Autonomous out-of-equilibrium Maxwell's demon for controlling the energy fluxes produced by thermal fluctuations

An autonomous out-of-equilibrium Maxwell's demon is used to reverse the natural direction of the heat flux between two electric circuits kept at different temperatures and coupled by the electric thermal noise. The demon does not process any information, but it achieves its goal by using a frequency-dependent coupling with the two reservoirs of the system. There is no mean energy flux between the demon and the system, but the total entropy production $(\mathrm{system}+\mathrm{demon})$ is positive. The demon can be power supplied by thermocouples. The system and the demon are ruled by equations similar to those of two coupled Brownian particles and of the Brownian gyrator. Thus our results pave the way to the application of autonomous out-of-equilibrium Maxwell's demons to coupled nanosystems at different temperatures.

Figure 1

Diagram of the system (gray box) and of the demon (yellow box). The system is constituted by the two resistances $R_1$ and $R_2$ kept respectively at temperature $T_1$ and $T_2$, with $T_2\ge T_1$. They are coupled via the capacitor $C$. The capacitors $C_1$ and $C_2$ schematize the capacitances of the cables and of the amplifier inputs. The demon (yellow box) is composed of two resistances ($R_{d1}$ and $R_{d2}$) kept at two different temperatures $T_{d1}$ and $T_{d2}$. Furthermore, the resistance $R_{d2}$ is driven by a voltage generator $V_s$ whose output is filtered by the low-pass filter composed of the resistance $R_s$ and the capacitance $C_s$. The four voltage generators ${\eta }_k$ ($k=1,2,d1,d2$) represent the Nyquist noise voltages of the resistances at the temperatures of the heat baths.

Figure 2

(a) Heat fluxes as a function of the demon temperature $T_d$ : $〈{\dot{Q}}_2〉$ (blue line) computed when the demon is on, using Eqs. (13, 14, 15, 16) and the condition for $V_t^2$ [Eq. (19)]; $〈{\dot{Q}}_1〉$ (horizontal red dashed line) computed [Eq. (6)] when the demon is “off”; $〈{\dot{Q}}_2〉$ (red circles) and $〈{\dot{Q}}_2〉+〈{\dot{Q}}_1〉$ (black stars) obtained from the direct numerical simulation of Eqs. (7)–(10). (b) Demon efficiencies as a function of $T_d$: ${\eta }_s$ (black line) and ${\eta }_Q$ (blue line, red circles) computed from Eqs. (14), (16), and (25) (continuous lines) and obtained from the direct numerical simulations of Eqs. (7)–(10) (red circles). The parameters used to compute the curves in (a) and (b) are $T_1=300\mathrm{K}$, $T_2=450\mathrm{K}$, $C=1$ nF, $C_1=C_2=100$ pF, $R_1=R_2=10\mathrm{M}\mathrm{\Omega }$, $R_d=3\mathrm{M}\mathrm{\Omega }$, and $C_d=50$ pF.