Large deviation analysis of a simple information engine

Information thermodynamics provides a framework for studying the effect of feedback loops on entropy production. It has enabled the understanding of novel thermodynamic systems such as the information engine, which can be seen as a modern version of “Maxwell's Dæmon,” whereby a feedback controller processes information gained by measurements in order to extract work. Here, we analyze a simple model of such an engine that uses feedback control based on measurements to obtain negative entropy production. We focus on the distribution and fluctuations of the information obtained by the feedback controller. Significantly, our model allows an analytic treatment for a two-state system with exact calculation of the large deviation rate function. These results suggest an approximate technique for larger systems, which is corroborated by simulation data.

Figure 1

Schematic of model. The feedback mechanism makes a measurement of the particle position and places a barrier according to the outcome of the measurement. If $y=x$ or $y=x+1$, then the barrier influences the particle movement by preventing certain jumps as in the top and middle right. For all other measurement values, the barrier has no effect and the particle moves freely as in the bottom right.

Figure 2

Numerical tests of (2), (3), and (18) in semilog scale. Parameter values are $L=3,p=0.2$, and $r=0.9$; data avergaed over ${10}^7$ realizations. Error bars indicate the standard error of mean of $\langle e^{-S_t-I_t}\rangle$ at representative points.

Figure 3

Cumulative density function (32) for $\mathrm{\Delta }{I}_s$ for different system sizes $L$ with $p=0.2$ and $r=0.9$. Dashed lines give theoretical values for $L=2$ as given by (29) and (30).

Figure 4

Large deviation rate function (22) for $L=2,p=0.2$, and $r=0.9$. The solid black line shows the theoretical curve obtained from (34) and (22). Points represent data sampled at different finishing times $t$.

Figure 5

Scatter plot of $\mathrm{\Delta }{I}_{s+1}$ against $\mathrm{\Delta }{I}_s$ for $L=10,p=0.5$, and $r=0.9$ to illustrate correlations as explained in the text. Histograms on the axes show the density of points on the plot.

Figure 6

Sample autocorrelation function of $\mathrm{\Delta }{I}_s$ (35) shows significant negative correlations after one time step. 95% confidence intervals plotted as blue dashed lines. $L=10$ and $r=0.9$ in both cases.

Figure 7

Large deviation rate function (22) for $L=10,p=0.5$, and $r=0.9$. Lines show rate functions obtained from the Markov approximation with different numbers of bins. The lines for 12 and 24 bins are not distinguishable at this scale. Points represent data sampled at different finishing times $t$. The cutoff for positive deviations is explained in the text and given in (36), and is represented by a vertical line.

Figure 8

Baseline values for an unbiased system $\left(p=q\right)$ for $r=0.9$. Points show the value of $\mathrm{\Delta }{I}_s$ for consecutive up jumps and blocked down jumps. Lines show the predictions from (B2) and (B6).

Figure 9

Baseline values for a biased system, $p=0.2$ and $r=0.9$. Points show the value of $\mathrm{\Delta }{I}_s$ for consecutive up jumps and blocked down jumps. Lines show the predictions from (B2) and (B6).

Figure 10

Trajectory of a biased system for $L=10,p=0.2$, and $r=0.9$. The trajectory $x_s$ is given by a gold line, and measurements by square boxes. The information gain $\mathrm{\Delta }{I}_s$, given by a blue line, is always bounded above by $ln(P\left({\mathbf{y}}_{s-1}\right)/P\left({\mathbf{y}}_s\right))$ given by a red line. Note that these quantities have different units. Note also that for $x_s$ and $y_s$ there are periodic boundary conditions between 0 and 10.

Figure 11

Trajectory of an unbiased system for $L=10,p=0.5$, and $r=0.9$. See Fig. 10 for details of the plot.

Figure 12

Trajectory of an unbiased system for $L=10,p=0.5$, and $r=0.9$. See Fig. 10 for details of the plot.

Figure 13

Points show numerical results for $\mathrm{Max}\left(\mathrm{\Delta }{I}_s\right)$ and $-\mathrm{Min}\left(\mathrm{\Delta }{I}_s\right)$ (from ${10}^7$ trajectory realizations of length $t=500$) and lines show values given by (D1) and (D2) for varying $L$ for $p=0.5$ and $r=0.9$.