A Programmable Mechanical Maxwell's Demon

Abstract

We introduce and investigate a simple and explicitly mechanical model of Maxwell's demon-a device that interacts with a memory register (a stream of bits), a thermal reservoir (an ideal gas) and a work reservoir (a mass that can be lifted or lowered). Our device is similar to one that we have briefly described elsewhere, but it has the additional feature that it can be programmed to recognize a chosen reference sequence, for instance, the binary representation of π . If the bits in the memory register match those of the reference sequence, then the device extracts heat from the thermal reservoir and converts it into work to lift a small mass. Conversely, the device can operate as a generalized Landauer's eraser (or copier), harnessing the energy of a dropping mass to write the chosen reference sequence onto the memory register, replacing whatever information may previously have been stored there. Our model can be interpreted either as a machine that autonomously performs a conversion between information and energy, or else as a feedback-controlled device that is operated by an external agent. We derive generalized second laws of thermodynamics for both pictures. We illustrate our model with numerical simulations, as well as analytical calculations in a particular, exactly solvable limit.

Keywords: Landauer’s principle; Maxwell’s demon; Shannon entropy; Szilard engine; information engine; second law of thermodynamics.

Figure 1

In our schematic conception of a programmable, autonomous Maxwell’s demon, a fixed set of binary gates defines a reference sequence. As the demon interacts one bit at a time with an incoming sequence of memory bits, it is able to lift a small mass against gravity if the incoming bit sequence matches the reference sequence. As the demon writes information onto the memory bits, the outgoing sequence becomes less correlated with the reference sequence. To highlight the correlation between each bit–gate pair, we use blue when the pair are in the same state and red when the pair are in the opposite state. Conversely, if the mass is large and falls against gravity, then this energy can be used to copy the reference sequence to the memory bits.

Figure 2

Alternatively, our model can illustrate a non-autonomous device operated via feedback control by an external agent. (a) In the engine mode, which resembles Szilard’s thought experiment [35], the agent measures each incoming memory bit and switches a gate accordingly. When these measurements are accurate, the procedure induces a bias toward counter-clockwise rotation that can be harnessed to lift a small mass against gravity; (b) If the mass is large and falls against gravity, the energy that is released can be used to write a sequence chosen by the agent, onto the outgoing bit stream. In this mode, the agent does not measure the incoming bits, but rather manipulates the gate to encode the desired sequence.

Figure 3

Snapshot of our programmable demon. A series of green paddles move down frictionlessly along the central axle. The paddles are separated by the red bars into binary states, left (0) and right (1)—see inset. Each bit passes by the rotational ring (the blue ring with two inward blades) for the same finite amount of time, during which it can change states. We claim that if the incoming bits ($000101\cdots$) are in agreement with the programmed gates ($\overline{0}\overline{0}\overline{0}\overline{1}\overline{0}\overline{1}\cdots$), then the ring favors CCW motion, which can be used to lift an external load. A top view of the system is shown in the inset. A video clip illustrating the dynamics of our demon is found at https://youtu.be/LkYljJ__-Cs.

Figure 4

Engine Mode. The ring prefers CCW rotation when the bit starts with the state that is in agreement with its corresponding gate. The blue dots represent the programmed gates.

Figure 5

The phase diagram of the programmable Maxwell’s demon in the limit ${\tau }^{\mathrm{int}}\to \infty$. Here, the behavior of the ring depends only on the sequence cleanness, $\delta$, and the external torque scaled by bath temperature, $\beta \Gamma$. For finite values of ${\tau }^{\mathrm{int}}$, the behavior of the ring depends separately on three quantities $\beta$, $\Gamma$ and $\delta$.

Figure 6

Eraser (Copier) Mode. Under a strong external load, CW rotation occurs until the bit becomes pinched between the engaging gate (shown as a blue dot situated on the gray dashed line) and a blade of the ring. The binary state of the memory bit then matches that of the reference bit.

Figure 7

Trajectories of the ring’s angular orientation for different values of $\delta$ at fixed load $\Gamma =0.05 k_{B}T$, with a bit renewal rate of 1 bit per 20 s (${\tau }^{\mathrm{int}}=20$). For $\delta =1$ and $\delta =0.8$, the ring performs work against the clockwise external torque, while for the other values of $\delta$, the external load dominates and work is dissipated into the heat bath. The cleanness of the outgoing bits for each trajectory, from $\delta =1$ to $\delta =-1$, is ${\delta }^{\prime }=0.1094, 0.1144, 0.1054\mathrm{}, 0.1244, 0.1154, 0.0854, 0.1154, 0.0864, 0.0814, 0.1064, 0.0754$.

Figure 8

Trajectories of the ring’s angular orientation for different values of CW external torque $\Gamma$, at fixed $\delta =0.2$. For each trajectory, the ring rotates in the CW direction and thus the energy of the falling mass is dissipated into the heat bath. With increasing external torque ($\Gamma = 0.1k_{B}T, 0.15k_{B}T, 0.2k_{B}T, 0.25k_{B}T$), the cleanness of the outgoing sequence of bits increases as well: ${\delta }^{\prime }=0.1884, 0.2694, 0.4062, 0.4742$. For $\Gamma \ge 0.15k_{B}T$, we obtained ${\delta }^{\prime }>0.2$, hence the ring acts as an eraser, removing randomness from the incoming sequence.

Figure 9

The configuration space of the ring and interacting bit. The tilted lines at ${\theta }_D-{\theta }_B=n\pi$ depict hard boundaries associated with a collision between the interacting bit paddle and either blade of the ring. The vertical solid lines correspond to the location of the engaging gate that blocks the paddle. This gate is located: at $\theta =0=2\pi$ when the reference bit is set to $\overline{0}$ (a); or at $\theta =\pi$ when the reference bit is set to $\overline{1}$ (b); The dashed lines in (b) represent periodic boundary conditions. The hard wall boundaries partition the configuration space into parallelogram-shaped cells, which are numbered as shown, with cell $\#0$ shaded in each panel.

Figure 10

The shaded regions indicate the distribution of the composite system right after renewal, for the case when the memory bit is correctly matched with the reference bit. For purpose of illustration, we assume that just before the renewal the system was found in either one of the shaded cells shown in Figure 9, both corresponding to $\#0$: (a) the new memory and reference bits are in the combined state $\left(0\overline{0}\right)$; and (b) the new memory and reference bits are in the combined state $\left(1\overline{1}\right)$. The marginal probability distribution of the ring’s angle, $p_D^{\mathrm{eq}}\left({\theta }_D\right)$, is unaffected by the renewal mapping.

Figure 11

Efficiency plot of the programmable demon, obtained analytically in the limit ${\tau }^{\mathrm{int}}\to \infty$. Since efficiency is defined only for the eraser and engine modes, the dud region is left blank.

Figure 12

A non-autonomous version of our model. The snapshot is taken right at the time of bit renewal. The agent observes the state of the new bit (state 1) and simultaneously sets the gate $\overline{1}$ to be effective. Thus, the new bit can switch state only through the unblocked gate $\overline{0}$. In this illustration, the agent’s measurement is faithful and thus the ring is able to work in the engine mode.