Thermodynamic Cost of Random-Time Protocols

TL;DR

This work resolves apparent violations of the second law in systems driven by randomly timed external protocols by treating the random time $T$ as temporal information stored in a memory. Through a non-autonomous Maxwell-demon framework and Landauer's erasure principle, the authors show that the average erasure cost bounds the extractable work, restoring the Kelvin and Carnot limits when memory costs are included. They introduce a random-time Szilard engine in discrete and continuous versions, analyze finite-memory detectors, and compare measurement strategies for first-passage times, establishing a consistent information-thermodynamics framework for random-time control. The study demonstrates that temporal information behaves analogously to spatial information in thermodynamics and has broad implications for stochastic resetting and cyclic heat engines using random-time protocols.

Abstract

Systems that are driven by a randomly timed, external protocol can seemingly violate the second law of thermodynamics. We show that this thermodynamic paradox is resolved if the outcome of the random time is stored in a memory device. Specifically, we show that the average work required to erase the memory is always larger than the average work gained from the protocol. We also discuss concrete setups that measure random times directly without continuous monitoring. Taken together, this paper discusses the relationship between temporal information and thermodynamics. This framework is relevant for external protocols employing random times, such as, stochastic resetting protocols and cyclically driven heat engines that use randomly timed protocols.

Figures (9)

• Figure 1: Random-time Szilard engine: An example of an engine that uses temporal information to convert heat into work. Panel (a): a particle diffuses in an interval with reflecting boundary conditions. At one end of the segment there is a detector (depicted as a trapezoid) that is in the inactive state ($0$), a non-autonomous Maxwell demon that is dormant, and a memory device in its initial state ($\phi$). Panel (b): The particle has hit the right boundary. This activates the detector (brings it into state $1$) that sends a signal to Maxwell's demon and awakens it. The demon stores the time $T$ in its memory and initiates a protocol of fixed duration to extract work from the system. In the present example, the demon inserts a moveable wall together with an external load at a distance $\tilde{\ell}$ from the right side. Panel (c): The demon quasi-statically moves the wall to the left until it reaches a distance $\tilde{\ell}$ from the left boundary. In the process, a weight is lifted and heat is converted into work. The detector has relaxed to its inactive state $0$. The demon erases its memory and returns to the sleeping state in Panel (a).
• Figure 2: Random-time Szilard engine in discrete time without measurement error (see Sec. \ref{['sec:5']}). Comparison between the work $-\beta\langle W\rangle=\ln(\ell)$ (green squares) done by the Szilard engine on the external load and the entropy $H(T)$ (blue circles) of the stored time $T$ as a function of the length $\ell$ of the lattice; the hopping probability $p_0=1$. The blue circles denote the entropy $H(T)$ of the probability mass function of $T={\rm min}\left\{t\geq 0: X(t)=\ell\right\}$ obtained from repeated simulations of the random walk process with a uniform initial distribution for $X(0)$. The (orange) star denotes the value $H(T) =2 \ln 2$ for $\ell=2$ (see Appendix \ref{['App:B']}), and the red dotted line plots the function $\ln \ell^2$.
• Figure 3: Random-time Szilard engine in continuous time without measurement error (see Sec. \ref{['sec:6']}). The ratio between the work $-\beta\langle W\rangle$ done by the system on the external load and the entropy $H(T)$ as a function of the length ratio $\tilde{\ell}/\ell$ for three given values of $\zeta = D\theta/\ell^2$. See Appendix \ref{['app:C']} for explicit expressions of the plotted functions.
• Figure 4: Random-time Szilard engine with measurement error (see Sec. \ref{['sec:7']}). Average work that a random-time Szilard engine exerts on an external load as a function of the parameter $\pi_\ell/p_0$ that characterises the quality of the detector. The other detector parameters are $\pi_0=0.0001$ and $\beta e_\uparrow = 4$. The hopping probability $p_0=0.1$ and $\ell$ is as given in the legend. The average work is obtained from the formula $-\beta \langle W\rangle = (2p_+-1) \ln (\ell-1)$, where $p_+$ is the probability that $X(T)=\ell$; this is Eq. (\ref{['eq:Waverage']}) with $\tilde{\ell}=1$. Each marker is an average over $1e+6$ simulated samples.
• Figure 5: Example of a setup that measures the first time when a particle reaches the end of a bounded domain without continuously monitoring the system. At the right end of the segment there is an electrical circuit that is usually turned off (see top panel). The diffusing colloidal particle is made of a material of high conductance. When the particle reaches the right end, it fills the gap between two electrodes and the electrical circuit is switched on (see bottom panel). Once the circuit is switched on, a bell rings that alerts the external observer (i.e., it "awakens" the sleeping demon).