On the efficiency of an autonomous dynamic Szilard engine operating on a single active particle

Abstract

The Szilard engine stands as a compelling illustration of the intricate interplay between information and thermodynamics. While at thermodynamic equilibrium, the apparent breach of the second law of thermodynamics was reconciled by Landauer and Bennett's insights into memory writing and erasure, recent extensions of these concepts into regimes featuring active fluctuations have unveiled the prospect of exceeding Landauer's bound, capitalizing on information to divert free energy from dissipation towards useful work. To explore this question further, we investigate an autonomous dynamic Szilard engine, addressing the thermodynamic consistency of work extraction and measurement costs by extending the phase space to incorporate an auxiliary system, which plays the role of an explicit measurement device. The nonreciprocal coupling between active particle and measurement device introduces a feedback control loop, and the cost of measurement is quantified through excess entropy production. The study considers different measurement scenarios, highlighting the role of measurement precision in determining engine efficiency.

Figures (5)

• Figure 1: Schematic representation of the dynamic Szilard engine considered in this work. A single active particle undergoing run-and-tumble motion is subject to a time-dependent force $F_{\rm ext}$ applied by an external controller. The force is modulated based on the current state of auxiliary processes $s$ or $Q$, which are coupled to the internal self-propulsion state $w$ and particle position $x$, respectively. The nonequilibrium driving of the auxiliary system results in an increase in the total mean entropy production rate by an amount $\dot{S}_{\rm exc}$, which captures the thermodynamic cost of operating this information engine.
• Figure 2: Average power output as a fraction of maximum achievable power (left) and efficiency (right) as a function of the error probability $\epsilon \in [0,1/2]$ for protocols relying on direct coupling to the RnT motility state $w$. Dotted and solid lines refer to results for the naïve and Bayesian protocols, as introduced in Sec. \ref{['ss:naive_direct']} and \ref{['ss:direct_bayesian']}, respectively. The power output is a monotonically decreasing function of $\epsilon$ in both cases, with the Bayesian protocol generically outperforming the naïve one. The efficiency, on the other hand, displays a maximum at finite power for the naïve protocol and at zero power for the Bayesian one. Here, we set $\dot{W}_{\rm max}=1$ and $m=1$.
• Figure 3: Average power output (left column) and efficiency (right column) as a function of Péclet number ${\rm Pe} \equiv \nu^2/(\alpha D_x)$ and measurement error $D_Q \in \mathbb{R}^+$ for the naïve protocol relying on inference of the RnT motility state $w$ introduced in Sec. \ref{['ss:naive_indirect']}, setting $\alpha=\beta=1$. Solid lines denote results for the full nonlinear protocol, Eq. \ref{['eq:ave_w_ind_naive']} and are obtained by numerical integration. Dashed lines are closed-form analytical results for the linearised protocol, Eqs. \ref{['eq:ave_w_ind_naive_lowPe']} and \ref{['eq:cf_eff_naive']}. The linearised protocol performs similarly to the full protocol in the regime $D_Q/\alpha < {\rm Pe} \ll 1$, as expected. In both cases, we observe a non-monotonic dependence of the efficiency on both ${\rm Pe}$ and $D_Q$. The black dashed line in the top-right panel indicates the maximum extractable power under the constraint of a hidden self-propulsion state obtained in Ref. cocconi2023optimal.
• Figure 4: Color map of the efficiency $\eta^{(n)}_{\rm lin}$ of Eq. \ref{['eq:cf_eff_naive']} as a function of the dimensionless parameters $\rm Pe$ and $D_Q/\alpha$ for the linearised naïve protocol with indirect measurement discussed in Sec. \ref{['ss:naive_indirect']}. The dashed line indicates the contour $\eta^{(b)}_{\rm lin}=0$. The white cross indicates the location of the global maximum, for $\rm Pe = 2.159...$ and $D_Q/\alpha = 2.797...$, where $\eta^{(n)}_{\rm lin} \simeq 0.0794$.
• Figure 5: Average power output (left column) and efficiency (right column) as a function of Péclet number ${\rm Pe} \equiv \nu^2/(\alpha D_x)$ and measurement error $D_Q \in \mathbb{R}^+$ for the Bayesian protocol relying on inference of the RnT motility state $w$ introduced in Sec. \ref{['ss:bayesian_indirect']}, Eq. \ref{['eq:indirect_bayes_prot']}. The results are obtained by numerical integration of exact expressions, setting $\alpha=\beta=1$, for which no closed form is available. Similarly to the naïve protocol, we observe a non-monotonic dependence of the efficiency on both ${\rm Pe}$ and $D_Q$. The black dashed line in the top-left panel indicates the maximum extractable power under the constraint of a hidden self-propulsion state (data from Ref. cocconi2023optimal).