the thermodynamic work. Our measurements provide the first evidence of the role of mutual information in the fluctuation theorem and thermodynamics of irreversible processes.

Figure 1

Device and operation. (a) The single electron box (SEB), highlighted in blue, is the system under study. It is monitored by a single electron transistor (SET), whose current $I_{\text{det}}$ depends on $n$, the number of excess electrons on the right island of the SEB. The SEB is controlled by gate voltage $V_g$. (b) Single trace histogram of detector signal for states $n=0$ (peaks to the left) and $n=1$ (peaks to the right) with filter cutoff frequencies 50 (black), 100 (green), 300 (yellow), and 1000 Hz (red), in the order of decreasing maxima. (c) A full trace of the feedback control. $I_{\text{det}}$ shows the measured occupation in the SEB. (d) Energy diagrams of the process. The rapid feedback (left) extracts work by lowering the energy. During the return back to degeneracy (middle), net work is extracted from the thermal excitations of $n$ entering the higher energy state. If the rapid feedback were performed incorrectly (right), excess work equal to $\mathrm{\Delta }E$ would be applied to add energy to the system. The return to degeneracy would again follow the behavior of middle panel.

Figure 2

Distributions of work with mutual information. [(a)–(c)] The measured $W/k_{B}T$ for different measurement error probabilities $ϵ=0.02$, 0.05, and 0.13 with $N=1177$, 925, and 1025, respectively. Black lines show the numerically obtained distributions. [(d)–(f)] The distributions with the mutual information $I$ added to $W/k_{B}T$. The numbers in the panel (f) refer to the four peaks in Eq. (9).

Figure 3

Efficiency and fluctuation relations. (a) Average extracted work (blue squares and solid line) and the feedback efficiency (red diamonds and solid line) as functions of measurement error. Experimental data are shown by symbols (in both panels), mutual information obtained from Eq. (7) is shown by the green circles and solid line, and numerical predictions for other quantities by solid lines (in both panels). The dashed brown line shows the fundamental maximum for ${\eta }_f$, $⟨I⟩$, and $-⟨W⟩$. The black dashed line shows the limit for ${\eta }_f$ and $-⟨W⟩$ below which the process is dissipative. The results are obtained from $N=1177$, 1734, 925, 1060, and 1025 repetitions, in the order of increasing $ϵ$. (b) Test of the generalized JE with mutual information. The error bars include the statistical error, as well as the uncertainity in the measured value of $E_C/k_{B}T$.