Thermodynamics of Gambling Demons

We introduce and realize demons that follow a customary gambling strategy to stop a nonequilibrium process at stochastic times. We derive second-law-like inequalities for the average work done in the presence of gambling, and universal stopping-time fluctuation relations for classical and quantum nonstationary stochastic processes. We test experimentally our results in a single-electron box, where an electrostatic potential drives the dynamics of individual electrons tunneling into a metallic island. We also discuss the role of coherence in gambling demons measuring quantum jump trajectories.

Figure 1

Illustration of a gambling demon. The demon spends work ($W$, silver coins) on a physical system (slot machine) hoping to collect free energy ($F$, gold coins) by executing a gambling strategy. In each time step, the demon does work on the system (introduces a coin in the machine) and decides whether to continue (“play” sign) or to quit gambling and collect the prize (“stop” sign) at a stochastic time $\mathcal{T}$ following a prescribed strategy. In the illustration, the demon plays the slot machine until a fixed time $\mathcal{T}=3$ (top row) unless the outcome of the game is beneficial at a previous time, e.g., $\mathcal{T}=2$ (bottom row). Under specific gambling schemes, the demon can extract on average more free energy than the work spent over many iterations, a scenario that is forbidden by the standard second-law inequality.

Figure 2

(a) Scanning electron micrograph of the single-electron box (SEB) with false-color highlight on the Cu island (red) and the Al superconducting lead (turquoise). The superconducting leads are tunnel coupled through thin oxide barriers (yellow) to the island. The dc SET electrometer is coupled capacitively to the box through a bottom electrode (blue) which detects the excess charge of the box $n(t)$. (b) Representative time traces of the current measured through the electrometer (red solid line) and its digitized version (black solid line). The blue dashed line corresponds to the driving protocol $n_g(t)$ of duration $\tau =0.05\mathrm{s}$. (c) Example traces of the stochastic work done on the box as a function of time. We execute the following gambling strategy: the process is stopped at $\mathcal{T}<\tau$ (black line) only when the work reaches a threshold value $W_{\mathrm{th}}$ (red dashed line) before $\tau$. On the contrary, the process is stopped at final protocol time $\mathcal{T}=\tau$ if the work threshold is never reached during the driving protocol (blue line).

Figure 3

Dissipated work $⟨W{⟩}_{\mathcal{T}}-⟨\mathrm{\Delta }F{⟩}_{\mathcal{T}}$ (blue) and stochastic indistinguishability at stopping times (red) $-k_{B}T⟨\delta (\mathcal{T})⟩$ (dots: experimental data; solid lines: simulation) in charging energy $E_c=109\mu \mathrm{eV}$ units averaged over many realizations for protocol durations $\tau =0.05\mathrm{s}$ (a) and $\tau =0.2\mathrm{s}$ (d) as a function of work threshold values. (b),(e) Test of the generalized work fluctuation relation and of Eq. (3) (dots: experimental data; solid lines: simulation) for $\tau =0.05\mathrm{s}$ (b) and $\tau =0.2\mathrm{s}$ (e). (c),(f) Histograms of stopping times $\mathcal{T}$ (c) and corresponding work values $W(\mathcal{T})$ (f) for a ramp time $\tau =0.05\mathrm{s}$ for work thresholds $W_{\mathrm{th}}=5\times {10}^{-4},3\times {10}^{-2}$ and ${10}^{-1}{E}_c$. The total uncertainty is shown by shadowed areas; it is the combination of the statistical uncertainty and error on temperature (about 10%).