Correlation-powered information engines and the thermodynamics of self-correction

Information engines can use structured environments as a resource to generate work by randomizing ordered inputs and leveraging the increased Shannon entropy to transfer energy from a thermal reservoir to a work reservoir. We give a broadly applicable expression for the work production of an information engine, generally modeled as a memoryful channel that communicates inputs to outputs as it interacts with an evolving environment. The expression establishes that an information engine must have more than one memory state in order to leverage input environment correlations. To emphasize this functioning, we designed an information engine powered solely by temporal correlations and not by statistical biases, as employed by previous engines. Key to this is the engine's ability to synchronize—the engine automatically returns to a desired dynamical phase when thrown into an unwanted, dissipative phase by corruptions in the input—that is, by unanticipated environmental fluctuations. This self-correcting mechanism is robust up to a critical level of corruption, beyond which the system fails to act as an engine. We give explicit analytical expressions for both work and critical corruption level and summarize engine performance via a thermodynamic-function phase diagram over engine control parameters. The results reveal a thermodynamic mechanism based on nonergodicity that underlies error correction as it operates to support resilient engineered and biological systems.

Figure 1

Thermal ratchet information engine: a ratchet and three reservoirs—work, heat, and information. The work reservoir is depicted as gravitational mass suspended by a pulley. The information reservoir consists of a string of cells, each a two-state classical system that encodes one bit of information. The ratchet moves unidirectionally along the string and exchanges energy between the heat and the work reservoirs. The ratchet reads the value of a single cell (highlighted in yellow) at a time from the input string (green, right), interacts with it, and writes a symbol to the cell in the output string (blue, left) of the information reservoir. Information exchange between the ratchet and the information reservoir is signified by the change in the information content of the output symbols with respect to the input symbols. Driven by the information exchange, the ratchet transduces the input string $Y_{0:\infty }=Y_0{Y}_1...$ into an output string $Y_{0:\infty }^{\prime }=Y_0^{\prime }{Y}_1^{\prime }...$. (Reprinted from Ref. [59] with permission.)

Figure 2

Computational mechanics of information engines. The input tape values are generated by a hidden Markov model (HMM) with, say, three hidden states—$A, B$, and $C$. Specifically, transitions among the hidden states produce $0\mathrm{s}$ and $1\mathrm{s}$ that form the input tape random variables $Y_{0:\infty }$. The ratchet acts as an informational transducer that converts the input HMM into an output process, that is also represented as an HMM. That is, the output tape $Y_{0:\infty }^{\prime }$ can be considered as having been generated by an effective HMM that is the composition of the input HMM and the ratchet's transducer [62].

Figure 3

Period-2 process hidden Markov model with a transient start state $D$ and two recurrent causal states $E$ and $F$. Starting from $D$, the process makes a transition either to $E$ or to $F$ with equal probabilities while emitting $y=0$ or $y=1$, respectively. This is indicated by the transition labels from $D$: $y:p$ says generate symbol $y$ when taking the transition with probability $p$. On arriving at states $E$ or $F$, the process alternates between two states, emitting $y=0$ for transitions $E\to F$ and $y=1$ for transitions $F\to E$. In effect, we get either of two infinite sequences, $y_{0:\infty }=0101...$ and $y_{0:\infty }=1010...,$ with equal probabilities.

Figure 4

State transition diagram of a ratchet that extracts work from temporal correlations in its environment: $A, B$, and $C$ denote the ratchet's internal states and 0 and 1 denote the values of the interacting cell. The joint dynamics of the demon and interacting cell take place over the space of six internal joint states: $\{A\otimes 0,...,C\otimes 1\}$. Arrows indicate the allowed transitions and their probabilities in terms of the ratchet control parameters $\delta$ and $\gamma$. Note that $e$ here refers to the base of natural logarithm, not a variable or parameter.

Figure 5

Ratchet dynamics absent the synchronizing state: Assuming system parameters $\delta$ and $\gamma$ are set to zero, state $C$ becomes inaccessible and the ratchet's joint state-symbol dynamics become restricted to that shown here—a truncated form of the dynamics of Fig. 4.

Figure 6

Two dynamical modes of the ratchet while driven by a period-2 input process: (a) Counterclockwise (left panel): ratchet is out of synchronization with the input tape and makes a steady counterclockwise rotation in the composite space of the demon and the interacting cell. Work is steadily dissipated at the rate $-k_{B}T$ per pair of input symbols and no information is exchanged between the ratchet and the information reservoir. (b) Clockwise (right panel): ratchet is synchronized with the input correlated symbols on the tape, information exchange is nonzero, and work is continually accumulated at the rate $k_{B}T/e$ per pair of input symbols.

Figure 7

Crossover from the dissipative, counterclockwise mode to the generative, clockwise mode via synchronizing state $C$: Even though the microscopic dynamics satisfy time-reversal symmetry, a crossover is possible only from the counterclockwise mode to the clockwise mode because of the topology of the joint state space. With transitions between $C$ and the other two modes, the engine becomes ergodic among its dynamic modes. The heats and transition probabilities are shown above each arrow.

Figure 8

Tradeoff between average work production, synchronization rate, and synchronization heat: Contour plot of average extracted work per symbol $\langle W\rangle$ as a function of rate of synchronization $R_{\text{sync}}$ and synchronization heat $Q_{\text{sync}}$ using Eq. (11). Work values are in the unit of $k_{B}T$. Numbers labeling contours denote the average extracted work $\langle W\rangle$. If we focus on any particular contour, increasing $R_{\text{sync}}$ leads to a decrease in $Q_{\text{sync}}$ and vice versa. Similarly, restricting to a fixed value of $R_{\text{sync}}$, say the vertical $R_{\text{sync}}=0.4$ line, increasing $Q_{\text{sync}}$ decreases values of $\langle W\rangle$. Restricting to a fixed vale of $Q_{\text{sync}}$, say the horizontal $Q_{\text{sync}}=0.5$ line, increasing $R_{\text{sync}}$ going to the right also decreases $\langle W\rangle$.

Figure 9

Noisy phase-slip period-2 (NPSP2) process: As with the exact period-2 process of Fig. 3, its HMM has a transient start state $D$ and two recurrent causal states $E$ and $F$. Starting from $D$, the process makes a transition either to $E$ or $F$ with equal probabilities while outputting 0 or 1, respectively. Once in state $E$, the process either stays with probability $c$ and outputs a 0 or makes a transition to state $F$ with probability $1-c$ and outputs a 1. If in state $F$, the process either stays with probability $c$ and outputs a 1 or makes a transition to state $E$ with probability $1-c$ and outputs a 0. For small nonzero $c$, the output is no longer a pure alternating sequence of $0\mathrm{s}$ and $1\mathrm{s}$, but instead randomly breaks the period-2 phase. For $c=1/2$, the generated sequences are flips of a fair coin. The process reduces to that in Fig. 3, if $c=0$.

Figure 10

Average work production per symbol versus parameter $\delta$ and phase slip rate $c$ at fixed $\gamma =1$. Labels on the curves give $c$ values. Long-dashed lines give the upper and lower bounds on work production: It is bounded above by $k_{B}T/e$ and below by $k_{B}T(1-e)/\left(2e\right)$. (See text.)

Figure 11

Maximum work production versus phase-slip rate $c$. Maximum work production decreases with $c$ from $k_{B}T/e$ at $c=0$ to 0 when $c\ge c^{*}=1/(1+e)$.

Figure 12

Ratchet thermodynamic-function phase diagram: In the leftmost (red) region, the ratchet behaves as an engine, producing positive work. In the lower right (gray) region, the ratchet behaves as either an eraser (dissipating work to erase information) or a dud (dissipating energy without erasing information). The solid (red) line indicates the parameter values that maximize the work output in the engine mode. The dashed (black) line indicates the critical value of $c$ above which the ratchet cannot act as an engine.

Figure 13

State variable interdependence: Input HMM has an autonomous dynamics with transitions $S_N\to S_{N+1}$ leading to input symbols $Y_N$. That is, $Y_N$ depends only on $S_N$. The joint dynamics of the transducer in state $X_N$ and the input symbol $Y_N$ leads to the output symbol $Y_N^{\prime }$. In other words, $Y_N^{\prime }$ depend on $X_N$ and $Y_N$ or, equivalently, on $X_N$ and $S_N$. Knowing the joint stationary distribution of $X_N$ and $S_N$, then determines the stationary distribution of $Y_N^{\prime }$. However, if $Y_N$ and $X_N$ are known, $Y_N^{\prime }$ is independent of $S_N$.