Information flows in macroscopic Maxwell's demons

A CMOS-based implementation of an autonomous Maxwell's demon was recently proposed [Phys. Rev. Lett. 129, 120602 (2022)] to demonstrate that a Maxwell demon can still work at macroscopic scales, provided that its power supply is scaled appropriately. Here we first provide a full analytical characterization of the nonautonomous version of that model. We then study system-demon information flows within generic autonomous bipartite setups displaying a macroscopic limit. By doing so, we can study the thermodynamic efficiency of both the measurement and the feedback process performed by the demon. We find that the information flow is an intensive quantity and that, as a consequence, any Maxwell's demon is bound to stop working above a finite scale if all parameters but the scale are fixed. However, this can be prevented by appropriately scaling the thermodynamic forces. These general results are applied to the autonomous CMOS-based demon.

Figure 1

(a) Common implementation of a NOT gate with CMOS technology. (b) Deterministic output voltage as a function of the input (for $V_{\text{1}}=-V_{\text{2}}=\mathrm{\Delta }{V}_S/2>0$)

Figure 2

Average electric current $\langle I_S\rangle$ at steady state for the feedback-controlled CMOS inverter as a function of the voltage bias $\mathrm{\Delta }{V}_S$ and the feedback parameter $\alpha$ ($v_e=0.1, n=1$). Negative values (blue) correspond to a current flowing against the voltage bias or, equivalently, in the upward direction. The dashed line marks the boundary between the two phases and is given by Eq. (19).

Figure 3

CMOS implementation of an autonomous Maxwell's demon. If we break the left-right symmetry of the circuit by choosing powering biases $V_3-V_4>V_1-V_2$, then we can consider that the right inverter monitors the thermal fluctuations in the output of the left inverter and, by appropriately acting back on its input, can make the electric current $I_S$ to flow against the bias $V_1-V_2$. For simplicity, all transistors are assumed to have identical parameters.

Figure 4

Average electric current $\langle I_S\rangle$ at steady state for the autonomous circuit of Fig. 3 as a function of the voltage biases $\mathrm{\Delta }{V}_S$ and $\mathrm{\Delta }{V}_D$ ($v_e=0.1, n=1$). Negative values (blue) correspond to a current flowing against the voltage bias of the first inverter. The dotted line indicates the actual boundary between positive and negative currents. The solid black line shows the approximation given by Eq. (32), while the dashed black line shows the estimation of Eq. (22) based on the nonautonomous model.

Figure 5

Histograms of the steady-state distribution $P_{\text{ss}}\left(\mathit{v}\right)$ for $v_e=0.1, \mathrm{\Delta }{V}_S=0.4$ and (a) $\mathrm{\Delta }{V}_D=2$ (${\alpha }^2<1$, monostable phase), (b) $\mathrm{\Delta }{V}_D\simeq 3.415$ (${\alpha }^2=1$, critical point), and (c) $\mathrm{\Delta }{V}_D=5$ (${\alpha }^2>1$, bistable phase). Lateral and upper plots show the partial distributions for $v$ and $v_{\text{in}}$, respectively. The arrows indicate the direction and magnitude of the flow $\mathit{u}\left(\mathit{v}\right)={\sum }_{\rho }{\mathit{\Delta }}_{\rho }{\omega }_{\rho }\left(\mathit{v}\right)P_{\text{ss}}\left(\mathit{v}\right)$ at each point in the state space.

Figure 6

Information flow as a function of the scale parameter $\mathrm{\Omega }=v_e^{-1}$ for the scaling strategy $\mathrm{\Delta }{V}_S=c{v}_e$ and $\mathrm{\Delta }{V}_D=2log(1+2{\alpha }^2/c{v}_e)$, for different values of ${\alpha }^2$. The solid lines show the analytical result of Eq. (46), while the points show numerical results obtained from the steady-state distribution and Eq. (37).