Characterizing autonomous Maxwell demons

We distinguish traditional implementations of autonomous Maxwell demons from related linear devices that were recently proposed, not relying on the notions of measurements and feedback control. In both cases a current seems to flow against its spontaneous direction (imposed, e.g., by a thermal or electric gradient) without external energy intake. However, in the latter case, this current inversion may only be apparent. Even if the currents exchanged between a system and its reservoirs are inverted (by creating additional independent currents between system and demon), this is not enough to conclude that the original current through the system has been inverted. We show that this distinction can be revealed locally by measuring the fluctuations of the system-reservoir currents.

Figure 1

(a) A system interacting with only two thermal reservoirs, and thus bounded to comply with the second law. (b) When the system interacts with an additional agent (demon), then local violations of the second law might be observed. (c) However, for that to be the case, the demon must dissipate energy, which in this case is provided by a thermal gradient.

Figure 2

A simple thermal circuit. The system is composed of just three linear thermal conductors of resistance $R_1, R_2$, and $R_S$, connected in series. The “demon” is composed of thermal conductors $R_a$ and $R_b$, and is connected to the system by closing the thermal switches $S_a$ and $S_b$. Here, black lines represent thermal conductors of negligible resistance. The green arrow indicates the direction of the central current when the demon is not connected (for $T_1>T_2$).

Figure 3

A mesoscopic model of an autonomous Maxwell demon satisfying the strict criteria. (a) System and demon are two-level systems with state-dependent rates, modeling the measurement and feedback processes. (b) Global state space with possible transitions and associated rates. The size of the dots indicates the relative stationary probabilities of each state, and the thickness of the lines the relative strength of the transition rates.

Figure 4

Current through the system as a function of the thermal bias $T_1-T_2$. The energies are ${\epsilon }_d=5{\epsilon }_s$. The fixed temperatures are $T_2={\epsilon }_s/k_b$ and $T_{3/4}$ as given in the text. The ratios between decay rates are ${\omega }_2^1\left(0\right)/{\omega }_1^1\left(0\right)={\omega }_1^1\left(1\right)/{\omega }_2^1\left(1\right)={\omega }_4^1\left(0\right)/{\omega }_3^1\left(0\right)={\omega }_3^1\left(1\right)/{\omega }_4^1\left(1\right)=10$, and ${\omega }_{3/4}^1\left(z\right)/{\omega }_{1/2}^1\left(z\right)=5$, for $z=0,1$. The other rates can be determined by the local detailed balance conditions.

Figure 5

System and demon stationary rates of entropy production as a function of the bias $T_1-T_2$. The parameters are the same as in Fig. 4, and $\delta =0.2$.

Figure 6

Pearson correlation coefficient $R_{{\overline{J}}_1,{\overline{J}}_2}$ (in absolute value) for the currents ${\overline{J}}_1$ and ${\overline{J}}_2$ as a function of the observation time $t$, for the model of Sec. 5. The curve was obtained from stochastic trajectories generated for the same parameters as Fig. 4 and $(T_1-T_2)/T_2=\delta =0.2$.