Operationally Accessible Bounds on Fluctuations and Entropy Production in Periodically Driven Systems

For periodically driven systems, we derive a family of inequalities that relate entropy production with experimentally accessible data for the mean, its dependence on driving frequency, and the variance of a large class of observables. With one of these relations, overall entropy production can be bounded by just observing the time spent in a set of states. Among further consequences, the thermodynamic efficiency both of isothermal cyclic engines like molecular motors under a periodic load and of cyclic heat engines can be bounded using experimental data without requiring knowledge of the specific interactions within the system. We illustrate these results for a driven three-level system and for a colloidal Stirling engine.

Figure 1

Ratio of estimator (15) and total entropy production (6) as a function of the driving frequency $\mathrm{\Omega }$ for different energy amplitudes $E_1^c$ for the driven three-state level (inset). The parameters $E_1^s=1.0$, $E_1^0=8.0$, $E_2^0=2.3$, $E_3^0=4.1$, $k_{12}^0=0.01$, $k_{23}^0=0.2$, $k_{13}^0=10.0$, and $\alpha =0.1$ are kept fixed.

Figure 2

Particle in a harmonic trap with time-dependent stiffness $\lambda (\tau )$ operating as a Stirling heat engine between temperatures ${\beta }_c>{\beta }_h$.

Figure 3

(a) Entropy production $\sigma$, its corresponding estimator ${\sigma }_{\mathrm{est}}$ (30), estimator for steady-state systems ${\sigma }_{\mathrm{est},\mathrm{TUR}}$ and output power $P_{\mathrm{out}}$ divided by its maximum $P_{\max }$ and (b) efficiency $\eta$, bound on the efficiency $\widehat{\eta }$ (25), and its steady-state analog ${\widehat{\eta }}_{\mathrm{ss}}$ as function of the cycle duration $\mathcal{T}$ with $\mu =10.0$, $k_{\max }=2.0$, $k_{\min }=1.0$, ${\beta }_c=2.0$, and ${\beta }_h=1.0$. For $\mathcal{T}\lesssim 0.2$, the machine no longer generates power.